Finite Group Databasebeta

← #11886540

Subgroups — #11886540

Order 1152.

1072 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5718C211yesmaximal, characteristic, complemented
2C2×#63933C212yesmaximal, complemented
3#10593906C211yesmaximal, characteristic, complemented
4#10592774C211yesmaximal, characteristic, complemented
5C3×#5073C211yesmaximal, characteristic, complemented
6#10591519C212yesmaximal, complemented
7#10591498C212yesmaximal, complemented
8#10591450C212yesmaximal, complemented
9C4×#3719C211yesmaximal, characteristic, complemented
10#10590364C211yesmaximal, characteristic, complemented
11C3⋊C8×D12C211yesmaximal, characteristic, complemented
12#83585—31nomaximal
13#81790—31nomaximal
14C2×C3×#237C2212yes—
15C2×#3795C2212yes—
16C32×#127C2211yescharacteristic
17C2×#3754C2212yes—
18C4×#634C2211yescharacteristic
19C3×#940—21no—
20C3×#939—22no—
21C3×#938C2212yes—
22C3×D4×Dic3—21no—
23C2×C3×#213C2212yes—
24C2×C3×#213C2212yes—
25C3×#874C2211yescharacteristic
26C3×C4×D12C2211yescharacteristic
27C3×C4×Dic6C2211yescharacteristic
28#63933—28no—
29#63914—21no—
30#63908C2212yes—
31#63898—21no—
32#63885—22no—
33C2×#3719C2211yescharacteristic
34C2×#3715C2212yes—
35C2×D3×C3⋊C8C2212yes—
36#63659C2212yes—
37C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
38C3×C4×C3⋊C8C2211yescharacteristic
39#63575C2211yescharacteristic
40#63567C2211yescharacteristic
41#63566C2211yescharacteristic
42#63562C2212yes—
43(C2×D3)⋊(C3⋊C8)C2212yes—
44C3×#550—31no—
45C2×#950—32no—
46C2×#912—32no—
47#5718—31no—
48#5718D311yescharacteristic, complemented
49#5553—31no—
50C3×#493—31no—
51#5238—32no—
52#5158—32no—
53#5158—32no—
54#5157—32no—
55#5075—31no—
56#5075—31no—
57#5075—31no—
58#5075—61no—
59#5073D311yescharacteristic, complemented
60#5038—32no—
61#5037—32no—
62C4×#216—31no—
63#5027—31no—
64D4×C3⋊C8—31no—
65C4×Dic12—31no—
66C8×D12—31no—
67C2×C32×D4—21no—
68C2×C32×D4—21no—
69C2×C32×D4C2312yes—
70C2×#634C2311yescharacteristic
71C2×C3×#67—22no—
72C2×C3×#67—22no—
73C22×C3×Dic3—22no—
74C3×#237—24no—
75C3×#237—42no—
76C2×C3×D12—21no—
77C2×C3×D12—21no—
78C2×C3×D12D411yescharacteristic
79C2×C3×D12C2312yes—
80C2×C3×C4×D3C2312yes—
81C2×C3×Dic6C2311yescharacteristic
82C2×#625—22no—
83C2×#625—22no—
84C2×D3×Dic3—22no—
85#3795—24no—
86#3795—42no—
87C32×C42C2311yescharacteristic
88#3754—24no—
89C4⋊(C3⋊Dic3)C2311yescharacteristic
90#3750—21no—
91C4×C3⋊Dic3—21no—
92C2×C3⋊(C3⋊C8)C2312yes—
93C3×#217—22no—
94C3×#213—24no—
95C3×#213—42no—
96C3×(C2×D3)⋊C4D412yes—
97C3×C4⋊Dic3D411yescharacteristic
98C3×C4⋊Dic3C2311yescharacteristic
99C3×Dic3⋊C4—21no—
100C3×C4×Dic3—21no—
101C2×C3×C3⋊C8C2311yescharacteristic
102C2×C3×C3⋊C8C2311yescharacteristic
103(C2×D3)⋊Dic3—22no—
104#3719D414yescomplemented
105#3715—24no—
106D3×C3⋊C8—24no—
107#3287—91noSylow
108C2×C3×#41—32no—
109C2×#237—32no—
110C2×#237C2×D312yes—
111C2×#235—32no—
112C2×C3×D8—32no—
113C2×C3×D8—32no—
114C3×#127—21no—
115C3×#127—31no—
116C3×#127—61no—
117C3×#127C2×D311yescharacteristic
118C3×#127C2×D311yescharacteristic
119C3×#121—32no—
120C3×#121—62no—
121C3×C4×Q8—31no—
122C3×C4×D4—31no—
123C3×C4×D4—61no—
124#950—68no—
125C2×#216—31no—
126C2×#215—32no—
127#940—21no—
128#940—61no—
129#940—61no—
130#939—22no—
131#939—62no—
132#938—32no—
133#938C2×D312yes—
134D4×Dic3—21no—
135D4×Dic3—61no—
136C2×#214—32no—
137C2×#214—32no—
138C2×#214—32no—
139C2×#214—62no—
140C2×#213C2×D312yes—
141C2×#213C2×D312yes—
142#930—32no—
143C22×C3⋊C8—32no—
144#912—68no—
145C2×Dic12—31no—
146C2×D3×C8—32no—
147#893—61no—
148#884—62no—
149#874C2×D311yescharacteristic
150C4×D12C2×D311yescharacteristic
151C4×Dic6—31no—
152C4×Dic6—31no—
153C4×Dic6—61no—
154C4×Dic6C2×D311yescharacteristic
155C3×C4×C8—31no—
156#835—31no—
157#832—32no—
158#832—32no—
159#832—32no—
160#832—62no—
161#830—32no—
162(C2×D3)⋊C8—32no—
163#818—31no—
164#816—31no—
165#816—31no—
166#810—31no—
167#809—32no—
168#807—31no—
169C4⋊(C3⋊C8)—31no—
170C4⋊(C3⋊C8)—31no—
171C4⋊(C3⋊C8)—61no—
172C4×C3⋊C8C2×D311yescharacteristic
173C23×C32—22no—
174C22×C3×D3—22no—
175C22×C3×D3C2×D412yes—
176C32×D4—22no—
177C32×D4—22no—
178C32×D4—42no—
179C32×D4—81no—
180C2×C32×C4C2×D411yescharacteristic
181C2×C32×C4C2×D411yescharacteristic
182C2×C32×C4C2411yescharacteristic
183C2×C3⋊Dic3—21no—
184C2×C3⋊Dic3—21no—
185#634—21no—
186#634C2×D412yes—
187C3×#67—44no—
188C3×#67—82no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—44no—
192C2×C3×Dic3C2×D412yes—
193C3×D12—22no—
194C3×D12—22no—
195C3×D12—22no—
196C3×D12—42no—
197C3×D12—81no—
198C3×C4×D3—22no—
199C3×C4×D3—22no—
200C3×Dic6—21no—
201C3×Dic6C2×D412yes—
202#625—44no—
203#625—82no—
204D3×Dic3—44no—
205C3⋊(C3⋊C8)—22no—
206C3×C3⋊C8—21no—
207C3×C3⋊C8C2×D412yes—
208C2×#135—92no—
209#550—31no—
210#550—91no—
211#493—31no—
212#466—92no—
213#460—91no—
214#460—91no—
215#449—92no—
216#449—92no—
217C4×Q16—91no—
218C8×D4—91no—
219C3×#41—64no—
220C3×#41—122no—
221C2×C3×Q8—31no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—22no—
227C2×C3×D4—31no—
228C2×C3×D4—32no—
229C2×C3×D4—41no—
230C2×C3×D4—41no—
231C2×C3×D4—62no—
232C2×C3×D4—62no—
233C2×C3×D4—61no—
234C2×C3×D4—61no—
235C2×C3×D4C22×D312yes—
236C2×C3×D4C22×D312yes—
237C22×C3×C4—32no—
238C22×C3×C4—62no—
239C2×#67—22no—
240C2×#67—22no—
241C2×#67—62no—
242C2×#67—62no—
243C2×#67—62no—
244C2×#67—62no—
245C22×Dic3—22no—
246C22×Dic3—62no—
247#237—24no—
248#237—42no—
249#237—64no—
250#237—122no—
251#235—64no—
252#235—122no—
253C2×D12—21no—
254C2×D12—21no—
255C2×D12#6711yescharacteristic
256C2×D12C22×D312yes—
257C2×C4×D3—62no—
258C2×C4×D3C22×D312yes—
259C2×Dic6—31no—
260C2×Dic6—31no—
261C2×Dic6—61no—
262C2×Dic6C22×D311yescharacteristic
263C3×D8—64no—
264C3×D8—122no—
265C2×C3×C8—31no—
266C2×C3×C8—31no—
267C3×C4⋊C4—31no—
268C3×C4⋊C4—31no—
269C3×C4⋊C4—61no—
270C3×C22⋊C4—32no—
271C3×C22⋊C4—62no—
272C3×C42—21no—
273C3×C42—61no—
274C3×C42C22×D311yescharacteristic
275C3×C42C22×D311yescharacteristic
276#217—22no—
277#217—62no—
278#216—34no—
279#215—64no—
280#214—64no—
281#214—64no—
282#214—64no—
283#214—124no—
284#213—24no—
285#213—42no—
286(C2×D3)⋊C4—62no—
287(C2×D3)⋊C4#6712yes—
288C4⋊Dic3—31no—
289C4⋊Dic3—31no—
290C4⋊Dic3—61no—
291C4⋊Dic3#6711yescharacteristic
292C4⋊Dic3C22×D311yescharacteristic
293Dic3⋊C4—21no—
294Dic3⋊C4—61no—
295Dic3⋊C4—61no—
296Dic3⋊C4—121no—
297C4×Dic3—21no—
298C4×Dic3—61no—
299C4×Dic3—61no—
300C4×Dic3—121no—
301C2×C3⋊C8—32no—
302C2×C3⋊C8—32no—
303C2×C3⋊C8—62no—
304C2×C3⋊C8—64no—
305C2×C3⋊C8C22×D311yescharacteristic
306C2×C3⋊C8C22×D311yescharacteristic
307Dic12—34no—
308D3×C8—64no—
309C22×C32—42no—
310C22×C32—42no—
311C22×C32—42no—
312C22×C32C22×D411yescharacteristic
313C2×C3×D3—22no—
314C2×C3×D3—22no—
315C2×C3×D3—22no—
316C2×C3×D3—42no—
317C2×C3×D3—42no—
318C2×C3×D3—42no—
319C32×C4—21no—
320C32×C4#13512yescomplemented
321C32×C4C22×D411yescharacteristic
322C32×C4H(5,𝔽2)11yescharacteristic
323C3⋊Dic3—22no—
324C3⋊Dic3—41no—
325C3×Dic3—22no—
326C3×Dic3—22no—
327C3×Dic3—41no—
328C2×#41—32no—
329C2×#41—92no—
330#135—188no—
331C2×Q16—91no—
332C2×SD16—92no—
333C2×SD16—92no—
334C2×D8—32no—
335C2×D8—32no—
336C22×C8—92no—
337#127—31no—
338#127—61no—
339#127—181no—
340#127D3211yescharacteristic
341#121—32no—
342#121—62no—
343#121—92no—
344#121—182no—
345C4×Q8—31no—
346C4×Q8—91no—
347C4×D4—31no—
348C4×D4—61no—
349C4×D4—181no—
350#107—91no—
351C4⋊C8—91no—
352#103—91no—
353#103—91no—
354#102—92no—
355#102—92no—
356C22⋊C8—92no—
357C4×C8—31no—
358C23×C3—22no—
359C23×C3—22no—
360C23×C3—32no—
361C23×C3—42no—
362C23×C3—62no—
363C22×D3—22no—
364C22×D3C2×#6712yes—
365C3×Q8—32no—
366C3×Q8—61no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—22no—
371C3×D4—42no—
372C3×D4—42no—
373C3×D4—42no—
374C3×D4—42no—
375C3×D4—62no—
376C3×D4—62no—
377C3×D4—62no—
378C3×D4—82no—
379C3×D4—81no—
380C3×D4—81no—
381C3×D4—82no—
382C3×D4—124no—
383C3×D4—122no—
384C3×D4—241no—
385C3×D4—242no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—32no—
390C2×C3×C4—61no—
391C2×C3×C4—62no—
392C2×C3×C4—62no—
393C2×C3×C4—61no—
394C2×C3×C4—124no—
395C2×C3×C4D3×D411yescharacteristic
396C2×C3×C4D3×D411yescharacteristic
397C2×C3×C4C2×#6711yescharacteristic
398C2×C3×C4C2×#6711yescharacteristic
399C2×C3×C4C23×D311yescharacteristic
400C2×C3×C4C23×D311yescharacteristic
401#67—44no—
402#67—82no—
403#67—124no—
404#67—124no—
405#67—242no—
406#67—242no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—44no—
410C2×Dic3—61no—
411C2×Dic3—61no—
412C2×Dic3—61no—
413C2×Dic3—61no—
414C2×Dic3—121no—
415C2×Dic3—121no—
416C2×Dic3—124no—
417C2×Dic3C2×#6712yes—
418D12—22no—
419D12—22no—
420D12—22no—
421D12—42no—
422D12—81no—
423C4×D3—22no—
424C4×D3—22no—
425C4×D3—124no—
426Dic6—21no—
427Dic6—32no—
428Dic6—32no—
429Dic6—62no—
430Dic6—61no—
431Dic6—61no—
432Dic6—121no—
433Dic6D3×D412yes—
434C3×C8—32no—
435C3×C8—61no—
436C3⋊C8—21no—
437C3⋊C8—62no—
438C3⋊C8—62no—
439C3⋊C8—122no—
440C3⋊C8D3×D412yes—
441C2×C32—82no—
442C2×C32D4211yescharacteristic
443C2×C32C2×#13511yescharacteristic
444C2×C32#57611yescharacteristic
445C3×D3—42no—
446C3×D3—82no—
447#41—64no—
448#41—122no—
449#41—184no—
450#41—362no—
451C2×Q8—31no—
452C2×Q8—91no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—32no—
456C2×D4—31no—
457C2×D4—62no—
458C2×D4—62no—
459C2×D4—61no—
460C2×D4—61no—
461C2×D4—182no—
462C2×D4—182no—
463C2×D4C2×D3212yes—
464C22×C4—32no—
465C22×C4—62no—
466C22×C4—182no—
467Q16—94no—
468SD16—184no—
469SD16—184no—
470D8—64no—
471D8—122no—
472C2×C8—31no—
473C2×C8—31no—
474C2×C8—92no—
475C2×C8—184no—
476C4⋊C4—31no—
477C4⋊C4—31no—
478C4⋊C4—61no—
479C4⋊C4—91no—
480C4⋊C4—181no—
481C22⋊C4—32no—
482C22⋊C4—62no—
483C22⋊C4—182no—
484C42—61no—
485C42—181no—
486C42C2×D3211yescharacteristic
487C22×C3—21no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—62no—
495C22×C3—62no—
496C22×C3—62no—
497C22×C3—82no—
498C22×C3—82no—
499C22×C3—82no—
500C22×C3—122no—
501C22×C3—122no—
502C22×C3—122no—
503C22×C3C2×D3×D411yescharacteristic
504C22×C3C22×#6711yescharacteristic
505C2×D3—22no—
506C2×D3—22no—
507C2×D3—22no—
508C2×D3—42no—
509C2×D3—42no—
510C2×D3—42no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—22no—
516C3×C4—41no—
517C3×C4—62no—
518C3×C4—62no—
519C3×C4—121no—
520C3×C4#91212yescomplemented
521C3×C4#95012yescomplemented
522C3×C4C2×D3×D411yescharacteristic
523C3×C4C2×D3×D411yescharacteristic
524C3×C4#100411yescharacteristic
525C3×C4#100411yescharacteristic
526Dic3—22no—
527Dic3—22no—
528Dic3—41no—
529Dic3—62no—
530Dic3—62no—
531Dic3—122no—
532Dic3—121no—
533Dic3—121no—
534Dic3—241no—
535C32#328711yescharacteristic, complemented, Sylow
536C23—22no—
537C23—32no—
538C23—62no—
539Q8—32no—
540Q8—61no—
541Q8—92no—
542Q8—181no—
543D4—22no—
544D4—22no—
545D4—42no—
546D4—62no—
547D4—62no—
548D4—62no—
549D4—81no—
550D4—124no—
551D4—122no—
552D4—241no—
553D4—242no—
554D4—364no—
555D4—722no—
556C2×C4—32no—
557C2×C4—61no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—61no—
561C2×C4—124no—
562C2×C4—181no—
563C2×C4—181no—
564C2×C4—364no—
565C2×C4D3×#6711yescharacteristic
566C2×C4D3×#6711yescharacteristic
567C2×C4C22×D3211yescharacteristic
568C8—32no—
569C8—61no—
570C8—182no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—82no—
575C2×C3—82no—
576C2×C3—122no—
577C2×C3—162no—
578C2×C3—242no—
579C2×C3#571711yescharacteristic
580C2×C3C2×#91211yescharacteristic
581C2×C3#575711yescharacteristic
582C2×C3D4×#6711yescharacteristic
583C2×C3C2×#95011yescharacteristic
584C2×C3#583511yescharacteristic
585D3—42no—
586D3—82no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—62no—
591C22—62no—
592C22—62no—
593C22—122no—
594C22—122no—
595C22—122no—
596C22C2×D3×#6711yescharacteristic
597C4—21no—
598C4—62no—
599C4—62no—
600C4—121no—
601C4—182no—
602C4—361no—
603C4#6393312yescomplemented
604C4D32×D411yescharacteristic
605C4#6431311yescharacteristic
606C3—21no—
607C3#8179011yescharacteristic, complemented
608C3#8358511yescharacteristic, complemented
609C2—82no—
610C2—122no—
611C2—242no—
612C2#1059509611yescharacteristic
613C2C2×#6393311yescharacteristic
614C2#1059529311yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×#127C22
socleC22×C32C22×D4
hypercenter#127D32
hypocenterC32#3287
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C3×#5718C2semidirect
2. C2×#63933C2semidirect
3. #10593906C2semidirect
4. #10592774C2semidirect
5. C3×#5073C2semidirect
6. #10591519C2semidirect
7. #10591498C2semidirect
8. #10591450C2semidirect
9. C4×#3719C2semidirect
10. #10590364C2semidirect
11. C3⋊C8×D12C2semidirect
48. #5718D3semidirect
59. #5073D3semidirect
104. #3719D4semidirect
320. C32×C4#135semidirect
520. C3×C4#912semidirect
521. C3×C4#950semidirect
535. C32#3287semidirect
603. C4#63933semidirect
607. C3#81790semidirect
608. C3#83585semidirect

Maximal subgroups

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

subgroupindex
C3×#57182
C2×#639332
#105939062
#105927742
C3×#50732
#105915192
#105914982
#105914502
C4×#37192
#105903642
C3⋊C8×D122
#835853
#817903