Finite Group Databasebeta

← #33093833

Subgroups — #33093833

Order 18688.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619715C211yesmaximal, characteristic
2#22618870C211yesmaximal, characteristic, complemented
3#22618357C211yesmaximal, characteristic
4#22610873C211yesmaximal, characteristic, complemented
5#22610703C211yesmaximal, characteristic, complemented
6#22610531C211yesmaximal, characteristic, complemented
7#22610527C211yesmaximal, characteristic, complemented
8#22610206C211yesmaximal, characteristic
9#22609949C211yesmaximal, characteristic
10#22609818C211yesmaximal, characteristic, complemented
11#22609809C211yesmaximal, characteristic, complemented
12#22609802C211yesmaximal, characteristic, complemented
13#22609778C211yesmaximal, characteristic
14#22609769C211yesmaximal, characteristic
15C73×#1866C211yesmaximal, characteristic
16C2×#14556598C2211yescharacteristic
17#17989140C2211yescharacteristic
18#41×Dic73C2211yescharacteristic
19#17988896C411yescharacteristic, complemented
20#17988892C411yescharacteristic, complemented
21#17988889C411yescharacteristic, complemented
22#17988885C411yescharacteristic, complemented
23#17988880C411yescharacteristic, complemented
24#17988875C411yescharacteristic, complemented
25#17988870C411yescharacteristic, complemented
26#17988865C411yescharacteristic, complemented
27#17988716—21no—
28#17988716—21no—
29#17988714—21no—
30#17988714—21no—
31C2×#14556526C2211yescharacteristic
32#17988625C2211yescharacteristic
33#17988623C2211yescharacteristic
34#17988613C2211yescharacteristic
35#17988610C2211yescharacteristic
36C2×#14556520C2211yescharacteristic
37C2×#14556518C2211yescharacteristic
38#17988583C2211yescharacteristic
39#17988545—21no—
40#17988545C2211yescharacteristic
41#17988545C2211yescharacteristic
42#17988543C2211yescharacteristic
43#17988542—21no—
44#17988542C2211yescharacteristic
45M16×Dic73—21no—
46M16×Dic73C2211yescharacteristic
47C2×#14556501—21no—
48C2×#14556501C2211yescharacteristic
49C2×#14556501C2211yescharacteristic
50C2×C4×#10596831C2211yescharacteristic
51C4×#11924007C2211yescharacteristic
52#17988352C2211yescharacteristic
53#17988351C2211yescharacteristic
54#17988348C2211yescharacteristic
55#17988348C2211yescharacteristic
56#17988347C2211yescharacteristic
57#17988347C2211yescharacteristic
58#17988346C2211yescharacteristic
59#17988341C2211yescharacteristic
60C2×#131×C73C2211yescharacteristic
61C2×#108×C73C2211yescharacteristic
62#424×C73C2211yescharacteristic
63C2×#104×C73C2211yescharacteristic
64#416×C73C2211yescharacteristic
65#339×C73C2211yescharacteristic
66#336×C73C2211yescharacteristic
67#14556598—22no—
68#14556598Q814yescomplemented
69C2×#11924012—21no—
70C2×#11924012C2311yescharacteristic
71C2×#11924012C2311yescharacteristic
72C22×#10596831—21no—
73C22×#10596831C2311yescharacteristic
74C22×#10596831C2311yescharacteristic
75#14556590C2×C411yescharacteristic
76#14556590C2×C411yescharacteristic
77#14556582C2×C411yescharacteristic
78#14556582C2×C411yescharacteristic
79#14556568—21no—
80#14556568—21no—
81Q8×Dic73C2×C411yescharacteristic
82Q8×Dic73C2×C411yescharacteristic
83C4×Dic146—21no—
84C4×Dic146—21no—
85D4×Dic73—21no—
86D4×Dic73—21no—
87D4×Dic73C2×C411yescharacteristic
88D4×Dic73C2×C411yescharacteristic
89#14556550—21no—
90#14556550C2311yescharacteristic
91#14556550C2311yescharacteristic
92C2×#11924007—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73C2311yescharacteristic
95C2×C4×Dic73C2311yescharacteristic
96#14556526—22no—
97#14556525C2×C411yescharacteristic
98#14556525C2×C411yescharacteristic
99#14556524C2×C411yescharacteristic
100#14556524C2×C411yescharacteristic
101#14556523C2×C411yescharacteristic
102#14556523C2×C411yescharacteristic
103#14556522C2×C411yescharacteristic
104#14556522C2×C411yescharacteristic
105#14556520—21no—
106#14556520—21no—
107#14556520—21no—
108#14556520—21no—
109#14556520C2×C411yescharacteristic
110#14556520C2×C411yescharacteristic
111#14556520C2×C411yescharacteristic
112#14556520C2×C411yescharacteristic
113#14556518—22no—
114#14556515C2×C411yescharacteristic
115#14556515C2×C411yescharacteristic
116#14556512C2×C411yescharacteristic
117#14556512C2×C411yescharacteristic
118#14556502—21no—
119#14556501—21no—
120#14556501—21no—
121#14556501—22no—
122#14556501—21no—
123#14556501—21no—
124#14556501D411yescharacteristic
125#14556501D411yescharacteristic
126#14556501D411yescharacteristic
127#14556501D411yescharacteristic
128C4×#10596831—21no—
129C4×#10596831—22no—
130C8×Dic73—21no—
131C8×Dic73—21no—
132C8×Dic73—21no—
133C8×Dic73—21no—
134#14556498—21no—
135C2×#41×C73C2311yescharacteristic
136#131×C73—22no—
137#131×C73—22no—
138C2×M16×C73—21no—
139C2×M16×C73C2311yescharacteristic
140C2×M16×C73C2311yescharacteristic
141C22×C8×C73—21no—
142C22×C8×C73C2311yescharacteristic
143C22×C8×C73C2311yescharacteristic
144#117×C73C2311yescharacteristic
145C2×C42×C73C2311yescharacteristic
146#108×C73—22no—
147C4⋊C8×C73—21no—
148C4⋊C8×C73—21no—
149#104×C73—22no—
150#103×C73C2×C411yescharacteristic
151#103×C73C2×C411yescharacteristic
152#102×C73C2×C411yescharacteristic
153#102×C73C2×C411yescharacteristic
154C2×Dic146—21no—
155C2×Dic146—21no—
156C22×Dic73—21no—
157C22×Dic73—21no—
158C22×Dic73—21no—
159C22×Dic73#4111yescharacteristic
160C22×Dic73#4111yescharacteristic
161#11924012—21no—
162#11924012—24no—
163#11924012—21no—
164#11924012—21no—
165#11924012C2×Q812yes—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831—24no—
170C2×#10596831—21no—
171C2×#10596831C22×C411yescharacteristic
172C2×#10596831C22×C411yescharacteristic
173C2×#10596831C2×D411yescharacteristic
174C2×#10596831C2×D411yescharacteristic
175C2×#10596831C2×Q812yes—
176C2×#10596831#4112yes—
177C4⋊Dic73—21no—
178C4⋊Dic73—21no—
179C4⋊Dic73—21no—
180C4⋊Dic73—21no—
181C4⋊Dic73C22×C411yescharacteristic
182C4⋊Dic73C22×C411yescharacteristic
183C4⋊Dic73C22×C411yescharacteristic
184C4⋊Dic73C22×C411yescharacteristic
185#11924007—21no—
186#11924007—21no—
187#11924007—21no—
188#11924007—21no—
189#11924007—42no—
190C22⋊Dic73—21no—
191C22⋊Dic73—21no—
192C22⋊Dic73—21no—
193C22⋊Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196C4×Dic73—21no—
197C4×Dic73—21no—
198C4×Dic73—21no—
199C4×Dic73—21no—
200C4×Dic73C22×C411yescharacteristic
201C4×Dic73C22×C411yescharacteristic
202C4×Dic73C2×D411yescharacteristic
203C4×Dic73C2×D411yescharacteristic
204#41×C73—21no—
205#41×C73—21no—
206#41×C73C2×Q812yes—
207#41×C73#4112yes—
208C2×Q8×C73C22×C411yescharacteristic
209C2×D4×C73—21no—
210C2×D4×C73C22×C411yescharacteristic
211C22×C4×C73—21no—
212C22×C4×C73—21no—
213C22×C4×C73C2411yescharacteristic
214M16×C73—24no—
215M16×C73—21no—
216M16×C73—21no—
217M16×C73—21no—
218M16×C73#4112yes—
219C2×C8×C73—24no—
220C2×C8×C73—21no—
221C2×C8×C73—21no—
222C2×C8×C73—21no—
223C2×C8×C73—21no—
224C2×C8×C73—21no—
225C2×C8×C73—21no—
226C2×C8×C73C22×C411yescharacteristic
227C2×C8×C73C22×C411yescharacteristic
228C2×C8×C73C2×D411yescharacteristic
229C2×C8×C73C2×D411yescharacteristic
230C4⋊C4×C73C22×C411yescharacteristic
231C4⋊C4×C73C22×C411yescharacteristic
232C22⋊C4×C73—21no—
233C42×C73—21no—
234C42×C73—21no—
235C42×C73—21no—
236C23×C73—21no—
237C23×C73C2×#4111yescharacteristic
238Q8×C73—21no—
239Q8×C73C4×Q812yes—
240D4×C73—22no—
241D4×C73—22no—
242D4×C73—21no—
243D4×C73C4×Q812yes—
244C2×C4×C73—21no—
245C2×C4×C73—22no—
246C2×C4×C73—22no—
247C2×C4×C73—22no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73C23×C411yescharacteristic
253C2×C4×C73C22×D411yescharacteristic
254C2×C4×C73C22×Q811yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—81no—
268C2×Dic73—81no—
269C2×Dic73C4×D411yescharacteristic
270C2×Dic73C4×D411yescharacteristic
271C2×Dic73C4×D411yescharacteristic
272C2×Dic73C4×D411yescharacteristic
273Dic146—81no—
274Dic146—81no—
275C8×C73—21no—
276C8×C73—21no—
277C8×C73—21no—
278C8×C73—22no—
279C8×C73—21no—
280C8×C73—41no—
281#10596831—21no—
282#10596831—21no—
283#10596831—22no—
284#10596831—41no—
285#10596831C22⋊Q812yes—
286#10596831C22⋊Q812yes—
287C22×C73—21no—
288C22×C73—21no—
289C22×C73—22no—
290C22×C73—41no—
291C22×C73—41no—
292C22×C73C2×C4×D411yescharacteristic
293C22×C73#52911yescharacteristic
294C22×C73#53311yescharacteristic
295C4×C73—22no—
296C4×C73—42no—
297C4×C73—41no—
298C4×C73—41no—
299C4×C73C2×C4×Q811yescharacteristic
300C4×C73C4×#4111yescharacteristic
301C4×C73C2×C22⋊Q811yescharacteristic
302C4×C73#52511yescharacteristic
303Dic73—21no—
304Dic73—21no—
305Dic73—21no—
306Dic73—21no—
307Dic73—41no—
308Dic73—41no—
309Dic73—41no—
310Dic73—41no—
311Dic73—81no—
312#21629—731nomaximal, Sylow
313C2×C73—21no—
314C2×C73—21no—
315C2×C73—42no—
316C2×C73C4×C22⋊Q811yescharacteristic
317C2×C73#295611yescharacteristic
318C2×C73#295711yescharacteristic
319#3076—731no—
320#2961—731no—
321#2875—731no—
322#1986—731no—
323#1950—731no—
324#1922—731no—
325#1922—731no—
326#1866—731no—
327#1866D7311yescharacteristic
328#1806—731no—
329#1773—731no—
330#1772—731no—
331#1771—731no—
332#1765—731no—
333#1764—731no—
334C73#2162911yescharacteristic, complemented, Sylow
335C2×#131—731no—
336C2×#131C2×D7311yescharacteristic
337#533—731no—
338C4×#41—731no—
339#479—731no—
340#479—731no—
341#478—731no—
342#478—731no—
343#477—731no—
344#476—731no—
345#475—731no—
346#474—731no—
347#446—1461no—
348#446—1461no—
349#446—1461no—
350#446—1461no—
351C2×#108—731no—
352C2×#108C2×D7311yescharacteristic
353#427—731no—
354#427—731no—
355#424—731no—
356#424C2×D7311yescharacteristic
357#423—731no—
358C2×C4⋊C8—731no—
359C2×#104—731no—
360C2×#104C2×D7311yescharacteristic
361#416—731no—
362#416C2×D7311yescharacteristic
363#405—731no—
364#405—731no—
365#405—731no—
366#405—731no—
367#405—1461no—
368#405—1461no—
369C4×M16—731no—
370C4×M16—1461no—
371C2×#97—731no—
372C2×#97—731no—
373C2×#97—1461no—
374C2×C4×C8—731no—
375C4×C4⋊C4—731no—
376#341—731no—
377#341—731no—
378#339—731no—
379#339—731no—
380#339—731no—
381#339—731no—
382#339—731no—
383#339C2×D7311yescharacteristic
384#336—731no—
385#336C2×D7311yescharacteristic
386C2×#41C22×D7311yescharacteristic
387#131—22no—
388#131—22no—
389#131—734no—
390#131—1462no—
391C2×M16—21no—
392C2×M16—731no—
393C2×M16—731no—
394C2×M16—1461no—
395C2×M16C22×D7311yescharacteristic
396C2×M16C22×D7311yescharacteristic
397C22×C8—21no—
398C22×C8—731no—
399C22×C8—731no—
400C22×C8—1461no—
401C22×C8C22×D7311yescharacteristic
402C22×C8C22×D7311yescharacteristic
403C4⋊Q8—731no—
404C4⋊Q8—731no—
405#125—731no—
406#125—731no—
407C22⋊Q8—1461no—
408C22⋊Q8—1461no—
409C4×Q8—731no—
410C4×Q8—731no—
411C4×Q8—1461no—
412C4×Q8—1461no—
413C4×D4—731no—
414C4×D4—731no—
415C4×D4—1461no—
416C4×D4—1461no—
417#117—731no—
418#117—731no—
419#117—1461no—
420#117C22×D7311yescharacteristic
421C2×C4⋊C4—1461no—
422C2×C42—731no—
423C2×C42—731no—
424C2×C42—1461no—
425C2×C42C22×D7311yescharacteristic
426#108—22no—
427#108—1462no—
428#107—731no—
429#107—731no—
430#107—731no—
431#107—731no—
432#106—731no—
433#106—731no—
434#106—731no—
435#106—731no—
436C4⋊C8—21no—
437C4⋊C8—21no—
438C4⋊C8—731no—
439C4⋊C8—731no—
440C4⋊C8—731no—
441C4⋊C8—731no—
442C4⋊C8—1461no—
443C4⋊C8—1461no—
444C4⋊C8—1461no—
445C4⋊C8—1461no—
446#104—22no—
447#104—1462no—
448#103—731no—
449#103—731no—
450#103C4×D7311yescharacteristic
451#103C4×D7311yescharacteristic
452#102—731no—
453#102—731no—
454#102C4×D7311yescharacteristic
455#102C4×D7311yescharacteristic
456#97—731no—
457#97—731no—
458#97—731no—
459#97—731no—
460#97—1461no—
461#97—1461no—
462#97—1462no—
463#97—1461no—
464#97—1461no—
465#97—1461no—
466C4×C8—1462no—
467C4×C8—1461no—
468C4×C8—1461no—
469C4×C8—1461no—
470C4×C8—1461no—
471C4×C8—1461no—
472#95—1461no—
473#41—21no—
474#41—21no—
475#41Q8×D7312yes—
476#41#1192402712yes—
477C2×Q8—1461no—
478C2×Q8—1461no—
479C2×Q8C2×C4×D7311yescharacteristic
480C2×D4—21no—
481C2×D4C2×C4×D7311yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—731no—
485C22×C4—731no—
486C22×C4—1461no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4C23×D7311yescharacteristic
490M16—24no—
491M16—21no—
492M16—21no—
493M16—21no—
494M16—732no—
495M16—1461no—
496M16—1461no—
497M16—1464no—
498M16—1461no—
499M16#1192402712yes—
500C2×C8—24no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—731no—
508C2×C8—731no—
509C2×C8—732no—
510C2×C8—732no—
511C2×C8—731no—
512C2×C8—731no—
513C2×C8—1461no—
514C2×C8—1464no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8C2×C4×D7311yescharacteristic
519C2×C8C2×C4×D7311yescharacteristic
520C2×C8D4×D7311yescharacteristic
521C2×C8D4×D7311yescharacteristic
522C4⋊C4—2922no—
523C4⋊C4—731no—
524C4⋊C4—731no—
525C4⋊C4—731no—
526C4⋊C4—731no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4C2×C4×D7311yescharacteristic
536C4⋊C4C2×C4×D7311yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—1461no—
539C22⋊C4—1461no—
540C22⋊C4—1461no—
541C22⋊C4—1461no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—731no—
546C42—731no—
547C42—731no—
548C42—731no—
549C42—1461no—
550C42—1461no—
551C42—1461no—
552C42—1461no—
553C42—1461no—
554C42—1461no—
555C23—21no—
556C23#41×D7311yescharacteristic
557Q8—21no—
558Q8—5841no—
559Q8—5841no—
560Q8#1455655612yes—
561D4—22no—
562D4—22no—
563D4—21no—
564D4#1455655612yes—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—5841no—
579C2×C4—5841no—
580C2×C4—731no—
581C2×C4—731no—
582C2×C4—731no—
583C2×C4—731no—
584C2×C4—1461no—
585C2×C4—1461no—
586C2×C4—1461no—
587C2×C4—1461no—
588C2×C4C22×C4×D7311yescharacteristic
589C2×C4C2×D4×D7311yescharacteristic
590C2×C4C2×Q8×D7311yescharacteristic
591C2×C4#41×D7311yescharacteristic
592C2×C4C2×#1192402711yescharacteristic
593C2×C4C2×#1192402711yescharacteristic
594C8—21no—
595C8—21no—
596C8—21no—
597C8—22no—
598C8—21no—
599C8—41no—
600C8—2921no—
601C8—732no—
602C8—732no—
603C8—1461no—
604C8—1461no—
605C8—1462no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22C4×D4×D7311yescharacteristic
612C22#1798911411yescharacteristic
613C22#1798913911yescharacteristic
614C4—22no—
615C4—42no—
616C4—41no—
617C4—41no—
618C4—2921no—
619C4—2921no—
620C4—2921no—
621C4—2921no—
622C4—5841no—
623C4—1461no—
624C4—1461no—
625C4—1461no—
626C4—1461no—
627C4C2×#1455655611yescharacteristic
628C4C4×#1192402711yescharacteristic
629C4C22⋊Q8×D7311yescharacteristic
630C4#1798909311yescharacteristic
631C2—21no—
632C2—21no—
633C2—42no—
634C2#2261246311yescharacteristic
635C2#2261883611yescharacteristic
636C2#2261884511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1866C2
socleC22×C73C2×C4×D4
prime radicalC2×#41×C73C23
cosocleC22×C4×C73C24
hypercenter#1866D73
hypocenterC73#21629
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1866

Complements

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

NH ≅ G/Ntype
2. #22618870C2semidirect
4. #22610873C2semidirect
5. #22610703C2semidirect
6. #22610531C2semidirect
7. #22610527C2semidirect
10. #22609818C2semidirect
11. #22609809C2semidirect
12. #22609802C2semidirect
19. #17988896C4semidirect
20. #17988892C4semidirect
21. #17988889C4semidirect
22. #17988885C4semidirect
23. #17988880C4semidirect
24. #17988875C4semidirect
25. #17988870C4semidirect
26. #17988865C4semidirect
68. #14556598Q8semidirect
334. C73#21629semidirect

Maximal subgroups

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

subgroupindex
#226197152
#226188702
#226183572
#226108732
#226107032
#226105312
#226105272
#226102062
#226099492
#226098182
#226098092
#226098022
#226097782
#226097692
C73×#18662
#2162973