Finite Group Databasebeta

← #33102857

Subgroups — #33102857

Order 18688.

1054 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989310C211yesmaximal, characteristic
2#22612888C211yesmaximal, characteristic, complemented
3#22610861C211yesmaximal, characteristic
4#22610861C211yesmaximal, characteristic
5#22610853C211yesmaximal, characteristic, complemented
6#22610751C211yesmaximal, characteristic, complemented
7#22610747C211yesmaximal, characteristic
8#22610742C211yesmaximal, characteristic
9#22609947C211yesmaximal, characteristic, complemented
10#22609942C211yesmaximal, characteristic
11#22609937C211yesmaximal, characteristic, complemented
12#22609762C211yesmaximal, characteristic, complemented
13#22609757C211yesmaximal, characteristic
14C4×#14556516C211yesmaximal, characteristic, complemented
15C2×#446×C73C211yesmaximal, characteristic, complemented
16#17989310—24no—
17#17989310C418yescomplemented
18C2×#14556591C2211yescharacteristic
19C2×#14556586—21no—
20C2×#14556586—21no—
21C2×#14556579C2211yescharacteristic
22C2×#14556579C2211yescharacteristic
23C2×#14556567—21no—
24C2×C4×Dic146C2211yescharacteristic
25C2×C4×Dic146C2211yescharacteristic
26C2×C4×D292C2211yescharacteristic
27C2×#14556549—21no—
28C2×#14556516C2211yescharacteristic
29C2×#14556516C2211yescharacteristic
30C2×#14556516C2211yescharacteristic
31C2×#14556516C2211yescharacteristic
32C2×#14556516C2211yescharacteristic
33C2×#14556516C2211yescharacteristic
34C2×#14556516C2211yescharacteristic
35C2×#14556516C2211yescharacteristic
36C2×#14556513C2211yescharacteristic
37C2×#14556513C2211yescharacteristic
38C2×#14556513C2211yescharacteristic
39C2×#14556513C2211yescharacteristic
40#17988494—21no—
41#17988473—21no—
42#17988473—21no—
43#17988457C2211yescharacteristic
44#17988453C2211yescharacteristic
45#17988453C2211yescharacteristic
46C4⋊C4×Dic73—21no—
47C4×C4⋊Dic73C2211yescharacteristic
48C4×C4⋊Dic73C2211yescharacteristic
49C4×C4⋊Dic73C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53#17988350C2211yescharacteristic
54C2×C4×Q8×C73C2211yescharacteristic
55#446×C73—24no—
56C2×C4⋊C8×C73C2211yescharacteristic
57C2×C4⋊C8×C73C2211yescharacteristic
58C2×C4⋊C8×C73C2211yescharacteristic
59C2×#97×C73C2211yescharacteristic
60C2×#97×C73C2211yescharacteristic
61C2×C4×C8×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C22×Dic146C2311yescharacteristic
64C22×D292C2311yescharacteristic
65C22×C4×D73—21no—
66#14556591—22no—
67#14556591C2×C414yes—
68#14556586—28no—
69#14556586—28no—
70#14556579—22no—
71#14556579—22no—
72#14556579C2×C414yes—
73#14556579C2×C414yes—
74#14556567—28no—
75C4×Dic146—22no—
76C4×Dic146—22no—
77C4×Dic146C2×C414yes—
78C4×Dic146C2×C414yes—
79C4×D292—22no—
80C4×D292C2×C414yes—
81#14556549—28no—
82C2×C4⋊Dic73—21no—
83C2×C4⋊Dic73—21no—
84C2×C4⋊Dic73—21no—
85C2×C4⋊Dic73—21no—
86C2×C4⋊Dic73C2311yescharacteristic
87C2×C4⋊Dic73C2311yescharacteristic
88C2×C4⋊Dic73C2311yescharacteristic
89C2×C4⋊Dic73C2311yescharacteristic
90C2×C4⋊Dic73C2311yescharacteristic
91C2×#11924007—21no—
92C2×#11924007—21no—
93C2×#11924005—21no—
94C2×#11924005—21no—
95C2×#11924005—21no—
96C2×#11924005—21no—
97C2×#11924005—21no—
98C2×C4×Dic73—21no—
99C2×C4×Dic73—21no—
100C2×C4×Dic73—21no—
101C2×C4×Dic73—21no—
102C2×C4×Dic73—21no—
103#14556516—22no—
104#14556516—22no—
105#14556516—22no—
106#14556516—22no—
107#14556516—41no—
108#14556516—41no—
109#14556516—41no—
110#14556516—41no—
111#14556513—22no—
112#14556513—22no—
113#14556513—41no—
114#14556513—41no—
115#14556498—21no—
116#14556498—21no—
117#14556498—21no—
118#14556498—21no—
119#14556498—21no—
120#14556498—21no—
121#14556498—21no—
122C22×Q8×C73D411yescharacteristic
123C22×C8×C73C2311yescharacteristic
124C22×C8×C73C2311yescharacteristic
125C22×C8×C73C2311yescharacteristic
126C22×C8×C73C2311yescharacteristic
127C4×Q8×C73C2×C414yes—
128C4×Q8×C73D414yes—
129C2×C4⋊C4×C73D411yescharacteristic
130C2×C4⋊C4×C73D411yescharacteristic
131C2×C4⋊C4×C73D411yescharacteristic
132C2×C42×C73C2311yescharacteristic
133C2×C42×C73C2311yescharacteristic
134C2×C42×C73C2311yescharacteristic
135C4⋊C8×C73—22no—
136C4⋊C8×C73—22no—
137C4⋊C8×C73—22no—
138#97×C73—22no—
139#97×C73—22no—
140C4×C8×C73—22no—
141C23×D73—21no—
142C2×Dic146—21no—
143C2×Dic146—21no—
144C2×Dic146—21no—
145C2×Dic146—21no—
146C2×Dic146—21no—
147C2×Dic146—21no—
148C2×Dic146—21no—
149C2×Dic146—21no—
150C2×Dic146—21no—
151C2×Dic146—21no—
152C2×Dic146C22×C412yes—
153C2×Dic146C22×C412yes—
154C2×D292—21no—
155C2×D292—21no—
156C2×D292—21no—
157C2×D292—21no—
158C2×D292—21no—
159C2×D292C22×C412yes—
160C22×Dic73—21no—
161C22×Dic73—21no—
162C22×Dic73—21no—
163C22×Dic73—21no—
164C22×Dic73—21no—
165C22×Dic73—21no—
166C22×Dic73—21no—
167C2×C4×D73—24no—
168C2×C4×D73—42no—
169C2×C4×D73—42no—
170C4⋊Dic73—24no—
171C4⋊Dic73—24no—
172C4⋊Dic73—21no—
173C4⋊Dic73—21no—
174C4⋊Dic73—21no—
175C4⋊Dic73—21no—
176C4⋊Dic73—21no—
177C4⋊Dic73—21no—
178C4⋊Dic73—21no—
179C4⋊Dic73—21no—
180C4⋊Dic73—42no—
181C4⋊Dic73—81no—
182C4⋊Dic73C22×C412yes—
183C4⋊Dic73#4112yes—
184#11924007—24no—
185#11924007—24no—
186#11924005—24no—
187#11924005—24no—
188#11924005—24no—
189#11924005—24no—
190#11924005—24no—
191C4×Dic73—24no—
192C4×Dic73—24no—
193C4×Dic73—42no—
194C4×Dic73—81no—
195C4×Dic73—81no—
196C2×Q8×C73—41no—
197C2×Q8×C73—41no—
198C2×Q8×C73C22⋊C414yes—
199C22×C4×C73C2×D411yescharacteristic
200C22×C4×C73C2×D411yescharacteristic
201C22×C4×C73C2×D411yescharacteristic
202C22×C4×C73C2×D411yescharacteristic
203C22×C4×C73C2×D411yescharacteristic
204C22×C4×C73C2×D411yescharacteristic
205C22×C4×C73C2411yescharacteristic
206C2×C8×C73—21no—
207C2×C8×C73—21no—
208C2×C8×C73—21no—
209C2×C8×C73—21no—
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#4112yes—
218C4⋊C4×C73C22⋊C414yes—
219C4⋊C4×C73C22⋊C414yes—
220C4⋊C4×C73C22⋊C414yes—
221C42×C73C22×C412yes—
222C42×C73C22×C412yes—
223C42×C73C22×C412yes—
224C42×C73C2×D412yes—
225C42×C73C2×D412yes—
226C42×C73C2×D412yes—
227C23×C73C22×D411yescharacteristic
228C22×D73—22no—
229C22×D73—41no—
230C22×D73—41no—
231C22×D73—41no—
232C22×D73—41no—
233C22×D73—41no—
234C22×D73—41no—
235Q8×C73—44no—
236C2×C4×C73—21no—
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—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73C2×C22⋊C412yes—
249C2×C4×C73C2×C22⋊C412yes—
250C2×C4×C73C2×C22⋊C412yes—
251C2×C4×C73C2×C22⋊C412yes—
252C2×C4×C73C2×C22⋊C412yes—
253C2×C4×C73C2×C22⋊C412yes—
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—22no—
261C2×Dic73—22no—
262C2×Dic73—22no—
263C2×Dic73—22no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
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—
281D292—22no—
282D292—22no—
283D292—41no—
284D292—41no—
285C4×D73—48no—
286Dic146—22no—
287Dic146—22no—
288Dic146—22no—
289Dic146—22no—
290Dic146—41no—
291Dic146—41no—
292Dic146—41no—
293Dic146—41no—
294C8×C73—22no—
295C8×C73—41no—
296C8×C73—41no—
297C8×C73—41no—
298C22×C73C22×C22⋊C411yescharacteristic
299C22×C73#54811yescharacteristic
300C22×C73#55011yescharacteristic
301C22×C73#57511yescharacteristic
302C22×C73#57511yescharacteristic
303C22×C73#57711yescharacteristic
304C22×C73#57811yescharacteristic
305C2×D73—42no—
306C2×D73—42no—
307C2×D73—42no—
308C2×D73—81no—
309C2×D73—81no—
310C2×D73—81no—
311C2×D73—81no—
312C4×C73—22no—
313C4×C73—22no—
314C4×C73—22no—
315C4×C73—22no—
316C4×C73—22no—
317C4×C73—22no—
318C4×C73C4×#4111yescharacteristic
319C4×C73#52011yescharacteristic
320C4×C73#54811yescharacteristic
321C4×C73#55011yescharacteristic
322Dic73—42no—
323Dic73—42no—
324Dic73—42no—
325Dic73—42no—
326Dic73—81no—
327Dic73—81no—
328Dic73—81no—
329#22548—731nomaximal, Sylow
330C2×C73#234111yescharacteristic
331C2×C73#289611yescharacteristic
332C2×C73#289811yescharacteristic
333C2×C73#337611yescharacteristic
334C2×C73#337711yescharacteristic
335C2×C73#340111yescharacteristic
336C2×C73#340211yescharacteristic
337D73—82no—
338C2×#556—731no—
339C2×#446D7311yescharacteristic, complemented
340#2355—731no—
341#1983—731no—
342#1983—731no—
343#1981—731no—
344#1960—731no—
345#1959—731no—
346#1958—731no—
347#1805—731no—
348#1804—731no—
349#1803—731no—
350#1763—731no—
351#1762—731no—
352C4×#103—731no—
353C73#2254811yescharacteristic, complemented, Sylow
354#556—738no—
355#556—1464no—
356C2×C4⋊Q8—731no—
357C2×#126—1461no—
358C2×#126—1461no—
359C2×#124—731no—
360C2×#124—731no—
361C2×C22⋊Q8—1461no—
362C2×C4×Q8—731no—
363C2×C4×Q8—731no—
364C2×C4×Q8C2×D7311yescharacteristic
365C2×C4×D4—731no—
366C2×#117—1461no—
367#446—24no—
368C2×C4⋊C8C2×D7311yescharacteristic
369C2×C4⋊C8C2×D7311yescharacteristic
370C2×C4⋊C8C2×D7311yescharacteristic
371C2×#103—731no—
372C2×#103—731no—
373C2×#103—731no—
374C2×#103—731no—
375C2×#103—731no—
376C2×#103—731no—
377C2×#103—731no—
378C2×#103—731no—
379C2×#102—731no—
380C2×#102—731no—
381C2×#102—731no—
382C2×#102—731no—
383C2×#97C2×D7311yescharacteristic
384C2×#97C2×D7311yescharacteristic
385C2×C4×C8C2×D7311yescharacteristic
386#391—1461no—
387#387—1461no—
388#387—1461no—
389#384—731no—
390#382—731no—
391#382—731no—
392C4×C4⋊C4—731no—
393C4×C4⋊C4—731no—
394C4×C4⋊C4—731no—
395C4×C4⋊C4—1461no—
396#340—731no—
397#340—731no—
398#340—731no—
399#340—731no—
400C22×Q8—731no—
401C22×Q8—731no—
402C22×Q8D29211yescharacteristic
403C22×D4—731no—
404C23×C4—1461no—
405C22×C8C22×D7311yescharacteristic
406C22×C8C22×D7311yescharacteristic
407C22×C8C22×D7311yescharacteristic
408C22×C8C22×D7311yescharacteristic
409C4⋊Q8—734no—
410C4⋊Q8—1462no—
411#126—1468no—
412#126—1468no—
413#124—734no—
414#124—734no—
415#124—1462no—
416#124—1462no—
417C22⋊Q8—1468no—
418C4×Q8—734no—
419C4×Q8—734no—
420C4×Q8—1462no—
421C4×Q8—1462no—
422C4×Q8C4×D7314yes—
423C4×Q8C22⋊D7314yes—
424C4×D4—734no—
425C4×D4—1462no—
426#117—1468no—
427C2×C4⋊C4—731no—
428C2×C4⋊C4—731no—
429C2×C4⋊C4—731no—
430C2×C4⋊C4—731no—
431C2×C4⋊C4—731no—
432C2×C4⋊C4—1461no—
433C2×C4⋊C4—1461no—
434C2×C4⋊C4—1461no—
435C2×C4⋊C4—1461no—
436C2×C4⋊C4—1461no—
437C2×C4⋊C4—1461no—
438C2×C4⋊C4D29211yescharacteristic
439C2×C4⋊C4D29211yescharacteristic
440C2×C4⋊C4D29211yescharacteristic
441C2×C22⋊C4—1461no—
442C2×C22⋊C4—1461no—
443C2×C22⋊C4—1461no—
444C2×C22⋊C4—1461no—
445C2×C22⋊C4—1461no—
446C2×C42—1461no—
447C2×C42—1461no—
448C2×C42—1461no—
449C2×C42—1461no—
450C2×C42—1461no—
451C2×C42C22×D7311yescharacteristic
452C2×C42C22×D7311yescharacteristic
453C2×C42C22×D7311yescharacteristic
454C4⋊C8—22no—
455C4⋊C8—22no—
456C4⋊C8—22no—
457#103—2921no—
458#103—2921no—
459#103—2921no—
460#103—2921no—
461#103—1462no—
462#103—1462no—
463#103—1462no—
464#103—1462no—
465#102—2921no—
466#102—2921no—
467#102—1462no—
468#102—1462no—
469#97—22no—
470#97—22no—
471C4×C8—22no—
472#95—1461no—
473#95—1461no—
474#95—1461no—
475#95—1461no—
476#95—1461no—
477#95—1461no—
478#95—1461no—
479C24—1461no—
480C2×Q8—41no—
481C2×Q8—41no—
482C2×Q8—732no—
483C2×Q8—732no—
484C2×Q8—1461no—
485C2×Q8—1461no—
486C2×Q8—1461no—
487C2×Q8—1461no—
488C2×Q8—1461no—
489C2×Q8—1461no—
490C2×Q8—1461no—
491C2×Q8—1461no—
492C2×Q8—1461no—
493C2×Q8—1461no—
494C2×Q8#1192400514yes—
495C2×D4—732no—
496C2×D4—1461no—
497C2×D4—1461no—
498C2×D4—1461no—
499C2×D4—1461no—
500C2×D4—1461no—
501C22×C4—2922no—
502C22×C4—2922no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4—1461no—
506C22×C4—1461no—
507C22×C4—1461no—
508C22×C4—1461no—
509C22×C4—1461no—
510C22×C4—1464no—
511C22×C4C2×D29211yescharacteristic
512C22×C4C2×D29211yescharacteristic
513C22×C4C2×D29211yescharacteristic
514C22×C4C2×D29211yescharacteristic
515C22×C4C2×D29211yescharacteristic
516C22×C4C2×D29211yescharacteristic
517C22×C4C23×D7311yescharacteristic
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8#1192402912yes—
530C4⋊C4—2922no—
531C4⋊C4—5841no—
532C4⋊C4—732no—
533C4⋊C4—732no—
534C4⋊C4—1461no—
535C4⋊C4—1461no—
536C4⋊C4—1461no—
537C4⋊C4—1461no—
538C4⋊C4—1461no—
539C4⋊C4—1461no—
540C4⋊C4—1464no—
541C4⋊C4—1464no—
542C4⋊C4—1464no—
543C4⋊C4—1464no—
544C4⋊C4—1461no—
545C4⋊C4—1461no—
546C4⋊C4#1192400514yes—
547C4⋊C4#1192400514yes—
548C4⋊C4#1192400514yes—
549C22⋊C4—1464no—
550C22⋊C4—1464no—
551C22⋊C4—1464no—
552C22⋊C4—1464no—
553C22⋊C4—1464no—
554C42—2922no—
555C42—5841no—
556C42—5841no—
557C42—1464no—
558C42—1464no—
559C42C2×C4×D7312yes—
560C42C2×C4×D7312yes—
561C42C2×C4×D7312yes—
562C42C2×C22⋊D7312yes—
563C42C2×C22⋊D7312yes—
564C42C2×C22⋊D7312yes—
565C23—2921no—
566C23—2921no—
567C23—2921no—
568C23—2921no—
569C23—2921no—
570C23—2921no—
571C23—1462no—
572C23C22×D29211yescharacteristic
573Q8—44no—
574Q8—2921no—
575Q8—2921no—
576Q8—2921no—
577Q8—2921no—
578Q8—1462no—
579Q8—1462no—
580Q8—1462no—
581Q8—1462no—
582D4—2921no—
583D4—2921no—
584D4—1462no—
585D4—1462no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—2928no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—2921no—
603C2×C4—2921no—
604C2×C4—2921no—
605C2×C4—2921no—
606C2×C4—2921no—
607C2×C4—2921no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—2921no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—2921no—
614C2×C4—2921no—
615C2×C4—2921no—
616C2×C4—1462no—
617C2×C4—1462no—
618C2×C4—1462no—
619C2×C4—1462no—
620C2×C4C2×#1192400512yes—
621C2×C4C2×#1192400512yes—
622C2×C4C2×#1192400512yes—
623C2×C4C2×#1192400512yes—
624C2×C4C2×#1192400512yes—
625C2×C4C2×#1192400512yes—
626C2×C4C22×C4×D7311yescharacteristic
627C2×C4C22×C22⋊D7311yescharacteristic
628C2×C4#41×D7311yescharacteristic
629C2×C4C2×#1192402911yescharacteristic
630C2×C4#1455663711yescharacteristic
631C2×C4#1455663911yescharacteristic
632C8—22no—
633C8—41no—
634C8—41no—
635C8—41no—
636C22—2922no—
637C22—2922no—
638C22—2922no—
639C22—5841no—
640C22—5841no—
641C22—5841no—
642C22—5841no—
643C22C22×#1192400511yescharacteristic
644C22#1798924911yescharacteristic
645C22#1798925911yescharacteristic
646C22#1798938711yescharacteristic
647C22#1798938711yescharacteristic
648C22#1798939911yescharacteristic
649C22#1798940411yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—2922no—
657C4—2922no—
658C4—2922no—
659C4—2922no—
660C4—5841no—
661C4—5841no—
662C4—5841no—
663C4C4×#1192402911yescharacteristic
664C4#1798906911yescharacteristic
665C4#1798924811yescharacteristic
666C4#1798925611yescharacteristic
667C2—5842no—
668C2#2261276911yescharacteristic
669C2#2261845111yescharacteristic
670C2#2261846511yescharacteristic
671C2#2262276011yescharacteristic
672C2#2262277511yescharacteristic
673C2#2262299911yescharacteristic
674C2#2262301211yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#446×C73C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenterC2×#446D73
hypocenterC73#22548
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
2. #22612888C2semidirect
5. #22610853C2semidirect
6. #22610751C2semidirect
9. #22609947C2semidirect
11. #22609937C2semidirect
12. #22609762C2semidirect
14. C4×#14556516C2semidirect
15. C2×#446×C73C2semidirect
17. #17989310C4semidirect
339. C2×#446D73semidirect
353. C73#22548semidirect

Maximal subgroups

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

subgroupindex
C2×#179893102
#226128882
#226108612
#226108532
#226107512
#226107472
#226107422
#226099472
#226099422
#226099372
#226097622
#226097572
C4×#145565162
C2×#446×C732
#2254873