Finite Group Databasebeta

← #33143121

Subgroups — #33143121

Order 18688.

990 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988873C211yesmaximal, characteristic, complemented
2C22×#14556516C211yesmaximal, characteristic
3#22613633C211yesmaximal, characteristic
4C22⋊C4×Dic146C211yesmaximal, characteristic
5#22611473C211yesmaximal, characteristic, complemented
6#22611311C211yesmaximal, characteristic, complemented
7#22611311C211yesmaximal, characteristic, complemented
8#22611073C211yesmaximal, characteristic, complemented
9#22610202C211yesmaximal, characteristic, complemented
10#22610202C211yesmaximal, characteristic, complemented
11#22610126C211yesmaximal, characteristic
12#22609956C211yesmaximal, characteristic
13#22609937C211yesmaximal, characteristic, complemented
14#22609867C211yesmaximal, characteristic
15C73×#1850C211yesmaximal, characteristic
16C23×Dic146C2211yescharacteristic
17C2×#14556591C2211yescharacteristic
18C2×#14556568—21no—
19C2×C4×Dic146C2211yescharacteristic
20C2×C4×Dic146C2211yescharacteristic
21C22×C4⋊Dic73C2211yescharacteristic
22#17988873—24no—
23C2×#14556523C2211yescharacteristic
24C2×#14556523C2211yescharacteristic
25C2×#14556523C2211yescharacteristic
26C2×#14556523C2211yescharacteristic
27C2×#14556516—24no—
28C2×#14556516C2211yescharacteristic
29C2×#14556516C2211yescharacteristic
30C2×#14556516C2211yescharacteristic
31C2×#14556516C2211yescharacteristic
32C2×#14556516C2211yescharacteristic
33C2×#14556516C2211yescharacteristic
34C2×#14556516C2211yescharacteristic
35C2×#14556516C2211yescharacteristic
36#17988519—21no—
37#17988508—21no—
38#17988493C2211yescharacteristic
39#17988493C2211yescharacteristic
40#17988491—21no—
41#17988491—21no—
42#17988485C2211yescharacteristic
43#17988485C2211yescharacteristic
44#17988462—21no—
45#17988457C2211yescharacteristic
46#17988447C2211yescharacteristic
47#17988447C2211yescharacteristic
48#17988445—21no—
49C22⋊C4×Dic73—21no—
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53#17988350C2211yescharacteristic
54C23×C8×C73C2211yescharacteristic
55C2×C4⋊C8×C73C2211yescharacteristic
56C2×C4⋊C8×C73C2211yescharacteristic
57C2×C22⋊C8×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60#336×C73C2211yescharacteristic
61C22×Dic146—22no—
62C22×Dic146—22no—
63C22×Dic146—21no—
64C22×Dic146—21no—
65C22×Dic146—22no—
66C22×Dic146—41no—
67C22×Dic146—41no—
68C22×Dic146Q814yescomplemented
69C22×Dic146C2311yescharacteristic
70C22×Dic146C2311yescharacteristic
71C23×Dic73—21no—
72C23×Dic73—21no—
73#14556591—22no—
74#14556591—41no—
75#14556568—44no—
76C4×Dic146—22no—
77C4×Dic146—22no—
78C4×Dic146—41no—
79C4×Dic146—41no—
80C2×C4⋊Dic73—22no—
81C2×C4⋊Dic73—21no—
82C2×C4⋊Dic73—21no—
83C2×C4⋊Dic73—21no—
84C2×C4⋊Dic73—21no—
85C2×C4⋊Dic73—41no—
86C2×C4⋊Dic73C2311yescharacteristic
87C2×C4⋊Dic73C2311yescharacteristic
88C2×C4⋊Dic73C2311yescharacteristic
89C2×C4⋊Dic73C2311yescharacteristic
90C2×C4⋊Dic73C2311yescharacteristic
91C2×C4⋊Dic73C2311yescharacteristic
92C2×#11924007—21no—
93C2×#11924007—21no—
94C2×#11924007—21no—
95C2×#11924007—21no—
96C2×C22⋊Dic73—21no—
97C2×C22⋊Dic73—21no—
98C2×C4×Dic73—21no—
99C2×C4×Dic73—21no—
100#14556523—22no—
101#14556523—22no—
102#14556523—41no—
103#14556523—41no—
104#14556516—22no—
105#14556516—22no—
106#14556516—22no—
107#14556516—22no—
108#14556516—22no—
109#14556516—41no—
110#14556516—41no—
111#14556516D414yes—
112#14556498—21no—
113#14556498—21no—
114#14556498—21no—
115#14556498—21no—
116#14556498—21no—
117#14556498—21no—
118#14556498—21no—
119#14556498—21no—
120C23×C4×C73C2311yescharacteristic
121C22×C8×C73—22no—
122C22×C8×C73—22no—
123C22×C8×C73—41no—
124C22×C8×C73C2311yescharacteristic
125C22×C8×C73C2311yescharacteristic
126C22×C8×C73C2311yescharacteristic
127C22×C8×C73C2311yescharacteristic
128C2×C22⋊C4×C73D411yescharacteristic
129C2×C22⋊C4×C73D411yescharacteristic
130C2×C42×C73C2311yescharacteristic
131C2×C42×C73C2311yescharacteristic
132C4⋊C8×C73—22no—
133C4⋊C8×C73—41no—
134C22⋊C8×C73—22no—
135#95×C73D411yescharacteristic
136#95×C73D411yescharacteristic
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146—21no—
143C2×Dic146—21no—
144C2×Dic146—28no—
145C2×Dic146—24no—
146C2×Dic146—21no—
147C2×Dic146—44no—
148C2×Dic146—44no—
149C2×Dic146—44no—
150C2×Dic146—42no—
151C2×Dic146—42no—
152C2×Dic146—42no—
153C2×Dic146—42no—
154C2×Dic146—81no—
155C2×Dic146C2×D412yes—
156C2×Dic146C2×Q812yes—
157C2×Dic146C2×Q812yes—
158C2×Dic146#4112yes—
159C22×Dic73—21no—
160C22×Dic73—24no—
161C22×Dic73—21no—
162C22×Dic73—21no—
163C22×Dic73—21no—
164C22×Dic73—21no—
165C22×Dic73—21no—
166C22×Dic73—21no—
167C22×Dic73—21no—
168C22×Dic73—42no—
169C22×Dic73—42no—
170C22×Dic73—42no—
171C22×Dic73—42no—
172C22×Dic73—81no—
173C4⋊Dic73—21no—
174C4⋊Dic73—21no—
175C4⋊Dic73—21no—
176C4⋊Dic73—21no—
177C4⋊Dic73—21no—
178C4⋊Dic73—21no—
179C4⋊Dic73—21no—
180C4⋊Dic73—21no—
181C4⋊Dic73—21no—
182C4⋊Dic73—21no—
183C4⋊Dic73—21no—
184C4⋊Dic73—42no—
185C4⋊Dic73—42no—
186C4⋊Dic73—42no—
187C4⋊Dic73—81no—
188C4⋊Dic73C2×D412yes—
189#11924007—42no—
190#11924007—42no—
191#11924007—42no—
192#11924007—42no—
193C22⋊Dic73—42no—
194C22⋊Dic73—42no—
195C4×Dic73—42no—
196C4×Dic73—42no—
197C24×C73C2×D411yescharacteristic
198C22×C4×C73—21no—
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73C2×D411yescharacteristic
203C22×C4×C73C2×D411yescharacteristic
204C22×C4×C73C2×D411yescharacteristic
205C22×C4×C73C2×D411yescharacteristic
206C22×C4×C73C2×D411yescharacteristic
207C22×C4×C73C2×Q812yes—
208C22×C4×C73#4112yes—
209C22×C4×C73C2411yescharacteristic
210C2×C8×C73—21no—
211C2×C8×C73—21no—
212C2×C8×C73—21no—
213C2×C8×C73—21no—
214C2×C8×C73—21no—
215C2×C8×C73—21no—
216C2×C8×C73—21no—
217C2×C8×C73—21no—
218C2×C8×C73—21no—
219C2×C8×C73—21no—
220C2×C8×C73—21no—
221C2×C8×C73—44no—
222C2×C8×C73C2×D412yes—
223C22⋊C4×C73—22no—
224C22⋊C4×C73SD1614yes—
225C42×C73—21no—
226C42×C73—21no—
227C42×C73—21no—
228C42×C73—21no—
229C23×C73—21no—
230C23×C73—21no—
231C23×C73—21no—
232C23×C73—21no—
233C23×C73C22⋊Q812yes—
234C23×C73C2×SD1612yes—
235C23×C73#13712yes—
236C23×C73C22×D411yescharacteristic
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—22no—
244C2×C4×C73—22no—
245C2×C4×C73—22no—
246C2×C4×C73—22no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
251C2×C4×C73—21no—
252C2×C4×C73—21no—
253C2×C4×C73C22⋊Q812yes—
254C2×C4×C73C2×SD1612yes—
255C2×C4×C73C2×SD1612yes—
256C2×C4×C73C22×D411yescharacteristic
257C2×C4×C73C22×Q811yescharacteristic
258C2×C4×C73C2×#4111yescharacteristic
259C2×C4×C73C2×#4111yescharacteristic
260C2×C4×C73H(5,𝔽2)11yescharacteristic
261C2×C4×C73H(5,𝔽2)11yescharacteristic
262C2×Dic73—22no—
263C2×Dic73—22no—
264C2×Dic73—22no—
265C2×Dic73—22no—
266C2×Dic73—22no—
267C2×Dic73—48no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—41no—
284C2×Dic73—41no—
285C2×Dic73—41no—
286C2×Dic73—41no—
287C2×Dic73—84no—
288Dic146—22no—
289Dic146—22no—
290Dic146—22no—
291Dic146—22no—
292Dic146—22no—
293Dic146—48no—
294Dic146—41no—
295Dic146—41no—
296Dic146—84no—
297Dic146C22⋊Q814yescomplemented
298C8×C73—22no—
299C8×C73—41no—
300C8×C73—41no—
301C8×C73—41no—
302C22×C73—22no—
303C22×C73—22no—
304C22×C73—22no—
305C22×C73—22no—
306C22×C73—22no—
307C22×C73—22no—
308C22×C73#47514yescomplemented
309C22×C73C2×C22⋊Q811yescharacteristic
310C22×C73D4211yescharacteristic
311C22×C73#54611yescharacteristic
312C22×C73C22×SD1611yescharacteristic
313C22×C73C2×#13711yescharacteristic
314C22×C73#57711yescharacteristic
315C22×C73#57811yescharacteristic
316C4×C73—22no—
317C4×C73—22no—
318C4×C73—22no—
319C4×C73—41no—
320C4×C73—41no—
321C4×C73C2×C22⋊Q811yescharacteristic
322C4×C73#54311yescharacteristic
323C4×C73#54611yescharacteristic
324C4×C73#55111yescharacteristic
325Dic73—24no—
326Dic73—42no—
327Dic73—42no—
328Dic73—81no—
329Dic73—81no—
330Dic73—81no—
331Dic73—81no—
332Dic73—81no—
333#25955—731nomaximal, Sylow
334C2×C73—24no—
335C2×C73#243711yescharacteristic
336C2×C73C2×#47511yescharacteristic
337C2×C73#308211yescharacteristic
338C2×C73#327011yescharacteristic
339C2×C73D4×SD1611yescharacteristic
340C2×C73#329811yescharacteristic
341C2×C73#331611yescharacteristic
342C2×#475—731no—
343C22×#103—731no—
344#2440—731no—
345Q8×C22⋊C4—731no—
346#2096—731no—
347#2066—731no—
348#2066—731no—
349#2024—731no—
350#1865—731no—
351#1865—731no—
352#1853—731no—
353#1850D7311yescharacteristic
354#1807—731no—
355#1803—731no—
356#1787—731no—
357C73#2595511yescharacteristic, complemented, Sylow
358C23×Q8—731no—
359C23×C8C2×D7311yescharacteristic
360C2×C4⋊Q8—731no—
361C2×C22⋊Q8—1461no—
362C2×C4×Q8—731no—
363C2×C4×Q8—731no—
364C22×C4⋊C4—731no—
365#475—1464no—
366C2×#106—731no—
367C2×#106—731no—
368C2×#106—731no—
369C2×#106—731no—
370C2×C4⋊C8C2×D7311yescharacteristic
371C2×C4⋊C8C2×D7311yescharacteristic
372C2×#103—731no—
373C2×#103—731no—
374C2×#103—731no—
375C2×#103—731no—
376C2×#103—731no—
377C2×#103—731no—
378C2×#103—731no—
379C2×#103—731no—
380C2×#103—1464no—
381C2×C22⋊C8C2×D7311yescharacteristic
382#398—1461no—
383#395—1461no—
384#391—731no—
385#391—731no—
386#391—1461no—
387#391—1461no—
388#389—731no—
389#389—731no—
390#385—1461no—
391#384—731no—
392#380—731no—
393#380—731no—
394#380—1461no—
395C4×C22⋊C4—1461no—
396C4×C22⋊C4C2×D7311yescharacteristic
397#340—731no—
398#340—731no—
399#340—731no—
400#340—731no—
401#336C2×D7311yescharacteristic
402#336C2×D7311yescharacteristic
403C22×Q8—2921no—
404C22×Q8—2921no—
405C22×Q8—734no—
406C22×Q8—731no—
407C22×Q8—731no—
408C22×Q8—1462no—
409C22×Q8—1462no—
410C22×Q8—1461no—
411C22×Q8—1461no—
412C22×Q8—1462no—
413C23×C4—1461no—
414C23×C4—1461no—
415C23×C4C22×D7311yescharacteristic
416C22×C8—22no—
417C22×C8—22no—
418C22×C8—41no—
419C22×C8C22×D7311yescharacteristic
420C22×C8C22×D7311yescharacteristic
421C22×C8C22×D7311yescharacteristic
422C22×C8C22×D7311yescharacteristic
423C4⋊Q8—2921no—
424C4⋊Q8—1462no—
425C22⋊Q8—2924no—
426C4×Q8—2921no—
427C4×Q8—2921no—
428C4×Q8—1462no—
429C4×Q8—1462no—
430C2×C4⋊C4—2921no—
431C2×C4⋊C4—731no—
432C2×C4⋊C4—731no—
433C2×C4⋊C4—731no—
434C2×C4⋊C4—731no—
435C2×C4⋊C4—731no—
436C2×C4⋊C4—731no—
437C2×C4⋊C4—1461no—
438C2×C4⋊C4—1461no—
439C2×C4⋊C4—1461no—
440C2×C4⋊C4—1461no—
441C2×C4⋊C4—1461no—
442C2×C4⋊C4—1461no—
443C2×C4⋊C4—1461no—
444C2×C4⋊C4—1462no—
445C2×C4⋊C4—1461no—
446C2×C22⋊C4—1461no—
447C2×C22⋊C4—1461no—
448C2×C22⋊C4D29211yescharacteristic
449C2×C22⋊C4D29211yescharacteristic
450C2×C42—1461no—
451C2×C42—1461no—
452C2×C42C22×D7311yescharacteristic
453C2×C42C22×D7311yescharacteristic
454#106—2921no—
455#106—2921no—
456#106—1462no—
457#106—1462no—
458C4⋊C8—22no—
459C4⋊C8—41no—
460#103—2921no—
461#103—2921no—
462#103—734no—
463#103—1462no—
464#103—1462no—
465#103—1462no—
466#103—1462no—
467#103—1462no—
468C22⋊C8—22no—
469#95—1461no—
470#95—1461no—
471#95—1461no—
472#95—1461no—
473#95—1461no—
474#95—1461no—
475#95—1461no—
476#95—1461no—
477#95D29211yescharacteristic
478#95D29211yescharacteristic
479C24C2×D29211yescharacteristic
480C2×Q8—2924no—
481C2×Q8—2924no—
482C2×Q8—2924no—
483C2×Q8—2922no—
484C2×Q8—2922no—
485C2×Q8—2922no—
486C2×Q8—2922no—
487C2×Q8—5841no—
488C2×Q8—732no—
489C2×Q8—732no—
490C2×Q8—732no—
491C2×Q8—732no—
492C2×Q8—1468no—
493C2×Q8—1461no—
494C2×Q8—1461no—
495C2×Q8—1461no—
496C2×Q8—1461no—
497C2×Q8—1464no—
498C2×Q8—1461no—
499C2×Q8—1461no—
500C2×Q8—1461no—
501C2×Q8—1461no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—2922no—
507C22×C4—2922no—
508C22×C4—2922no—
509C22×C4—2922no—
510C22×C4—5841no—
511C22×C4—1461no—
512C22×C4—1461no—
513C22×C4—1461no—
514C22×C4—1461no—
515C22×C4—1461no—
516C22×C4—1464no—
517C22×C4—1461no—
518C22×C4—1461no—
519C22×C4—1461no—
520C22×C4C2×D29211yescharacteristic
521C22×C4C2×D29211yescharacteristic
522C22×C4C2×D29211yescharacteristic
523C22×C4C2×D29211yescharacteristic
524C22×C4C2×D29211yescharacteristic
525C22×C4Q8×D7312yes—
526C22×C4#1192402712yes—
527C22×C4C23×D7311yescharacteristic
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—44no—
540C2×C8D4×D7312yes—
541C4⋊C4—2922no—
542C4⋊C4—2922no—
543C4⋊C4—2922no—
544C4⋊C4—2922no—
545C4⋊C4—2922no—
546C4⋊C4—2922no—
547C4⋊C4—2922no—
548C4⋊C4—5841no—
549C4⋊C4—732no—
550C4⋊C4—1461no—
551C4⋊C4—1461no—
552C4⋊C4—1461no—
553C4⋊C4—1461no—
554C4⋊C4—1461no—
555C4⋊C4—1461no—
556C4⋊C4—1461no—
557C4⋊C4—1461no—
558C4⋊C4—1461no—
559C4⋊C4—1461no—
560C4⋊C4—1461no—
561C22⋊C4—22no—
562C22⋊C4—2922no—
563C22⋊C4—2922no—
564C22⋊C4#1192401714yes—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—2922no—
570C42—2922no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23#1455657112yes—
576C23C2×#1192401712yes—
577C23#1455662412yes—
578C23C22×D29211yescharacteristic
579Q8—2928no—
580Q8—2921no—
581Q8—2921no—
582Q8—5844no—
583Q8—734no—
584Q8—1462no—
585Q8—1462no—
586Q8—1462no—
587Q8—1462no—
588Q8—1462no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—2921no—
606C2×C4—2921no—
607C2×C4—2921no—
608C2×C4—2921no—
609C2×C4—2928no—
610C2×C4—2921no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—2921no—
614C2×C4—2921no—
615C2×C4—2921no—
616C2×C4—2921no—
617C2×C4—2921no—
618C2×C4—2921no—
619C2×C4—2921no—
620C2×C4—2921no—
621C2×C4—2921no—
622C2×C4—2921no—
623C2×C4—2921no—
624C2×C4—2921no—
625C2×C4—5844no—
626C2×C4—1462no—
627C2×C4—1462no—
628C2×C4—1462no—
629C2×C4—1462no—
630C2×C4—1462no—
631C2×C4#1455657112yes—
632C2×C4C2×#1192401712yes—
633C2×C4C2×#1192401712yes—
634C2×C4C2×D4×D7311yescharacteristic
635C2×C4C2×Q8×D7311yescharacteristic
636C2×C4#41×D7311yescharacteristic
637C2×C4C2×#1192402711yescharacteristic
638C2×C4#1455663611yescharacteristic
639C2×C4#1455663711yescharacteristic
640C8—22no—
641C8—41no—
642C8—41no—
643C8—41no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22#1798887314yescomplemented
651C22C2×#1455657111yescharacteristic
652C22D4×D29211yescharacteristic
653C22#1798923211yescharacteristic
654C22C22×#1192401711yescharacteristic
655C22C2×#1455662411yescharacteristic
656C22#1798939911yescharacteristic
657C22#1798940411yescharacteristic
658C4—22no—
659C4—22no—
660C4—22no—
661C4—41no—
662C4—41no—
663C4—2922no—
664C4—2922no—
665C4—5841no—
666C4—5841no—
667C4—5841no—
668C4—5841no—
669C4—5841no—
670C4—1464no—
671C4C22⋊Q8×D7311yescharacteristic
672C4#1798921211yescharacteristic
673C4#1798922411yescharacteristic
674C4#1798926011yescharacteristic
675C2—24no—
676C2#2261359511yescharacteristic
677C2C2×#1798887311yescharacteristic
678C2#2261977411yescharacteristic
679C2#2262150611yescharacteristic
680C2D4×#1192401711yescharacteristic
681C2#2262179711yescharacteristic
682C2#2262204411yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1850C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#1850D73
hypocenterC73#25955
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#17988873C2semidirect
5. #22611473C2semidirect
6. #22611311C2semidirect
7. #22611311C2semidirect
8. #22611073C2semidirect
9. #22610202C2semidirect
10. #22610202C2semidirect
13. #22609937C2semidirect
68. C22×Dic146Q8semidirect
297. Dic146C22⋊Q8semidirect
308. C22×C73#475semidirect
357. C73#25955semidirect
650. C22#17988873semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179888732
C22×#145565162
#226136332
C22⋊C4×Dic1462
#226114732
#226113112
#226110732
#226102022
#226101262
#226099562
#226099372
#226098672
C73×#18502
#2595573