Finite Group Databasebeta

← #11887051

Subgroups — #11887051

Order 1152.

1047 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5726C211yesmaximal, characteristic, complemented
2D122C211yesmaximal, characteristic, complemented
3#10592787C211yesmaximal, characteristic, complemented
4C3×#5087C211yesmaximal, characteristic, complemented
5#10590595C212yesmaximal, complemented
6D3×#809C212yesmaximal, complemented
7#10590576C212yesmaximal, complemented
8#10590422C212yesmaximal, complemented
9#10590349C211yesmaximal, characteristic, complemented
10C4×#3713C211yesmaximal, characteristic, complemented
11#10590329C211yesmaximal, characteristic, complemented
12#83525—31nomaximal
13#80980—31nomaximal
14C2×C3×#239—21no—
15C2×D3×D12—21no—
16C2×D3×D12C2212yes—
17C32×C4⋊Q8C2211yescharacteristic, complemented
18#64083C2211yescharacteristic
19C3×#946—21no—
20C3×#895C2212yescomplemented
21C3×#894—22no—
22C3×D3×C4⋊C4C2212yescomplemented
23C3×C4×D12C2211yescharacteristic
24C3×C4×D12C2211yescharacteristic
25C3×C4×D12C2211yescharacteristic, complemented
26#63923—22no—
27#63909—21no—
28#63909C2212yes—
29C2×#3714C2212yescomplemented
30C2×#3713C2211yescharacteristic
31C2×D3×C3⋊C8C2212yescomplemented
32#63636C2212yescomplemented
33#63636C2212yescomplemented
34C4×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
35C3×#809C2212yes—
36C3×#807C2212yescomplemented
37C3×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
38#63576C2211yescharacteristic, complemented
39#63564C2212yescomplemented
40#63562C2211yescharacteristic
41#63562C2211yescharacteristic, complemented
42(C2×D3)⋊(C3⋊C8)C2212yescomplemented
43C3×#547—31no—
44#5726D311yescharacteristic, complemented
45D4×D12—31no—
46D4×D12—31no—
47C3×#474—31no—
48#5088—31no—
49#5088—31no—
50#5088—61no—
51#5087D311yescharacteristic, complemented
52#5038—32no—
53#5036—32no—
54#5015—31no—
55C4×#213—31no—
56C4×#213—31no—
57D4×C3⋊C8—31no—
58#4967—32no—
59C2×#809—32no—
60#4883—32no—
61#4882—32no—
62#4843—31no—
63#4834—31no—
64#4773—32no—
65D3×#102—32no—
66C22×D32—22no—
67C2×C32×Q8—21no—
68C2×C4⋊(C3⋊D3)—21no—
69C2×C4⋊(C3⋊D3)—21no—
70C2×C4⋊(C3⋊D3)C2312yes—
71C3×#239—44no—
72C2×C3×D12—22no—
73C2×C3×D12D411yescharacteristic
74C2×C3×D12C2311yescharacteristic
75C2×C3×D12C2311yescharacteristic
76C2×C3×C4×D3—22no—
77C2×C3×C4×D3—21no—
78C2×C3×C4×D3—21no—
79C2×C3×C4×D3C2312yes—
80C2×#626—22no—
81C2×#626—22no—
82D3×D12—24no—
83D3×D12—42no—
84D3×D12—44no—
85C32×C4⋊C4—21no—
86C32×C4⋊C4C2312yes—
87C32×C42C2311yescharacteristic
88C2×C3⋊(C3⋊C8)C2311yescharacteristic
89C2×C3⋊(C3⋊C8)C2311yescharacteristic
90C3×(C2×D3)⋊C4—21no—
91C3×(C2×D3)⋊C4—21no—
92C3×(C2×D3)⋊C4—22no—
93C3×(C2×D3)⋊C4D412yes—
94C3×C4⋊Dic3D411yescharacteristic
95C3×C4⋊Dic3C2311yescharacteristic
96C3×C4⋊Dic3C2311yescharacteristic
97C3×Dic3⋊C4—22no—
98C2×C3×C3⋊C8C2312yes—
99#3722—22no—
100#3714—42no—
101#3713—22no—
102D3×C3⋊C8—42no—
103#3294—91noSylow
104C2×C3×#41—61no—
105C2×#239—21no—
106C2×D3×D4—32no—
107C2×D3×D4—61no—
108C22×D12—32no—
109C22×D12—61no—
110C3×C4⋊Q8—21no—
111C3×C4⋊Q8C2×D311yescharacteristic, complemented
112C3×C4⋊Q8C2×D311yescharacteristic, complemented
113C3×#123—62no—
114C3×C22⋊Q8—61no—
115C3×#121—32no—
116C3×C4×D4—31no—
117C3×C4×D4—31no—
118C3×C4×D4—31no—
119C2×C3×C4⋊C4—32no—
120#946—21no—
121C2×#213—31no—
122C2×#213—31no—
123C2×#213—32no—
124#930—32no—
125#930—61no—
126C22×C3⋊C8—32no—
127C2×D24—32no—
128C2×D3×C8—32no—
129#895—32no—
130#895—61no—
131#895C2×D312yescomplemented
132#894—22no—
133D3×C4⋊C4C2×D312yescomplemented
134#882—62no—
135#882—62no—
136#874—31no—
137#874—31no—
138#874—61no—
139C4×D12C2×D311yescharacteristic
140C4×D12C2×D311yescharacteristic
141C4×D12C2×D311yescharacteristic, complemented
142C3×#107—32no—
143C3×C4⋊C8—31no—
144C3×#102—32no—
145#832—31no—
146#832—31no—
147#830—32no—
148#821—32no—
149(C2×D3)⋊C8—32no—
150#809—31no—
151#809—31no—
152#809—32no—
153#809—32no—
154#809—32no—
155#809—32no—
156#809—32no—
157#809—62no—
158#809—62no—
159#809C2×D312yes—
160#807—31no—
161#807—31no—
162#807C2×D312yescomplemented
163C4⋊(C3⋊C8)C2×D311yescharacteristic, complemented
164C4×C3⋊C8—31no—
165C4×C3⋊C8—31no—
166C4×C3⋊C8—61no—
167C22×C3⋊D3—22no—
168C22×C3×D3—21no—
169C22×C3×D3—21no—
170C22×C3×D3C2×D412yes—
171C2×D32—44no—
172C2×D32—44no—
173C2×D32—82no—
174C32×Q8—42no—
175C2×C32×C4—22no—
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4C2×D411yescharacteristic
178C2×C32×C4C2411yescharacteristic
179C4⋊(C3⋊D3)—22no—
180C4⋊(C3⋊D3)—22no—
181C4⋊(C3⋊D3)—42no—
182C4⋊(C3⋊D3)—81no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3C2×D412yes—
186C3×D12—21no—
187C3×D12—21no—
188C3×D12—21no—
189C3×D12—44no—
190C3×D12D814yescomplemented
191C3×D12#4112yescomplemented
192C3×C4×D3—22no—
193C3×C4×D3—22no—
194C3×C4×D3—44no—
195C3×C4×D3—42no—
196C3×C4×D3—42no—
197#626—44no—
198#626—44no—
199C3⋊(C3⋊C8)—21no—
200C3⋊(C3⋊C8)—21no—
201C3×C3⋊C8—22no—
202#547—31no—
203D42—91no—
204#486—91no—
205#480—92no—
206#474—31no—
207#466—92no—
208#459—91no—
209#447—92no—
210C4×D8—91no—
211C8×D4—91no—
212C2×#102—92no—
213C23×D3—62no—
214C23×D3—62no—
215C3×#41—124no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×Q8—41no—
219C2×C3×D4—31no—
220C2×C3×D4—31no—
221C2×C3×D4—31no—
222C2×C3×D4—62no—
223C22×C3×C4—32no—
224C22×C3×C4—62no—
225C22×C3×C4—61no—
226C22×C3×C4—61no—
227C2×#67—62no—
228C2×#67—62no—
229#239—44no—
230D3×D4—64no—
231D3×D4—122no—
232D3×D4—124no—
233C2×D12—22no—
234C2×D12—32no—
235C2×D12—32no—
236C2×D12—62no—
237C2×D12—64no—
238C2×D12—62no—
239C2×D12—62no—
240C2×D12—61no—
241C2×D12—61no—
242C2×D12—61no—
243C2×D12—61no—
244C2×D12—121no—
245C2×D12—121no—
246C2×D12—122no—
247C2×D12—124no—
248C2×D12#6711yescharacteristic
249C2×D12C22×D311yescharacteristic
250C2×D12C22×D311yescharacteristic
251C2×C4×D3—22no—
252C2×C4×D3—21no—
253C2×C4×D3—21no—
254C2×C4×D3C22×D312yes—
255C2×C3×C8—32no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—22no—
259C3×C4⋊C4—31no—
260C3×C4⋊C4—31no—
261C3×C4⋊C4—31no—
262C3×C4⋊C4—41no—
263C3×C4⋊C4—62no—
264C3×C4⋊C4C22×D312yes—
265C3×C4⋊C4C22×D312yes—
266C3×C22⋊C4—32no—
267C3×C22⋊C4—62no—
268C3×C22⋊C4—61no—
269C3×C22⋊C4—61no—
270C3×C42—21no—
271C3×C42C22×D311yescharacteristic
272C3×C42C22×D311yescharacteristic
273#213—62no—
274#213—62no—
275#213—122no—
276(C2×D3)⋊C4—21no—
277(C2×D3)⋊C4—21no—
278(C2×D3)⋊C4—22no—
279(C2×D3)⋊C4—62no—
280(C2×D3)⋊C4—62no—
281(C2×D3)⋊C4#6712yes—
282C4⋊Dic3#6711yescharacteristic
283C4⋊Dic3C22×D311yescharacteristic
284C4⋊Dic3C22×D311yescharacteristic
285Dic3⋊C4—22no—
286C2×C3⋊C8—31no—
287C2×C3⋊C8—31no—
288C2×C3⋊C8—31no—
289C2×C3⋊C8—31no—
290C2×C3⋊C8—61no—
291C2×C3⋊C8—61no—
292C2×C3⋊C8—122no—
293C2×C3⋊C8C22×D312yes—
294D24—122no—
295D3×C8—122no—
296C22×C32C22×D411yescharacteristic
297C2×C3⋊D3—42no—
298C2×C3⋊D3—42no—
299C2×C3⋊D3—42no—
300C2×C3×D3—22no—
301C2×C3×D3—22no—
302C2×C3×D3—22no—
303C2×C3×D3—41no—
304C2×C3×D3—41no—
305C2×C3×D3—41no—
306C2×C3×D3—41no—
307C2×C3×D3—41no—
308C2×C3×D3C2×D814yes—
309D32—88no—
310C32×C4—21no—
311C32×C4—42no—
312C32×C4C2×D812yes—
313C32×C4C2×#4111yescharacteristic
314C32×C4H(5,𝔽2)11yescharacteristic
315C3×Dic3—22no—
316C3×Dic3—41no—
317C3×Dic3—41no—
318C2×#41—61no—
319C22×D4—92no—
320C22×D4—181no—
321C2×D8—92no—
322C2×D8—91no—
323C22×C8—92no—
324C4⋊Q8D3211yescharacteristic, complemented
325#127—91no—
326#123—62no—
327C22⋊Q8—61no—
328#121—32no—
329#121—92no—
330#121—181no—
331#120—182no—
332C4×D4—31no—
333C4×D4—31no—
334C4×D4—31no—
335C2×C4⋊C4—32no—
336#107—32no—
337#107—91no—
338C4⋊C8—31no—
339#102—32no—
340#102—92no—
341#102—91no—
342#102—91no—
343#102—92no—
344#102—92no—
345C22⋊C8—92no—
346C4×C8—91no—
347C23×C3—32no—
348C23×C3—61no—
349C23×C3—61no—
350C22×D3—21no—
351C22×D3—21no—
352C22×D3—62no—
353C22×D3—62no—
354C22×D3—122no—
355C22×D3—124no—
356C22×D3—124no—
357C22×D3—124no—
358C22×D3—124no—
359C22×D3—242no—
360C22×D3—242no—
361C22×D3C2×#6712yes—
362C3×Q8—42no—
363C3×Q8—42no—
364C3×Q8—82no—
365C3×D4—34no—
366C3×D4—32no—
367C3×D4—61no—
368C3×D4—61no—
369C3×D4—61no—
370C3×D4—124no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—32no—
377C2×C3×C4—42no—
378C2×C3×C4—62no—
379C2×C3×C4—62no—
380C2×C3×C4—61no—
381C2×C3×C4—61no—
382C2×C3×C4—124no—
383C2×C3×C4—122no—
384C2×C3×C4—122no—
385C2×C3×C4D3×D411yescharacteristic
386C2×C3×C4D3×D411yescharacteristic
387C2×C3×C4C2×#6711yescharacteristic
388C2×C3×C4C2×#6711yescharacteristic
389C2×C3×C4C23×D311yescharacteristic
390C2×C3×C4C23×D311yescharacteristic
391#67—124no—
392#67—124no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3C2×#6712yes—
396D12—21no—
397D12—21no—
398D12—21no—
399D12—44no—
400D12—62no—
401D12—62no—
402D12—62no—
403D12—62no—
404D12—122no—
405D12—122no—
406D12—124no—
407D12—124no—
408D12—122no—
409D12—122no—
410D12—242no—
411D12—241no—
412D12—241no—
413D12—242no—
414D12#21314yescomplemented
415D12#23912yescomplemented
416C4×D3—22no—
417C4×D3—22no—
418C4×D3—44no—
419C4×D3—42no—
420C4×D3—42no—
421C3×C8—62no—
422C3⋊C8—22no—
423C3⋊C8—61no—
424C3⋊C8—61no—
425C3⋊C8—61no—
426C3⋊C8—61no—
427C3⋊C8—121no—
428C3⋊C8—121no—
429C2×C32#54611yescharacteristic
430C2×C32C22×D811yescharacteristic
431C2×C32#57711yescharacteristic
432C3⋊D3—82no—
433C3×D3—24no—
434C3×D3—42no—
435C3×D3—81no—
436C24—182no—
437#41—124no—
438C2×Q8—21no—
439C2×D4—31no—
440C2×D4—31no—
441C2×D4—31no—
442C2×D4—62no—
443C2×D4—92no—
444C2×D4—182no—
445C2×D4—184no—
446C2×D4—182no—
447C2×D4—181no—
448C2×D4—181no—
449C2×D4—362no—
450C2×D4—364no—
451C22×C4—32no—
452C22×C4—62no—
453C22×C4—61no—
454C22×C4—61no—
455D8—182no—
456D8—362no—
457C2×C8—32no—
458C2×C8—91no—
459C2×C8—91no—
460C2×C8—362no—
461C4⋊C4—21no—
462C4⋊C4—31no—
463C4⋊C4—31no—
464C4⋊C4—31no—
465C4⋊C4—62no—
466C4⋊C4C2×D3212yes—
467C22⋊C4—32no—
468C22⋊C4—62no—
469C22⋊C4—61no—
470C22⋊C4—61no—
471C22⋊C4—182no—
472C42C2×D3211yescharacteristic
473C22×C3—21no—
474C22×C3—34no—
475C22×C3—62no—
476C22×C3—62no—
477C22×C3—62no—
478C22×C3—121no—
479C22×C3—121no—
480C22×C3—121no—
481C22×C3—121no—
482C22×C3—121no—
483C22×C3C2×D3×D411yescharacteristic
484C22×C3C22×#6711yescharacteristic
485C2×D3—22no—
486C2×D3—22no—
487C2×D3—22no—
488C2×D3—41no—
489C2×D3—41no—
490C2×D3—41no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—122no—
494C2×D3—122no—
495C2×D3—122no—
496C2×D3—122no—
497C2×D3—122no—
498C2×D3—122no—
499C2×D3—242no—
500C2×D3—242no—
501C2×D3—242no—
502C2×D3—248no—
503C2×D3—248no—
504C2×D3C2×#21314yes—
505C3×C4—22no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—21no—
510C3×C4—22no—
511C3×C4—42no—
512C3×C4—42no—
513C3×C4—62no—
514C3×C4—82no—
515C3×C4—121no—
516C3×C4—121no—
517C3×C4D3×D812yes—
518C3×C4C2×#21312yes—
519C3×C4C2×#23911yescharacteristic
520C3×C4D3×#4111yescharacteristic
521C3×C4#100911yescharacteristic
522C3×C4#100911yescharacteristic
523Dic3—22no—
524Dic3—41no—
525Dic3—41no—
526C32#329411yescharacteristic, complemented, Sylow
527C23—32no—
528C23—61no—
529C23—61no—
530C23—182no—
531C23—364no—
532C23—364no—
533C23—722no—
534Q8—42no—
535D4—32no—
536D4—34no—
537D4—61no—
538D4—61no—
539D4—61no—
540D4—124no—
541D4—182no—
542D4—182no—
543D4—364no—
544D4—364no—
545D4—362no—
546D4—721no—
547C2×C4—22no—
548C2×C4—32no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—124no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4D3×#6711yescharacteristic
557C2×C4D3×#6711yescharacteristic
558C2×C4C22×D3211yescharacteristic
559C8—62no—
560C8—181no—
561C8—181no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—64no—
566C2×C3—122no—
567C2×C3—241no—
568C2×C3#563011yescharacteristic
569C2×C3C2×D3×D811yescharacteristic
570C2×C3#577811yescharacteristic
571C2×C3C22×#21311yescharacteristic
572C2×C3#582911yescharacteristic
573C2×C3#583711yescharacteristic
574D3—24no—
575D3—42no—
576D3—81no—
577D3—242no—
578D3—242no—
579D3—482no—
580C22—34no—
581C22—62no—
582C22—62no—
583C22—62no—
584C22—121no—
585C22—121no—
586C22—121no—
587C22—121no—
588C22—121no—
589C22—362no—
590C22—362no—
591C22—362no—
592C22—728no—
593C22C2×D3×#6711yescharacteristic
594C4—21no—
595C4—42no—
596C4—62no—
597C4—121no—
598C4—121no—
599C4D3×#21312yes—
600C4D3×#23911yescharacteristic
601C4#6432011yescharacteristic
602C3—21no—
603C3#8098011yescharacteristic, complemented
604C3#8352511yescharacteristic, complemented
605C2—64no—
606C2—122no—
607C2—241no—
608C2—722no—
609C2C2×D3×#21311yescharacteristic
610C2#1059531911yescharacteristic
611C2#1059542711yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
hypercenterC4⋊Q8D32
hypocenterC32#3294
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#5726C2semidirect
2. D122C2semidirect
3. #10592787C2semidirect
4. C3×#5087C2semidirect
5. #10590595C2semidirect
6. D3×#809C2semidirect
7. #10590576C2semidirect
8. #10590422C2semidirect
9. #10590349C2semidirect
10. C4×#3713C2semidirect
11. #10590329C2semidirect
17. C32×C4⋊Q8C22semidirect
20. C3×#895C22semidirect
22. C3×D3×C4⋊C4C22semidirect
25. C3×C4×D12C22semidirect
29. C2×#3714C22semidirect
31. C2×D3×C3⋊C8C22semidirect
32. #63636C22semidirect
33. #63636C22semidirect
34. C4×C3⋊(C3⋊C8)C22semidirect
36. C3×#807C22semidirect
37. C3×C4⋊(C3⋊C8)C22semidirect
38. #63576C22semidirect
39. #63564C22semidirect
41. #63562C22semidirect
42. (C2×D3)⋊(C3⋊C8)C22semidirect
44. #5726D3semidirect
51. #5087D3semidirect
111. C3×C4⋊Q8C2×D3semidirect
112. C3×C4⋊Q8C2×D3semidirect
131. #895C2×D3semidirect
133. D3×C4⋊C4C2×D3semidirect
141. C4×D12C2×D3semidirect
162. #807C2×D3semidirect
163. C4⋊(C3⋊C8)C2×D3semidirect
190. C3×D12D8semidirect
191. C3×D12#41semidirect
324. C4⋊Q8D32semidirect
414. D12#213semidirect
415. D12#239semidirect
526. C32#3294semidirect
603. C3#80980semidirect
604. C3#83525semidirect

Maximal subgroups

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

subgroupindex
C3×#57262
D1222
#105927872
C3×#50872
#105905952
D3×#8092
#105905762
#105904222
#105903492
C4×#37132
#105903292
#835253
#809803