Finite Group Databasebeta

← #33108596

Subgroups — #33108596

Order 18688.

982 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988826C211yesmaximal, characteristic
2C2×#17988679C211yesmaximal, characteristic, complemented
3C22×#14556524C211yesmaximal, characteristic
4#22612940C211yesmaximal, characteristic
5#22612936C211yesmaximal, characteristic
6#22610789C211yesmaximal, characteristic, complemented
7#22610663C211yesmaximal, characteristic, complemented
8#22610575C211yesmaximal, characteristic, complemented
9#22610570C211yesmaximal, characteristic, complemented
10#22610236C211yesmaximal, characteristic
11#22610186C211yesmaximal, characteristic, complemented
12#22610183C211yesmaximal, characteristic, complemented
13#22609823C211yesmaximal, characteristic, complemented
14#22609705C211yesmaximal, characteristic
15C73×#1872C211yesmaximal, characteristic
16C22×#11924018C2211yescharacteristic
17C23×#10596831C2211yescharacteristic
18C2×#14556570C2211yescharacteristic
19C2×Q8×Dic73C2211yescharacteristic
20C2×#14556556C2211yescharacteristic
21C22×C4⋊Dic73C2211yescharacteristic
22#17988826C418yescomplemented
23#17988679—24no—
24C2×#14556524—24no—
25C2×#14556524C2211yescharacteristic
26C2×#14556524C2211yescharacteristic
27C2×#14556524C2211yescharacteristic
28C2×#14556524C2211yescharacteristic
29C2×#14556524C2211yescharacteristic
30C2×#14556516C2211yescharacteristic
31C2×#14556516C2211yescharacteristic
32C2×#14556515C2211yescharacteristic
33C2×#14556514C2211yescharacteristic
34C2×C8×Dic73C2211yescharacteristic
35C2×C8×Dic73C2211yescharacteristic
36#17988494C2211yescharacteristic
37#17988492C2211yescharacteristic
38#17988486—21no—
39#17988486C2211yescharacteristic
40#17988483C2211yescharacteristic
41#17988482—21no—
42#17988473—21no—
43#17988468—21no—
44#17988462—21no—
45#17988460—21no—
46C4⋊C4×Dic73C2211yescharacteristic
47C4⋊C4×Dic73C2211yescharacteristic
48C22⋊C4×Dic73—21no—
49C4×C22⋊Dic73—21no—
50C4×C22⋊Dic73C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53#17988341C2211yescharacteristic
54#17988341C2211yescharacteristic
55C2×C22⋊Q8×C73C2211yescharacteristic
56C22×C4⋊C4×C73C2211yescharacteristic
57C2×#103×C73C2211yescharacteristic
58C2×#103×C73C2211yescharacteristic
59C2×C22⋊C8×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#340×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C23×Dic73—21no—
64C2×#11924018—22no—
65C2×#11924018—22no—
66C2×#11924018C2×C414yescomplemented
67C22×#10596831—22no—
68C22×#10596831—22no—
69C22×#10596831C2×C414yes—
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72#14556570—22no—
73#14556570C2×C414yes—
74Q8×Dic73—22no—
75Q8×Dic73—41no—
76#14556556—22no—
77#14556556—41no—
78C2×C4⋊Dic73—22no—
79C2×C4⋊Dic73—21no—
80C2×C4⋊Dic73—21no—
81C2×C4⋊Dic73—21no—
82C2×C4⋊Dic73—41no—
83C2×C4⋊Dic73C2311yescharacteristic
84C2×C4⋊Dic73C2311yescharacteristic
85C2×C4⋊Dic73C2311yescharacteristic
86C2×#11924007—21no—
87C2×#11924007—21no—
88C2×#11924007—21no—
89C2×C22⋊Dic73—21no—
90C2×C22⋊Dic73D411yescharacteristic
91C2×C22⋊Dic73D411yescharacteristic
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73—21no—
95C2×C4×Dic73—21no—
96C2×C4×Dic73C2311yescharacteristic
97C2×C4×Dic73C2311yescharacteristic
98#14556524—22no—
99#14556524—22no—
100#14556524—22no—
101#14556524C2×C414yescomplemented
102#14556524D414yes—
103#14556516—22no—
104#14556516—41no—
105#14556515C2×C414yescomplemented
106#14556514C2×C414yescomplemented
107C8×Dic73—22no—
108C8×Dic73—22no—
109#14556498—21no—
110#14556498—21no—
111#14556498—21no—
112#14556498—21no—
113#14556498—21no—
114#14556498—21no—
115#14556498—21no—
116#14556498—21no—
117#14556498D411yescharacteristic
118#14556498D411yescharacteristic
119C22×Q8×C73C2311yescharacteristic
120C23×C4×C73—21no—
121C23×C4×C73C2311yescharacteristic
122C22×C8×C73C2311yescharacteristic
123C22×C8×C73C2311yescharacteristic
124C22⋊Q8×C73—22no—
125C22⋊Q8×C73C2×C414yes—
126C2×C4⋊C4×C73—21no—
127C2×C4⋊C4×C73—22no—
128C2×C4⋊C4×C73—21no—
129C2×C4⋊C4×C73—41no—
130C2×C4⋊C4×C73C2311yescharacteristic
131C2×C4⋊C4×C73C2311yescharacteristic
132C2×C4⋊C4×C73C2311yescharacteristic
133C2×C22⋊C4×C73—21no—
134#103×C73—22no—
135#103×C73—41no—
136C22⋊C8×C73—22no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146C22×C412yes—
143C22×Dic73—21no—
144C22×Dic73—21no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147C22×Dic73—42no—
148C22×Dic73—42no—
149C22×Dic73—81no—
150C22×Dic73C2×D411yescharacteristic
151C22×Dic73C2×D411yescharacteristic
152C22×Dic73C2×D411yescharacteristic
153C22×Dic73C2×D411yescharacteristic
154#11924018—28no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—28no—
158C2×#10596831C22×C412yes—
159C2×#10596831C22×C412yes—
160C2×#10596831C2×D412yes—
161C2×#10596831#4112yes—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—42no—
167C4⋊Dic73—42no—
168C4⋊Dic73—81no—
169C4⋊Dic73C22×C412yes—
170C4⋊Dic73C2×D412yes—
171#11924007—24no—
172#11924007—42no—
173#11924007—81no—
174C22⋊Dic73—24no—
175C22⋊Dic73—22no—
176C22⋊Dic73Q1614yescomplemented
177C4×Dic73—21no—
178C4×Dic73—21no—
179C4×Dic73—21no—
180C4×Dic73—21no—
181C4×Dic73—42no—
182C4×Dic73—42no—
183C4×Dic73—81no—
184C4×Dic73—81no—
185C24×C73C2×D411yescharacteristic
186C2×Q8×C73—21no—
187C2×Q8×C73—21no—
188C2×Q8×C73—21no—
189C2×Q8×C73—21no—
190C2×Q8×C73—21no—
191C2×Q8×C73C22×C412yes—
192C22×C4×C73—21no—
193C22×C4×C73—21no—
194C22×C4×C73—21no—
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73—21no—
198C22×C4×C73—21no—
199C22×C4×C73—21no—
200C22×C4×C73—42no—
201C22×C4×C73—42no—
202C22×C4×C73—81no—
203C22×C4×C73C22×C412yes—
204C22×C4×C73C2×D411yescharacteristic
205C22×C4×C73#4112yes—
206C22×C4×C73C2411yescharacteristic
207C2×C8×C73—21no—
208C2×C8×C73—21no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73—21no—
212C2×C8×C73—21no—
213C4⋊C4×C73—24no—
214C4⋊C4×C73—21no—
215C4⋊C4×C73—21no—
216C4⋊C4×C73—21no—
217C4⋊C4×C73—21no—
218C4⋊C4×C73—42no—
219C4⋊C4×C73C22×C412yes—
220C4⋊C4×C73C2×D412yes—
221C22⋊C4×C73—24no—
222C23×C73—21no—
223C23×C73—21no—
224C23×C73—21no—
225C23×C73—21no—
226C23×C73C4×D412yes—
227C23×C73C2×Q1612yes—
228C23×C73#13512yes—
229C23×C73C22×D411yescharacteristic
230Q8×C73—22no—
231Q8×C73—22no—
232Q8×C73—41no—
233Q8×C73—41no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—22no—
237C2×C4×C73—22no—
238C2×C4×C73—22no—
239C2×C4×C73—22no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—84no—
253C2×C4×C73C4×D412yes—
254C2×C4×C73C23×C411yescharacteristic
255C2×C4×C73C22×D411yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73H(5,𝔽2)11yescharacteristic
259C2×C4×C732−1+411yescharacteristic
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—22no—
265C2×Dic73—22no—
266C2×Dic73—21no—
267C2×Dic73—22no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—21no—
271C2×Dic73—21no—
272C2×Dic73—21no—
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—84no—
283C2×Dic73C2×Q1612yes—
284C2×Dic73C2×Q1612yes—
285Dic146—22no—
286Dic146—22no—
287Dic146—41no—
288Dic146—41no—
289C8×C73—41no—
290C8×C73—41no—
291#10596831—22no—
292#10596831C4×D414yes—
293C22×C73—22no—
294C22×C73—22no—
295C22×C73—22no—
296C22×C73—22no—
297C22×C73—22no—
298C22×C73—22no—
299C22×C73C4×Q1614yescomplemented
300C22×C73C2×C4×D411yescharacteristic
301C22×C73D4211yescharacteristic
302C22×C73#54711yescharacteristic
303C22×C73C22×Q1611yescharacteristic
304C22×C73C2×#13511yescharacteristic
305C22×C73#57611yescharacteristic
306C22×C73#57811yescharacteristic
307C4×C73—22no—
308C4×C73—42no—
309C4×C73—42no—
310C4×C73—42no—
311C4×C73—81no—
312C4×C73C2×C4×D411yescharacteristic
313C4×C73#52011yescharacteristic
314C4×C73#54811yescharacteristic
315C4×C73#55311yescharacteristic
316Dic73—22no—
317Dic73—22no—
318Dic73—41no—
319Dic73—42no—
320Dic73—42no—
321Dic73—41no—
322Dic73—42no—
323Dic73—81no—
324#23024—731nomaximal, Sylow
325C2×C73—24no—
326C2×C73#236111yescharacteristic
327C2×C73C2×C4×Q1611yescharacteristic
328C2×C73#295111yescharacteristic
329C2×C73#328011yescharacteristic
330C2×C73D4×Q1611yescharacteristic
331C2×C73#333611yescharacteristic
332C2×C73#333811yescharacteristic
333C2×#467—731no—
334C2×C4×Q16—731no—
335C22×#107—731no—
336#2360—731no—
337#2360—731no—
338#1969—731no—
339#1942—731no—
340#1928—731no—
341#1928—731no—
342#1872—731no—
343#1872D7311yescharacteristic
344#1863—731no—
345#1863—731no—
346C4×#107—731no—
347C8×C22⋊C4—731no—
348C73#2302411yescharacteristic, complemented, Sylow
349C22×Q16—731no—
350C23×C8—731no—
351C2×C22⋊Q8—731no—
352C2×C22⋊Q8C2×D7311yescharacteristic
353C2×C4×Q8—731no—
354C2×C4×Q8—731no—
355C22×C4⋊C4—731no—
356C22×C4⋊C4C2×D7311yescharacteristic
357#467—738no—
358C4×Q16—1464no—
359C2×#107—731no—
360C2×#107—731no—
361C2×#107—731no—
362C2×#107—731no—
363C2×#107—731no—
364C2×#107—1464no—
365C2×#103—731no—
366C2×#103—731no—
367C2×#103—731no—
368C2×#103—731no—
369C2×#103C2×D7311yescharacteristic
370C2×#103C2×D7311yescharacteristic
371C2×C22⋊C8C2×D7311yescharacteristic
372C2×C4×C8—731no—
373C2×C4×C8—731no—
374#391—731no—
375#391—731no—
376#389—731no—
377#389—731no—
378#389—1461no—
379#389—1461no—
380#387—1461no—
381#387—1461no—
382#385—1461no—
383#385—1461no—
384C4×C4⋊C4—731no—
385C4×C4⋊C4—731no—
386C4×C22⋊C4—731no—
387C4×C22⋊C4—1461no—
388C4×C22⋊C4—1461no—
389#340—731no—
390#340—731no—
391#340C2×D7311yescharacteristic
392#340C2×D7311yescharacteristic
393#336—731no—
394#336—731no—
395C22×Q8—731no—
396C22×Q8C22×D7311yescharacteristic
397C23×C4—21no—
398C23×C4—1461no—
399C23×C4C22×D7311yescharacteristic
400C2×Q16—734no—
401C2×Q16—1462no—
402C2×Q16—1462no—
403C22×C8—731no—
404C22×C8—734no—
405C22×C8—731no—
406C22×C8—1462no—
407C22×C8—1462no—
408C22×C8C22×D7311yescharacteristic
409C22×C8C22×D7311yescharacteristic
410C22⋊Q8—22no—
411C22⋊Q8—734no—
412C22⋊Q8—1462no—
413C22⋊Q8C4×D7314yes—
414C4×Q8—2921no—
415C4×Q8—2921no—
416C4×Q8—1462no—
417C4×Q8—1462no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—22no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—41no—
422C2×C4⋊C4—2921no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1462no—
430C2×C4⋊C4—1461no—
431C2×C4⋊C4—1461no—
432C2×C4⋊C4—1461no—
433C2×C4⋊C4C22×D7311yescharacteristic
434C2×C4⋊C4C22×D7311yescharacteristic
435C2×C4⋊C4C22×D7311yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—731no—
438C2×C22⋊C4—731no—
439C2×C22⋊C4—1461no—
440C2×C42—731no—
441C2×C42—731no—
442C2×C42—1461no—
443C2×C42—1461no—
444C2×C42—1461no—
445C2×C42—1461no—
446#107—734no—
447#107—734no—
448#107—1462no—
449#107—1462no—
450#107—1462no—
451#103—22no—
452#103—41no—
453#103—2921no—
454#103—734no—
455#103—734no—
456#103—1462no—
457C22⋊C8—22no—
458C4×C8—1462no—
459C4×C8—1462no—
460#95—731no—
461#95—731no—
462#95—1461no—
463#95—1461no—
464#95—1461no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470C24D4×D7311yescharacteristic
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—732no—
477C2×Q8—1461no—
478C2×Q8—1461no—
479C2×Q8—1461no—
480C2×Q8—1461no—
481C2×Q8—1461no—
482C2×Q8C2×C4×D7312yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—81no—
494C22×C4—2922no—
495C22×C4—2922no—
496C22×C4—5841no—
497C22×C4—731no—
498C22×C4—731no—
499C22×C4—731no—
500C22×C4—731no—
501C22×C4—1461no—
502C22×C4—1461no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4C2×C4×D7312yes—
506C22×C4D4×D7311yescharacteristic
507C22×C4#1192402712yes—
508C22×C4C23×D7311yescharacteristic
509Q16—1468no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—732no—
517C2×C8—732no—
518C2×C8—732no—
519C2×C8—732no—
520C2×C8—1468no—
521C2×C8—1461no—
522C2×C8—1461no—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—2922no—
530C4⋊C4—2922no—
531C4⋊C4—2922no—
532C4⋊C4—5841no—
533C4⋊C4—5841no—
534C4⋊C4—732no—
535C4⋊C4—732no—
536C4⋊C4—1461no—
537C4⋊C4—1461no—
538C4⋊C4—1464no—
539C4⋊C4—1461no—
540C4⋊C4—1461no—
541C4⋊C4C2×C4×D7312yes—
542C4⋊C4D4×D7312yes—
543C22⋊C4—24no—
544C22⋊C4—734no—
545C22⋊C4—1464no—
546C22⋊C4—1462no—
547C42—2922no—
548C42—2922no—
549C42—5841no—
550C42—5841no—
551C42—1461no—
552C42—1461no—
553C42—1461no—
554C42—1461no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23#1455655112yes—
560C23Q16×D7312yes—
561C23#1455661212yes—
562C23C2×D4×D7311yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—2921no—
568Q8—2921no—
569Q8—1462no—
570Q8—1462no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—5844no—
600C2×C4—732no—
601C2×C4—732no—
602C2×C4—1461no—
603C2×C4—1461no—
604C2×C4—1462no—
605C2×C4—1461no—
606C2×C4—1461no—
607C2×C4—1462no—
608C2×C4—1462no—
609C2×C4—1461no—
610C2×C4—1461no—
611C2×C4—1461no—
612C2×C4—1461no—
613C2×C4—1461no—
614C2×C4—1461no—
615C2×C4#1455655112yes—
616C2×C4C22×C4×D7311yescharacteristic
617C2×C4C2×D4×D7311yescharacteristic
618C2×C4#41×D7311yescharacteristic
619C2×C4C2×#1192402711yescharacteristic
620C2×C4#1455663611yescharacteristic
621C2×C4#1455663811yescharacteristic
622C8—41no—
623C8—41no—
624C8—734no—
625C8—1462no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22#1798867914yescomplemented
633C22C2×#1455655111yescharacteristic
634C22#1798921911yescharacteristic
635C22#1798923511yescharacteristic
636C22C2×Q16×D7311yescharacteristic
637C22C2×#1455661211yescharacteristic
638C22#1798939111yescharacteristic
639C22#1798940311yescharacteristic
640C4—22no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—81no—
645C4—2921no—
646C4—2922no—
647C4—2921no—
648C4—2922no—
649C4—2922no—
650C4—5841no—
651C4—1462no—
652C4—1462no—
653C4C4×D4×D7311yescharacteristic
654C4#1798906611yescharacteristic
655C4#1798924311yescharacteristic
656C4#1798928011yescharacteristic
657C2—24no—
658C2#2261294211yescharacteristic
659C2C2×#1798867911yescharacteristic
660C2#2261880211yescharacteristic
661C2#2262160911yescharacteristic
662C2#2262167611yescharacteristic
663C2#2262229711yescharacteristic
664C2#2262231811yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1872C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#1872D73
hypocenterC73#23024
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
2. C2×#17988679C2semidirect
6. #22610789C2semidirect
7. #22610663C2semidirect
8. #22610575C2semidirect
9. #22610570C2semidirect
11. #22610186C2semidirect
12. #22610183C2semidirect
13. #22609823C2semidirect
22. #17988826C4semidirect
66. C2×#11924018C2×C4semidirect
101. #14556524C2×C4semidirect
105. #14556515C2×C4semidirect
106. #14556514C2×C4semidirect
176. C22⋊Dic73Q16semidirect
299. C22×C73C4×Q16semidirect
348. C73#23024semidirect
632. C22#17988679semidirect

Maximal subgroups

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

subgroupindex
C2×#179888262
C2×#179886792
C22×#145565242
#226129402
#226129362
#226107892
#226106632
#226105752
#226105702
#226102362
#226101862
#226101832
#226098232
#226097052
C73×#18722
#2302473