Finite Group Databasebeta

← #11069477

Subgroups — #11069477

Order 768.

1954 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5583C212yesmaximal, complemented
2C3×#2351C211yesmaximal, characteristic, complemented
3#82499C211yesmaximal, characteristic
4#82494C212yesmaximal, complemented
5C4×#944C211yesmaximal, characteristic, complemented
6#76251C212yesmaximal, complemented
7#76216C212yesmaximal, complemented
8#76130C214yesmaximal
9#76126C214yesmaximal, complemented
10#74240C211yesmaximal, characteristic
11#74220C211yesmaximal, characteristic, complemented
12#74175C214yesmaximal, complemented
13#73699C214yesmaximal, complemented
14#73637C211yesmaximal, characteristic, complemented
15#73617C211yesmaximal, characteristic
16#39373—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2212yes—
19C2×#946C2212yes—
20C2×#944C2211yescharacteristic
21#5583—216no—
22C2×#898C2214yes—
23C2×#897C2212yes—
24C2×#890C2212yes—
25C2×#889C2214yes—
26C2×#876C2214yes—
27C2×C4×D12C2211yescharacteristic
28C2×#872C2212yes—
29C2×#870C2214yes—
30C2×C4×Dic6C2211yescharacteristic
31C2×C4×Dic6C2212yes—
32C3×#389C2214yes—
33C3×#389C2212yes—
34C3×#382C2211yescharacteristic
35C3×#382C2212yes—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C4⋊C4C2212yes—
38#5229C2211yescharacteristic
39#4994C2214yes—
40#4993C2214yes—
41#4993C2214yes—
42#4991C2212yes—
43#4988C2212yes—
44#4987C2212yes—
45#4987C2212yes—
46#4983C2214yes—
47#4983C2214yes—
48#4982C2214yes—
49#4981C2212yes—
50#4976C2214yes—
51#4976C2214yes—
52#4975C2211yescharacteristic
53#4975C2212yes—
54#4973C2211yescharacteristic
55#4972C2212yes—
56#4940C2214yes—
57#4940C2214yes—
58C4×(C2×D3)⋊C4C2212yes—
59#4936C2211yescharacteristic
60#4933C2214yes—
61C4×Dic3⋊C4C2214yes—
62C42×Dic3C2211yescharacteristic
63#4928C2212yes—
64#4676C2214yes—
65#4672C2212yes—
66#4670C2214yes—
67#4669C2214yes—
68#4669C2212yes—
69#4669C2212yes—
70#4661C2212yes—
71#4657C2214yes—
72#4655C2214yes—
73#4650C2212yes—
74#4649C2212yes—
75#4647C2214yes—
76#4647C2214yes—
77C2×#541—32no—
78#2859—31no—
79#2727—34no—
80#2726—34no—
81#2629—32no—
82#2536—31no—
83#2536—31no—
84#2519—34no—
85#2519—34no—
86#2423—32no—
87#2423—32no—
88#2351—31no—
89#2351D311yescharacteristic, complemented
90C4×C4⋊Q8—31no—
91#2293—31no—
92C22×C3×Q8C2311yescharacteristic
93C22×D12C2311yescharacteristic
94C22×C4×D3C2312yes—
95C22×Dic6C2311yescharacteristic
96C3×C4×Q8—28no—
97C3×C4×Q8—24no—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2314yes—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C42C2312yes—
105C2×C3×C42C2312yes—
106C2×C3×C42C2314yes—
107C2×C3×C42C2311yescharacteristic
108#946—28no—
109#944D418yescomplemented
110C2×(C2×D3)⋊C4C2312yes—
111C2×(C2×D3)⋊C4C2312yes—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2312yes—
114C2×(C2×D3)⋊C4C2312yes—
115C2×(C2×D3)⋊C4C2312yes—
116C2×(C2×D3)⋊C4C2314yes—
117C2×(C2×D3)⋊C4C2314yes—
118C2×(C2×D3)⋊C4C2314yes—
119C2×C4⋊Dic3C2314yes—
120C2×C4⋊Dic3C2314yes—
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4⋊Dic3C2311yescharacteristic
123C2×C4⋊Dic3C2312yes—
124C2×Dic3⋊C4C2314yes—
125C2×Dic3⋊C4C2314yes—
126C2×Dic3⋊C4C2314yes—
127C2×Dic3⋊C4C2312yes—
128C2×Dic3⋊C4C2312yes—
129C2×Dic3⋊C4C2312yes—
130C2×Dic3⋊C4C2314yes—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2314yes—
134C2×Dic3⋊C4C2314yes—
135C2×C4×Dic3C2312yes—
136C2×C4×Dic3C2312yes—
137C2×C4×Dic3C2314yes—
138C2×C4×Dic3C2311yescharacteristic
139C2×C4×Dic3C2312yes—
140C2×C4×Dic3C2311yescharacteristic
141#898—216no—
142#897—28no—
143#890—28no—
144#889—216no—
145#876—216no—
146C4×D12—24no—
147#872—28no—
148#870—216no—
149C4×Dic6—28no—
150C4×Dic6—24no—
151C3×#95C2312yes—
152C3×#95C2312yes—
153C3×#95C2311yescharacteristic
154C3×#95C2311yescharacteristic
155C3×#95C2314yes—
156C3⋊#95C2312yes—
157C3⋊#95C2312yes—
158C3⋊#95C2314yes—
159C3⋊#95C2312yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162C3⋊#95C2314yes—
163C3⋊#95C2312yes—
164C3⋊#95C2314yes—
165#541—616no—
166C2×C4⋊Q8—31no—
167C2×#126—34no—
168C2×#126—34no—
169C2×#125—32no—
170C2×#125—34no—
171C2×#125—34no—
172C2×C22⋊Q8—32no—
173C2×C22⋊Q8—32no—
174C2×C4×Q8—31no—
175C2×C4×Q8—32no—
176C2×C4×Q8C2×D311yescharacteristic
177C2×C4×Q8C2×D312yes—
178C2×C4×D4—31no—
179C2×#117—32no—
180#399—34no—
181#399—34no—
182#398—34no—
183#398—32no—
184#398—34no—
185#398—34no—
186#398—34no—
187#398—32no—
188#396—34no—
189#396—34no—
190#394—32no—
191#394—32no—
192#391—31no—
193#391—31no—
194#390—32no—
195#389—34no—
196#389—34no—
197#389—32no—
198#389—32no—
199#389—34no—
200#389—34no—
201#389C2×D312yes—
202#389C2×D314yes—
203#388—34no—
204#388—34no—
205#388—34no—
206#388—32no—
207#388—32no—
208#388—34no—
209#387—34no—
210#387—34no—
211#384—31no—
212#384—32no—
213#382—31no—
214#382—32no—
215#382C2×D311yescharacteristic
216#382C2×D312yes—
217#380—32no—
218#380—32no—
219C4×C4⋊C4—34no—
220C4×C4⋊C4—32no—
221C4×C4⋊C4C2×D311yescharacteristic
222C4×C4⋊C4C2×D312yes—
223C4×C22⋊C4—32no—
224C43—31no—
225C23×D3C2412yes—
226C2×C3×Q8—22no—
227C2×C3×Q8—22no—
228C2×C3×Q8C2×D414yes—
229C22×C3×C4C2412yes—
230C22×C3×C4C2412yes—
231C22×C3×C4C2412yes—
232C22×C3×C4C2414yes—
233C22×C3×C4C2412yes—
234C22×C3×C4C2411yescharacteristic
235C22×C3×C4C2411yescharacteristic
236C22×C3×C4C2411yescharacteristic
237C22×Dic3C2412yes—
238C22×Dic3C2412yes—
239C22×Dic3C2414yes—
240C22×Dic3C2412yes—
241C22×Dic3C2412yes—
242C22×Dic3C2412yes—
243C2×D12—22no—
244C2×D12—42no—
245C2×C4×D3—24no—
246C2×C4×D3—24no—
247C2×C4×D3—24no—
248C2×Dic6—22no—
249C2×Dic6—22no—
250C2×Dic6C2×D414yes—
251C3×C4⋊C4—28no—
252C3×C4⋊C4—24no—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—22no—
255C3×C4⋊C4—42no—
256C3×C4⋊C4—44no—
257C3×C42—24no—
258C3×C42—28no—
259C3×C42—24no—
260C3×C42#4114yes—
261(C2×D3)⋊C4—28no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—28no—
266(C2×D3)⋊C4—24no—
267(C2×D3)⋊C4—24no—
268(C2×D3)⋊C4—44no—
269(C2×D3)⋊C4—44no—
270C4⋊Dic3—24no—
271C4⋊Dic3—22no—
272C4⋊Dic3—22no—
273C4⋊Dic3—28no—
274C4⋊Dic3—28no—
275Dic3⋊C4—28no—
276Dic3⋊C4—24no—
277Dic3⋊C4—28no—
278Dic3⋊C4—24no—
279Dic3⋊C4—24no—
280Dic3⋊C4—28no—
281Dic3⋊C4—28no—
282Dic3⋊C4—44no—
283Dic3⋊C4—44no—
284Dic3⋊C4—44no—
285Dic3⋊C4C2×D4116yes—
286C4×Dic3—24no—
287C4×Dic3—22no—
288C4×Dic3—28no—
289C4×Dic3—24no—
290C4×Dic3—42no—
291C4×Dic3C2×D414yes—
292C22×Q8—31no—
293C22×Q8C22×D311yescharacteristic
294C22×D4—31no—
295C23×C4—32no—
296C4⋊Q8—38no—
297#126—616no—
298#126—616no—
299#125—616no—
300#125—68no—
301#125—616no—
302C22⋊Q8—68no—
303C22⋊Q8—68no—
304C4×Q8—28no—
305C4×Q8—24no—
306C4×Q8—68no—
307C4×Q8—64no—
308C4×D4—64no—
309#117—68no—
310C2×C4⋊C4—34no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—32no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—34no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—32no—
318C2×C4⋊C4—34no—
319C2×C4⋊C4—34no—
320C2×C4⋊C4—34no—
321C2×C4⋊C4—34no—
322C2×C4⋊C4—34no—
323C2×C4⋊C4—34no—
324C2×C4⋊C4—34no—
325C2×C4⋊C4—34no—
326C2×C4⋊C4C22×D312yes—
327C2×C4⋊C4C22×D314yes—
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C4⋊C4C22×D312yes—
330C2×C4⋊C4C22×D312yes—
331C2×C4⋊C4C22×D314yes—
332C2×C22⋊C4—32no—
333C2×C22⋊C4—32no—
334C2×C22⋊C4—32no—
335C2×C22⋊C4—34no—
336C2×C22⋊C4—34no—
337C2×C22⋊C4—32no—
338C2×C22⋊C4—32no—
339C2×C22⋊C4—34no—
340C2×C22⋊C4—34no—
341C2×C42—31no—
342C2×C42—34no—
343C2×C42—32no—
344C2×C42—31no—
345C2×C42—32no—
346C2×C42—32no—
347C2×C42C22×D312yes—
348C2×C42C22×D314yes—
349C2×C42C22×D312yes—
350C2×C42C22×D311yescharacteristic
351#95—32no—
352#95—32no—
353#95—32no—
354#95—34no—
355#95—34no—
356#95—34no—
357#95—34no—
358#95—34no—
359#95—32no—
360#95C22×D311yescharacteristic
361#95C22×D311yescharacteristic
362#95C22×D312yes—
363#95C22×D314yes—
364#95C22×D312yes—
365C23×C3C2511yescharacteristic
366C22×D3—22no—
367C22×D3—22no—
368C22×D3—22no—
369C22×D3—22no—
370C22×D3—22no—
371C22×D3—22no—
372C22×D3—22no—
373C3×Q8—28no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—24no—
381C2×C3×C4—22no—
382C2×C3×C4—24no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—24no—
387C2×C3×C4—24no—
388C2×C3×C4—22no—
389C2×C3×C4C22×D414yes—
390C2×C3×C4C22×D412yes—
391C2×C3×C4C2×#4112yes—
392C2×C3×C4C2×#4112yes—
393C2×C3×C4C2×#4112yes—
394C2×C3×C42−1+412yes—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—24no—
402C2×Dic3—22no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—24no—
406C2×Dic3—22no—
407C2×Dic3—24no—
408C2×Dic3—24no—
409C2×Dic3C22×D414yes—
410C2×Dic3C22×D414yes—
411C2×Dic32−1+414yes—
412D12—44no—
413C4×D3—48no—
414Dic6—28no—
415C24—32no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—34no—
419C2×Q8—62no—
420C2×Q8—62no—
421C2×Q8D3×D414yes—
422C2×D4—62no—
423C2×D4—122no—
424C22×C4—32no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—32no—
428C22×C4—32no—
429C22×C4—34no—
430C22×C4—64no—
431C22×C4—64no—
432C22×C4—64no—
433C22×C4C23×D312yes—
434C22×C4C23×D312yes—
435C22×C4C23×D312yes—
436C22×C4C23×D314yes—
437C22×C4C23×D311yescharacteristic
438C22×C4C23×D311yescharacteristic
439C22×C4C23×D312yes—
440C22×C4C23×D311yescharacteristic
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—28no—
445C4⋊C4—316no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—64no—
449C4⋊C4—64no—
450C4⋊C4—68no—
451C4⋊C4—68no—
452C4⋊C4—68no—
453C4⋊C4—68no—
454C4⋊C4—68no—
455C4⋊C4—62no—
456C4⋊C4—64no—
457C4⋊C4—62no—
458C4⋊C4—64no—
459C4⋊C4—68no—
460C4⋊C4—124no—
461C4⋊C4—124no—
462C4⋊C4—124no—
463C22⋊C4—64no—
464C22⋊C4—64no—
465C22⋊C4—68no—
466C22⋊C4—64no—
467C22⋊C4—64no—
468C22⋊C4—64no—
469C22⋊C4—68no—
470C22⋊C4—124no—
471C22⋊C4—124no—
472C42—24no—
473C42—24no—
474C42—28no—
475C42—34no—
476C42—68no—
477C42—64no—
478C42—64no—
479C42—62no—
480C42—122no—
481C42#23514yes—
482C22×C3C23×D411yescharacteristic
483C22×C3C22×#4111yescharacteristic
484C22×C3C2×2−1+411yescharacteristic
485C22×C3C2×2−1+412yes—
486C22×C3#58512yes—
487C2×D3—42no—
488C2×D3—42no—
489C2×D3—42no—
490C2×D3—42no—
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—42no—
494C3×C4—22no—
495C3×C4—24no—
496C3×C4—22no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—44no—
501C3×C4#54714yes—
502Dic3—24no—
503Dic3—24no—
504Dic3—24no—
505Dic3—42no—
506Dic3—42no—
507Dic3—44no—
508C23—62no—
509C23—62no—
510C23—62no—
511C23—62no—
512C23—62no—
513C23—62no—
514C23—62no—
515C23C24×D311yescharacteristic
516Q8—28no—
517Q8—68no—
518D4—124no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—34no—
535C2×C4—34no—
536C2×C4—34no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—64no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—128no—
552C2×C4C2×#23512yes—
553C2×C4C2×#23512yes—
554C2×C4C2×#23512yes—
555C2×C4C2×D3×D414yes—
556C2×C4C2×D3×D412yes—
557C2×C4#100712yes—
558C2×C3#345511yescharacteristic
559C2×C3C2×#54711yescharacteristic
560C2×C3#349712yes—
561C2×C3#350911yescharacteristic
562C2×C3#356312yes—
563D3—82no—
564C22—122no—
565C22—122no—
566C22—122no—
567C22—122no—
568C22—122no—
569C22—122no—
570C22—122no—
571C22C22×#23511yescharacteristic
572C22C22×D3×D411yescharacteristic
573C22C2×#100711yescharacteristic
574C22#596412yes—
575C22D3×2−1+412yes—
576C4—22no—
577C4—22no—
578C4—24no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—44no—
583C4—64no—
584C4—64no—
585C4—64no—
586C4—122no—
587C4—124no—
588C4—122no—
589C4#550714yes—
590C3#3937311yescharacteristic, complemented, Sylow
591C2—242no—
592C2C2×#550711yescharacteristic
593C2#8583711yescharacteristic
594C2#8587611yescharacteristic
595C2#8673712yes—
596C2#8703812yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2351C2
prime radicalC22×D12C23
hypercenter#2351D3
hypocenterC3#39373
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
1. C2×#5583C2semidirect
2. C3×#2351C2semidirect
4. #82494C2semidirect
5. C4×#944C2semidirect
6. #76251C2semidirect
7. #76216C2semidirect
9. #76126C2semidirect
11. #74220C2semidirect
12. #74175C2semidirect
13. #73699C2semidirect
14. #73637C2semidirect
89. #2351D3semidirect
109. #944D4semidirect
590. C3#39373semidirect

Maximal subgroups

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

subgroupindex
C2×#55832
C3×#23512
#824992
#824942
C4×#9442
#762512
#762162
#761302
#761262
#742402
#742202
#741752
#736992
#736372
#736172
#393733