Finite Group Databasebeta

← #33108070

Subgroups — #33108070

Order 18688.

692 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619717C211yesmaximal, characteristic
2#22618877C211yesmaximal, characteristic, complemented
3#22618498C211yesmaximal, characteristic
4#22610910C211yesmaximal, characteristic, complemented
5#22610703C211yesmaximal, characteristic, complemented
6#22610683C211yesmaximal, characteristic, complemented
7#22610532C211yesmaximal, characteristic, complemented
8#22610527C211yesmaximal, characteristic, complemented
9#22610205C211yesmaximal, characteristic
10#22610203C211yesmaximal, characteristic
11#22609789C211yesmaximal, characteristic, complemented
12#22609772C211yesmaximal, characteristic
13#22609768C211yesmaximal, characteristic
14#22609740C211yesmaximal, characteristic
15C73×#1945C211yesmaximal, characteristic, complemented
16C2×#14556599C2211yescharacteristic
17C2×#14556597C2211yescharacteristic
18#17989140C2211yescharacteristic
19#17989055C2211yescharacteristic
20#17988897C411yescharacteristic, complemented
21#17988893C411yescharacteristic, complemented
22#17988890C411yescharacteristic, complemented
23#17988886C411yescharacteristic, complemented
24#17988881C411yescharacteristic, complemented
25#17988876C411yescharacteristic, complemented
26#17988871C411yescharacteristic, complemented
27#17988866C411yescharacteristic, complemented
28#17988716—21no—
29#17988716—21no—
30#17988714—21no—
31#17988711—21no—
32C2×#14556526C2211yescharacteristic
33#17988623C2211yescharacteristic
34#17988615C2211yescharacteristic
35#17988614C2211yescharacteristic
36#17988612—21no—
37#17988612—21no—
38#17988610C2211yescharacteristic
39#17988609—21no—
40C2×#14556520C2211yescharacteristic
41C2×#14556520C2211yescharacteristic
42C4⋊C8×D73—21no—
43C2×#14556519C2211yescharacteristic
44C2×#14556517C2211yescharacteristic
45#17988584C2211yescharacteristic
46#17988581C2211yescharacteristic
47#17988543C2211yescharacteristic
48#17988542C2211yescharacteristic
49M16×Dic73C2211yescharacteristic
50C2×#14556502C2211yescharacteristic
51#17988433C2211yescharacteristic
52#17988352C2211yescharacteristic
53#17988351C2211yescharacteristic
54#17988348C2211yescharacteristic
55#17988347C2211yescharacteristic
56#17988346C2211yescharacteristic
57#17988346C2211yescharacteristic
58#17988341C2211yescharacteristic
59#17988341C2211yescharacteristic
60C2×#108×C73C2211yescharacteristic
61#427×C73C2211yescharacteristic
62#424×C73C2211yescharacteristic
63#424×C73C2211yescharacteristic
64C2×C4×C8×C73C2211yescharacteristic
65#339×C73C2211yescharacteristic
66#339×C73C2211yescharacteristic
67C2×#11924028C2311yescharacteristic
68#14556599—22no—
69#14556599—41no—
70#14556597—22no—
71#14556597—41no—
72C2×#11924012C2311yescharacteristic
73C2×#11924012C2311yescharacteristic
74C2×#11924011—21no—
75C2×#11924011—21no—
76C22×#10596831C2311yescharacteristic
77C22×#10596831C2311yescharacteristic
78C2×C8×D73—21no—
79C2×C8×D73—21no—
80#14556590C2×C411yescharacteristic
81#14556590C2×C411yescharacteristic
82#14556582C2×C411yescharacteristic
83#14556582C2×C411yescharacteristic
84#14556568—21no—
85#14556568—21no—
86C4×Dic146—21no—
87C4×Dic146—21no—
88#14556556C2×C411yescharacteristic
89#14556556C2×C411yescharacteristic
90#14556553C2×C411yescharacteristic
91#14556553C2×C411yescharacteristic
92#14556551—21no—
93#14556551—21no—
94#14556550C2311yescharacteristic
95#14556548—21no—
96C2×C4×Dic73C2311yescharacteristic
97C2×C4×Dic73C2311yescharacteristic
98C42×D73—21no—
99#14556526—41no—
100#14556524C2×C411yescharacteristic
101#14556524C2×C411yescharacteristic
102#14556522C2×C411yescharacteristic
103#14556522C2×C411yescharacteristic
104C4⋊#10596831—21no—
105C4⋊#10596831—21no—
106#14556520—21no—
107#14556520—21no—
108#14556520—21no—
109#14556520—21no—
110#14556520C2×C411yescharacteristic
111#14556520C2×C411yescharacteristic
112#14556520C2×C411yescharacteristic
113#14556520C2×C411yescharacteristic
114#14556520D411yescharacteristic
115#14556520D411yescharacteristic
116#14556520D411yescharacteristic
117#14556520D411yescharacteristic
118#14556519Q814yescomplemented
119#14556517—22no—
120#14556516C2×C411yescharacteristic
121#14556516C2×C411yescharacteristic
122#14556514C2×C411yescharacteristic
123#14556514C2×C411yescharacteristic
124#14556513C2×C411yescharacteristic
125#14556513C2×C411yescharacteristic
126#14556511C2×C411yescharacteristic
127#14556511C2×C411yescharacteristic
128#14556502—21no—
129#14556502—41no—
130#14556501—21no—
131#14556501—21no—
132C4×#10596831—21no—
133C8×Dic73—21no—
134C8×Dic73—21no—
135#14556498—21no—
136#14556498—21no—
137C2×M16×C73C2311yescharacteristic
138C2×M16×C73C2311yescharacteristic
139C22×C8×C73C2311yescharacteristic
140C22×C8×C73C2311yescharacteristic
141#117×C73C2311yescharacteristic
142#117×C73C2311yescharacteristic
143C2×C42×C73C2311yescharacteristic
144#108×C73—41no—
145#107×C73C2×C411yescharacteristic
146#107×C73C2×C411yescharacteristic
147#106×C73C2×C411yescharacteristic
148#106×C73C2×C411yescharacteristic
149C4⋊C8×C73—21no—
150C4⋊C8×C73—21no—
151C4⋊C8×C73—21no—
152C4⋊C8×C73—21no—
153C4×C8×C73—41no—
154#11924028—21no—
155#11924028—21no—
156#11924028C2×Q812yes—
157#11924028#4112yes—
158C2×Dic146—21no—
159C2×Dic146—21no—
160C2×Dic146C22×C411yescharacteristic
161C2×D292C22×C411yescharacteristic
162C2×C22⋊D73—21no—
163C22×Dic73—21no—
164C22×Dic73#4111yescharacteristic
165C22×Dic73#4111yescharacteristic
166C2×C4×D73—21no—
167C2×C4×D73—21no—
168C2×C4×D73—21no—
169#11924012—21no—
170#11924012—21no—
171#11924012—21no—
172#11924012C2×Q812yes—
173#11924011—42no—
174#11924011—42no—
175C2×#10596831—21no—
176C2×#10596831—21no—
177C2×#10596831—21no—
178C2×#10596831—21no—
179C2×#10596831—21no—
180C2×#10596831—21no—
181C2×#10596831C22×C411yescharacteristic
182C2×#10596831C22×C411yescharacteristic
183C2×#10596831C2×D411yescharacteristic
184C2×#10596831C2×D411yescharacteristic
185C8×D73—42no—
186C8×D73—42no—
187C4⋊Dic73—21no—
188C4⋊Dic73—21no—
189C4⋊Dic73C22×C411yescharacteristic
190C4⋊Dic73C22×C411yescharacteristic
191#11924007—21no—
192#11924007—21no—
193#11924007—21no—
194#11924007—21no—
195#11924007—21no—
196#11924007—21no—
197C22⋊Dic73—21no—
198#11924005—21no—
199#11924005—21no—
200C4×Dic73—21no—
201C4×Dic73—21no—
202C4×Dic73—21no—
203C4×Dic73—21no—
204C4×Dic73—21no—
205C4×Dic73C22×C411yescharacteristic
206C4×Dic73C22×C411yescharacteristic
207C4×Dic73C2×D411yescharacteristic
208C4×Dic73C2×D411yescharacteristic
209C22×C4×C73—21no—
210C22×C4×C73C2411yescharacteristic
211M16×C73—21no—
212M16×C73—21no—
213M16×C73—21no—
214M16×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×C73C22×C411yescharacteristic
222C2×C8×C73C22×C411yescharacteristic
223C2×C8×C73C2×D411yescharacteristic
224C2×C8×C73C2×D411yescharacteristic
225C4⋊C4×C73C22×C411yescharacteristic
226C4⋊C4×C73C22×C411yescharacteristic
227C4⋊C4×C73C22×C411yescharacteristic
228C4⋊C4×C73C22×C411yescharacteristic
229C22⋊C4×C73—21no—
230C22⋊C4×C73—21no—
231C42×C73—21no—
232C42×C73—21no—
233C42×C73—21no—
234C42×C73C2×Q812yes—
235C23×C73C2×#4111yescharacteristic
236C22×D73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—22no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73C23×C411yescharacteristic
245C2×C4×C73C22×D411yescharacteristic
246C2×C4×C73C22×Q811yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73H(5,𝔽2)11yescharacteristic
249C2×C4×C732−1+411yescharacteristic
250C22⋊D73—22no—
251C22⋊D73—22no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73C4×D411yescharacteristic
261C2×Dic73C4×D411yescharacteristic
262C2×Dic73C4×D411yescharacteristic
263C2×Dic73C4×D411yescharacteristic
264D292—21no—
265D292C4×Q812yes—
266C4×D73—22no—
267C4×D73—22no—
268C4×D73—42no—
269C4×D73—81no—
270Dic146—21no—
271Dic146—42no—
272Dic146—81no—
273Dic146C4×Q812yes—
274C8×C73—21no—
275C8×C73—21no—
276C8×C73—21no—
277C8×C73—21no—
278C8×C73—41no—
279C8×C73—41no—
280#10596831—21no—
281#10596831—21no—
282#10596831—21no—
283#10596831—21no—
284#10596831—22no—
285#10596831—41no—
286C22×C73—21no—
287C22×C73—21no—
288C22×C73C2×C4×D411yescharacteristic
289C22×C73#55211yescharacteristic
290C22×C73#55411yescharacteristic
291C2×D73—22no—
292C2×D73—41no—
293C2×D73—41no—
294C4×C73—42no—
295C4×C73—41no—
296C4×C73—41no—
297C4×C73—41no—
298C4×C73—41no—
299C4×C73C2×C4×Q811yescharacteristic
300C4×C73#52011yescharacteristic
301C4×C73#54611yescharacteristic
302C4×C73D4×Q811yescharacteristic
303Dic73—22no—
304Dic73—21no—
305Dic73—21no—
306Dic73—21no—
307Dic73—21no—
308Dic73—41no—
309Dic73—41no—
310Dic73—81no—
311#22984—731nomaximal, Sylow
312C2×C73—21no—
313C2×C73—21no—
314C2×C73#236011yescharacteristic
315C2×C73#295511yescharacteristic
316C2×C73#295711yescharacteristic
317D73—42no—
318#3076—731no—
319#2962—731no—
320#2906—731no—
321#1995—731no—
322#1950—731no—
323#1946—731no—
324#1945D7311yescharacteristic, complemented
325#1922—731no—
326#1922—731no—
327#1866—731no—
328#1866—731no—
329#1768—731no—
330#1765—731no—
331#1764—731no—
332#1759—731no—
333C73#2298411yescharacteristic, complemented, Sylow
334C2×#131—731no—
335C2×#131—731no—
336#533—731no—
337C4×#41—731no—
338#479—731no—
339#479—731no—
340#478—731no—
341#478—731no—
342#477—731no—
343#476—731no—
344#475—731no—
345#474—731no—
346#446—1461no—
347#446—1461no—
348#446—1461no—
349C8×Q8—1461no—
350C2×#108—731no—
351C2×#108C2×D7311yescharacteristic
352#427—731no—
353#427C2×D7311yescharacteristic
354#424—731no—
355#424—731no—
356#424—1461no—
357#424—1461no—
358#424C2×D7311yescharacteristic
359#424C2×D7311yescharacteristic
360#423—731no—
361#423—1461no—
362C2×C4⋊C8—731no—
363C2×C4⋊C8—731no—
364C2×C4⋊C8—1461no—
365C2×#104—731no—
366C2×#104—731no—
367#416—731no—
368#416—731no—
369#405—731no—
370#405—731no—
371C4×M16—731no—
372C2×#97—731no—
373C2×C4×C8C2×D7311yescharacteristic
374#376—731no—
375#341—731no—
376#341—731no—
377#339—731no—
378#339—731no—
379#339—731no—
380#339—731no—
381#339C2×D7311yescharacteristic
382#339C2×D7311yescharacteristic
383#336—731no—
384#336—731no—
385C2×#41—731no—
386#131—2921no—
387#131—2921no—
388#131—1462no—
389#131—1462no—
390C2×M16—731no—
391C2×M16—731no—
392C2×M16—1461no—
393C2×M16—1461no—
394C2×M16C22×D7311yescharacteristic
395C2×M16C22×D7311yescharacteristic
396C22×C8—731no—
397C22×C8—731no—
398C22×C8—1461no—
399C22×C8—1461no—
400C22×C8C22×D7311yescharacteristic
401C22×C8C22×D7311yescharacteristic
402C4⋊Q8—731no—
403C4⋊Q8—731no—
404#125—731no—
405#125—731no—
406C22⋊Q8—1461no—
407C22⋊Q8—1461no—
408C4×Q8—731no—
409C4×Q8—731no—
410C4×Q8—1461no—
411C4×Q8—1461no—
412C4×D4—731no—
413C4×D4—731no—
414C4×D4—1461no—
415C4×D4—1461no—
416#117—731no—
417#117—1461no—
418#117C22×D7311yescharacteristic
419#117C22×D7311yescharacteristic
420C2×C42—731no—
421C2×C42—731no—
422C2×C42—1461no—
423C2×C42C22×D7311yescharacteristic
424#108—41no—
425#108—2921no—
426#107—731no—
427#107—731no—
428#107C4×D7311yescharacteristic
429#107C4×D7311yescharacteristic
430#106—731no—
431#106—731no—
432#106C4×D7311yescharacteristic
433#106C4×D7311yescharacteristic
434C4⋊C8—21no—
435C4⋊C8—21no—
436C4⋊C8—21no—
437C4⋊C8—21no—
438C4⋊C8—731no—
439C4⋊C8—731no—
440C4⋊C8—731no—
441C4⋊C8—731no—
442C4⋊C8—731no—
443C4⋊C8—731no—
444C4⋊C8—731no—
445C4⋊C8—731no—
446C4⋊C8—1461no—
447C4⋊C8—1461no—
448C4⋊C8—1461no—
449C4⋊C8—1461no—
450C4⋊C8—1461no—
451C4⋊C8—1461no—
452#104—734no—
453#104—1462no—
454#103—731no—
455#103—731no—
456#103—731no—
457#103—731no—
458#102—731no—
459#102—731no—
460#102—731no—
461#102—731no—
462#97—2921no—
463#97—1461no—
464#97—1461no—
465#97—1461no—
466C4×C8—41no—
467C4×C8—1461no—
468C4×C8—1461no—
469C4×C8—1461no—
470#95—1461no—
471#95—1461no—
472#41—732no—
473#41—732no—
474#41—1461no—
475#41—1461no—
476C2×Q8—731no—
477C2×Q8—1461no—
478C2×Q8—1461no—
479C2×D4—731no—
480C2×D4—1461no—
481C22×C4—21no—
482C22×C4—731no—
483C22×C4—731no—
484C22×C4—1461no—
485C22×C4—1461no—
486C22×C4—1461no—
487C22×C4—1461no—
488C22×C4C23×D7311yescharacteristic
489M16—21no—
490M16—21no—
491M16—21no—
492M16—21no—
493M16—2922no—
494M16—2922no—
495M16—732no—
496M16—1461no—
497M16—1461no—
498M16—1461no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—2922no—
506C2×C8—2922no—
507C2×C8—731no—
508C2×C8—731no—
509C2×C8—731no—
510C2×C8—731no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8C2×C4×D7311yescharacteristic
518C2×C8C2×C4×D7311yescharacteristic
519C2×C8D4×D7311yescharacteristic
520C2×C8D4×D7311yescharacteristic
521C4⋊C4—731no—
522C4⋊C4—731no—
523C4⋊C4—1461no—
524C4⋊C4—1461no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4C2×C4×D7311yescharacteristic
532C4⋊C4C2×C4×D7311yescharacteristic
533C4⋊C4C2×C4×D7311yescharacteristic
534C4⋊C4C2×C4×D7311yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—1461no—
538C22⋊C4—1461no—
539C22⋊C4—1461no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—731no—
544C42—731no—
545C42—731no—
546C42—731no—
547C42—1461no—
548C42—1461no—
549C42—1461no—
550C42—1461no—
551C42—1461no—
552C42Q8×D7312yes—
553C23—1461no—
554C23#41×D7311yescharacteristic
555Q8—2922no—
556Q8—5841no—
557Q8—732no—
558Q8—1461no—
559D4—732no—
560D4—1462no—
561D4—1462no—
562D4—1461no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—2922no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—5841no—
577C2×C4—731no—
578C2×C4—731no—
579C2×C4—731no—
580C2×C4—731no—
581C2×C4—1461no—
582C2×C4—1461no—
583C2×C4—1461no—
584C2×C4—1462no—
585C2×C4—1462no—
586C2×C4C22×C4×D7311yescharacteristic
587C2×C4C2×D4×D7311yescharacteristic
588C2×C4C2×Q8×D7311yescharacteristic
589C2×C4#41×D7311yescharacteristic
590C2×C4#1455663611yescharacteristic
591C2×C4#1455663911yescharacteristic
592C8—21no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—41no—
597C8—41no—
598C8—2921no—
599C8—1461no—
600C8—1461no—
601C8—1462no—
602C8—1461no—
603C8—1461no—
604C22—21no—
605C22—21no—
606C22—2921no—
607C22—2921no—
608C22—1462no—
609C22C4×D4×D7311yescharacteristic
610C22#1798926911yescharacteristic
611C22#1798929111yescharacteristic
612C4—42no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—2921no—
618C4—2921no—
619C4—5841no—
620C4—1462no—
621C4—1461no—
622C4—1461no—
623C4—1461no—
624C4—1461no—
625C4C4×Q8×D7311yescharacteristic
626C4#1798906811yescharacteristic
627C4#1798922611yescharacteristic
628C4#1798925211yescharacteristic
629C2—21no—
630C2—21no—
631C2—2922no—
632C2#2261293311yescharacteristic
633C2#2261882911yescharacteristic
634C2#2261884311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1945C2
socleC22×C73C2×C4×D4
prime radicalC2×#11924028C23
cosocleC22×C4×C73C24
hypercenter#1945D73
hypocenterC73#22984
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1945

Complements

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

NH ≅ G/Ntype
2. #22618877C2semidirect
4. #22610910C2semidirect
5. #22610703C2semidirect
6. #22610683C2semidirect
7. #22610532C2semidirect
8. #22610527C2semidirect
11. #22609789C2semidirect
15. C73×#1945C2semidirect
20. #17988897C4semidirect
21. #17988893C4semidirect
22. #17988890C4semidirect
23. #17988886C4semidirect
24. #17988881C4semidirect
25. #17988876C4semidirect
26. #17988871C4semidirect
27. #17988866C4semidirect
118. #14556519Q8semidirect
324. #1945D73semidirect
333. C73#22984semidirect

Maximal subgroups

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

subgroupindex
#226197172
#226188772
#226184982
#226109102
#226107032
#226106832
#226105322
#226105272
#226102052
#226102032
#226097892
#226097722
#226097682
#226097402
C73×#19452
#2298473