Finite Group Databasebeta

← #33149604

Subgroups — #33149604

Order 18688.

746 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22613622C211yesmaximal, characteristic
2#22611499C211yesmaximal, characteristic, complemented
3#22611427C211yesmaximal, characteristic, complemented
4#22611326C211yesmaximal, characteristic, complemented
5#22611072C211yesmaximal, characteristic, complemented
6#22610883C211yesmaximal, characteristic, complemented
7#22610516C211yesmaximal, characteristic, complemented
8#22610355C211yesmaximal, characteristic
9#22610244C211yesmaximal, characteristic
10#22610199C211yesmaximal, characteristic, complemented
11#22610130C211yesmaximal, characteristic
12#22610111C211yesmaximal, characteristic
13#22609946C211yesmaximal, characteristic, complemented
14#22609647C211yesmaximal, characteristic
15C73×#2362C211yesmaximal, characteristic
16C23×#10596831C2211yescharacteristic
17C2×#14556591C2211yescharacteristic
18C2×#14556550C2211yescharacteristic
19C2×#14556524C2211yescharacteristic
20C2×#14556522C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×C22⋊#10596831C2211yescharacteristic
28#17988510—21no—
29#17988499—21no—
30#17988491—21no—
31#17988485C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988474—21no—
34#17988453C2211yescharacteristic
35#17988447C2211yescharacteristic
36#17988349C2211yescharacteristic
37#17988349C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×C22⋊Q8×C73C2211yescharacteristic
49C2×C4×Q8×C73C2211yescharacteristic
50C2×#117×C73C2211yescharacteristic
51#391×C73C2211yescharacteristic
52#389×C73C2211yescharacteristic
53#388×C73—21no—
54#388×C73—21no—
55#387×C73—21no—
56C4×C4⋊C4×C73—21no—
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×Dic146—21no—
60C22×#10596831—22no—
61C22×#10596831—41no—
62C22×#10596831—41no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556591—22no—
68#14556591—41no—
69#14556550—22no—
70#14556550—41no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×C22⋊Dic73—21no—
79C2×C22⋊Dic73—21no—
80C2×C4×Dic73—21no—
81#14556524—41no—
82#14556522—22no—
83C4⋊#10596831—41no—
84C4⋊#10596831—41no—
85#14556515—22no—
86#14556515—41no—
87#14556515—41no—
88#14556515—41no—
89C22⋊#10596831—22no—
90#14556498—21no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95C22×Q8×C73C2311yescharacteristic
96C23×C4×C73C2311yescharacteristic
97C22⋊Q8×C73—22no—
98C22⋊Q8×C73Q814yescomplemented
99C4×Q8×C73—22no—
100C4×Q8×C73—41no—
101#117×C73—22no—
102#117×C73—41no—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73C2311yescharacteristic
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C22⋊C4×C73—21no—
110C2×C22⋊C4×C73—21no—
111C2×C22⋊C4×C73D411yescharacteristic
112C2×C22⋊C4×C73D411yescharacteristic
113C2×C42×C73—21no—
114C2×C42×C73—21no—
115C2×C42×C73—21no—
116C2×C42×C73C2311yescharacteristic
117C2×C42×C73C2311yescharacteristic
118#95×C73—21no—
119#95×C73—21no—
120#95×C73—21no—
121#95×C73—21no—
122#95×C73D411yescharacteristic
123#95×C73D411yescharacteristic
124C2×Dic146—42no—
125C2×Dic146—42no—
126C2×Dic146—81no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—44no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—42no—
152C4⋊Dic73#4112yes—
153#11924007—42no—
154#11924007—81no—
155C22⋊Dic73—42no—
156C22⋊Dic73—81no—
157C4×Dic73—42no—
158C24×C73C2×D411yescharacteristic
159C2×Q8×C73—21no—
160C2×Q8×C73—21no—
161C2×Q8×C73—21no—
162C2×Q8×C73—21no—
163C2×Q8×C73—21no—
164C2×Q8×C73C2×Q812yes—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×Q812yes—
179C22×C4×C73#4112yes—
180C22×C4×C73C2411yescharacteristic
181C4⋊C4×C73—24no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—42no—
187C4⋊C4×C73—42no—
188C4⋊C4×C73C2×Q812yes—
189C4⋊C4×C73#4112yes—
190C22⋊C4×C73—24no—
191C22⋊C4×C73—22no—
192C22⋊C4×C73—41no—
193C22⋊C4×C73—42no—
194C42×C73—21no—
195C42×C73—21no—
196C42×C73—21no—
197C42×C73—21no—
198C42×C73—42no—
199C42×C73—42no—
200C42×C73—42no—
201C23×C73—21no—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73C22⋊Q812yes—
208C23×C73C22×D411yescharacteristic
209Q8×C73—22no—
210Q8×C73—22no—
211Q8×C73—41no—
212Q8×C73—41no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—22no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73C22⋊Q812yes—
243C2×C4×C73#13512yes—
244C2×C4×C73C22×Q811yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73H(5,𝔽2)11yescharacteristic
249C2×C4×C73H(5,𝔽2)11yescharacteristic
250C2×Dic73—22no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262Dic146—84no—
263#10596831—41no—
264#10596831—41no—
265#10596831—41no—
266#10596831—41no—
267C22×C73—22no—
268C22×C73—22no—
269C22×C73—22no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73C2×C22⋊Q811yescharacteristic
275C22×C73#54611yescharacteristic
276C22×C73#54611yescharacteristic
277C22×C73C2×#13511yescharacteristic
278C22×C73#57511yescharacteristic
279C22×C73#57711yescharacteristic
280C22×C73#57811yescharacteristic
281C4×C73—22no—
282C4×C73—22no—
283C4×C73—42no—
284C4×C73—42no—
285C4×C73—41no—
286C4×C73—42no—
287C4×C73—41no—
288C4×C73—41no—
289C4×C73—81no—
290C4×C73#52911yescharacteristic
291C4×C73#53811yescharacteristic
292C4×C73#55411yescharacteristic
293C4×C73#55411yescharacteristic
294Dic73—42no—
295Dic73—81no—
296Dic73—81no—
297Dic73—81no—
298#26413—731nomaximal, Sylow
299C2×C73—42no—
300C2×C73#245011yescharacteristic
301C2×C73#307611yescharacteristic
302C2×C73#308211yescharacteristic
303C2×C73#329611yescharacteristic
304C2×C73#330211yescharacteristic
305C2×C73#331311yescharacteristic
306C2×C73#332111yescharacteristic
307#2439—731no—
308#2362D7311yescharacteristic
309#2101—731no—
310#2087—731no—
311#2068—731no—
312#2024—731no—
313#1988—731no—
314#1920—731no—
315#1892—731no—
316#1873—731no—
317#1865—731no—
318#1854—731no—
319#1850—731no—
320#1805—731no—
321#1735—731no—
322C73#2641311yescharacteristic, complemented, Sylow
323C23×C8—731no—
324C2×C4⋊Q8—731no—
325C2×C22⋊Q8C2×D7311yescharacteristic
326C2×C4×Q8C2×D7311yescharacteristic
327C2×#117—731no—
328C2×#117C2×D7311yescharacteristic
329C2×#107—731no—
330C2×#106—731no—
331C2×C4⋊C8—731no—
332C2×C4⋊C8—731no—
333C2×#103—731no—
334C2×#103—731no—
335C2×#103—731no—
336C2×#103—731no—
337C2×C22⋊C8—731no—
338#396—1461no—
339#393—1461no—
340#391—1461no—
341#391C2×D7311yescharacteristic
342#389—731no—
343#389—731no—
344#389C2×D7311yescharacteristic
345#388—21no—
346#388—21no—
347#387—21no—
348#387—1461no—
349#382—731no—
350#380—731no—
351C4×C4⋊C4—21no—
352C4×C4⋊C4C2×D7311yescharacteristic
353C4×C22⋊C4C2×D7311yescharacteristic
354#340—731no—
355#340—731no—
356#340—731no—
357#340—731no—
358#340—731no—
359#340—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#336—731no—
365#336—731no—
366C22×Q8—1461no—
367C22×Q8C22×D7311yescharacteristic
368C23×C4C22×D7311yescharacteristic
369C22×C8—2921no—
370C22×C8—2921no—
371C22×C8—731no—
372C22×C8—731no—
373C22×C8—731no—
374C22×C8—731no—
375C22×C8—1462no—
376C4⋊Q8—2921no—
377C4⋊Q8—1462no—
378C22⋊Q8—22no—
379C22⋊Q8Dic14614yescomplemented
380C4×Q8—22no—
381C4×Q8—41no—
382#117—22no—
383#117—41no—
384#117—2921no—
385#117—1462no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4C22×D7311yescharacteristic
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C4⋊C4C22×D7311yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1461no—
402C2×C22⋊C4—1461no—
403C2×C22⋊C4C22⋊D7311yescharacteristic
404C2×C22⋊C4C22⋊D7311yescharacteristic
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—1461no—
409C2×C42C22×D7311yescharacteristic
410C2×C42C22×D7311yescharacteristic
411#107—2921no—
412#106—1462no—
413C4⋊C8—2921no—
414C4⋊C8—2921no—
415#103—2921no—
416#103—2921no—
417#103—2921no—
418#103—1462no—
419C22⋊C8—1462no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—1461no—
425#95—1461no—
426#95—1461no—
427#95—1461no—
428#95—1461no—
429#95C22⋊D7311yescharacteristic
430#95C22⋊D7311yescharacteristic
431C24C2×C22⋊D7311yescharacteristic
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—2922no—
438C2×Q8—2922no—
439C2×Q8—5841no—
440C2×Q8C2×Dic14612yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4C2×C22⋊D7311yescharacteristic
454C22×C4C2×C22⋊D7311yescharacteristic
455C22×C4C2×C22⋊D7311yescharacteristic
456C22×C4C2×C22⋊D7311yescharacteristic
457C22×C4C2×C22⋊D7311yescharacteristic
458C22×C4C2×Dic14612yes—
459C22×C4#1192402812yes—
460C22×C4C23×D7311yescharacteristic
461C2×C8—2924no—
462C2×C8—1461no—
463C2×C8—1461no—
464C2×C8—1461no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—2922no—
482C4⋊C4—2922no—
483C4⋊C4—5841no—
484C4⋊C4—732no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4C2×Dic14612yes—
493C4⋊C4#1192402712yes—
494C22⋊C4—24no—
495C22⋊C4—22no—
496C22⋊C4—41no—
497C22⋊C4—42no—
498C22⋊C4—2922no—
499C22⋊C4—5841no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—2922no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23#1455657012yes—
515C23C22×C22⋊D7311yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520Q8—5844no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—1462no—
562C2×C4#1455657012yes—
563C2×C4#1455661112yes—
564C2×C4C22×Dic14611yescharacteristic
565C2×C4#41×D7311yescharacteristic
566C2×C4C2×#1192402711yescharacteristic
567C2×C4C2×#1192402811yescharacteristic
568C2×C4#1455663611yescharacteristic
569C2×C4#1455663711yescharacteristic
570C8—2921no—
571C8—2921no—
572C8—2921no—
573C8—2921no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C2×#1455657011yescharacteristic
582C22#1798922511yescharacteristic
583C22#1798923111yescharacteristic
584C22C2×#1455661111yescharacteristic
585C22#1798938511yescharacteristic
586C22#1798939511yescharacteristic
587C22#1798940211yescharacteristic
588C4—22no—
589C4—22no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—42no—
594C4—41no—
595C4—42no—
596C4—81no—
597C4—2922no—
598C4—5841no—
599C4—5841no—
600C4—5841no—
601C4#1798911711yescharacteristic
602C4#1798917011yescharacteristic
603C4#1798929411yescharacteristic
604C4#1798929611yescharacteristic
605C2—42no—
606C2#2261374911yescharacteristic
607C2#2261971911yescharacteristic
608C2#2261977311yescharacteristic
609C2#2262176011yescharacteristic
610C2#2262184811yescharacteristic
611C2#2262199111yescharacteristic
612C2#2262211711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2362C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2362D73
hypocenterC73#26413
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
2. #22611499C2semidirect
3. #22611427C2semidirect
4. #22611326C2semidirect
5. #22611072C2semidirect
6. #22610883C2semidirect
7. #22610516C2semidirect
10. #22610199C2semidirect
13. #22609946C2semidirect
98. C22⋊Q8×C73Q8semidirect
322. C73#26413semidirect
379. C22⋊Q8Dic146semidirect

Maximal subgroups

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

subgroupindex
#226136222
#226114992
#226114272
#226113262
#226110722
#226108832
#226105162
#226103552
#226102442
#226101992
#226101302
#226101112
#226099462
#226096472
C73×#23622
#2641373