Finite Group Databasebeta

← #10829937

Subgroups — #10829937

Order 768.

1150 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5584C211yesmaximal, characteristic, complemented
2C3×#2538C211yesmaximal, characteristic, complemented
3C2×#5019C211yesmaximal, characteristic, complemented
4C2×C4×#213C211yesmaximal, characteristic
5#71634C211yesmaximal, characteristic, complemented
6#69660C212yesmaximal, complemented
7#69659C212yesmaximal
8#69631C212yesmaximal, complemented
9#69628C212yesmaximal, complemented
10#69584C211yesmaximal, characteristic, complemented
11#69579C211yesmaximal, characteristic, complemented
12#24185—31nomaximal, Sylow
13C2×C3×C22⋊Q8—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#946—21no—
17C22×#214C2212yes—
18C22×#213C2211yescharacteristic
19#5584—24no—
20#5584—24no—
21C2×#898—22no—
22C2×#892—21no—
23C2×#875C2212yes—
24C2×C4×D12C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C4×Dic6C2212yes—
27C3×#396—22no—
28C3×#390C2212yes—
29C3×#389C2212yes—
30C3×#384C2211yescharacteristic
31C3×#380—21no—
32C2×#832C2211yescharacteristic
33#5019—24no—
34C4×#213—24no—
35C2×#810C2212yes—
36C2×#810C2212yes—
37C2×#810C2212yes—
38C2×#809C2211yescharacteristic
39C2×#809C2212yes—
40C2×#807C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#556—31no—
47C2×#461—31no—
48C2×C4×D8—31no—
49#2538D311yescharacteristic, complemented
50#2042—32no—
51#2012—32no—
52#1984—31no—
53#1956—31no—
54#1880—32no—
55#1863—32no—
56#1802—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C22×Dic6C2312yes—
63C3×C22⋊Q8—44no—
64C3×C4×Q8—22no—
65C3×C4×Q8—22no—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4D412yes—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C42C2312yes—
79C2×C3×C42C2311yescharacteristic
80#946—28no—
81C2×#214—24no—
82C2×#214—24no—
83C2×#214—42no—
84C2×#213—22no—
85C2×#213—22no—
86C2×#213—22no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—22no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×Dic3⋊C4—22no—
94C2×C4×Dic3—22no—
95C22×C3⋊C8C2311yescharacteristic
96C22×C3⋊C8C2312yes—
97C22×C3⋊C8C2311yescharacteristic
98#898—216no—
99#892—28no—
100#875—24no—
101#875—24no—
102C4×D12—22no—
103C4×D12D414yes—
104#869—22no—
105#869D414yes—
106C4×Dic6—24no—
107C4×Dic6—24no—
108C3×#95—22no—
109#832—22no—
110#810—24no—
111#810—24no—
112#810—42no—
113#809—22no—
114#809—42no—
115#807—22no—
116C4⋊(C3⋊C8)—22no—
117C4×C3⋊C8—22no—
118C22×SD16—32no—
119C22×D8—31no—
120#556—64no—
121#556—64no—
122C2×C4⋊Q8—31no—
123C2×#126—62no—
124C2×#124—32no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—61no—
127C2×C4×Q8—32no—
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×D4—31no—
130C2×C4×D4C2×D311yescharacteristic
131C2×#117—61no—
132#461—64no—
133C4×D8—64no—
134C2×#107—31no—
135C2×C4⋊C8—31no—
136C2×#103—32no—
137C2×#103—32no—
138C2×#103—32no—
139C2×#102—31no—
140C2×#102—31no—
141C2×#102—32no—
142C2×C4×C8—31no—
143#396—22no—
144#390C2×D312yes—
145#389C2×D312yes—
146#384C2×D311yescharacteristic
147#380—21no—
148#340—32no—
149#336—32no—
150#336—32no—
151C24×C3—21no—
152C23×D3—21no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4#4112yes—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—21no—
173C22×Dic3—22no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12C2×D412yes—
180C2×C4×D3—24no—
181C2×C4×D3—42no—
182C2×C4×D3—42no—
183C2×Dic6—22no—
184C2×Dic6—22no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6—22no—
188C2×Dic6C2×D414yes—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—44no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4—81no—
198C3×C22⋊C4—44no—
199C3×C22⋊C4—42no—
200C3×C22⋊C4—82no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42C2×D412yes—
204C3×C42C2×D412yes—
205#214—48no—
206#213—28no—
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4—28no—
209(C2×D3)⋊C4—24no—
210C4⋊Dic3—24no—
211C4⋊Dic3—24no—
212C4⋊Dic3—21no—
213C4⋊Dic3C2×D412yes—
214Dic3⋊C4—28no—
215C4×Dic3—28no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8#4112yes—
225C22×Q8—32no—
226C22×Q8#6711yescharacteristic
227C22×D4—31no—
228C22×D4C22×D311yescharacteristic
229C23×C4—21no—
230C23×C4—61no—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—122no—
234C2×D8—62no—
235C2×D8—62no—
236C2×D8—62no—
237C22×C8—31no—
238C22×C8—32no—
239C22×C8—31no—
240C4⋊Q8—34no—
241C4⋊Q8—62no—
242#126—616no—
243#124—64no—
244#124—64no—
245C22⋊Q8—44no—
246C22⋊Q8—68no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—64no—
250C4×Q8—64no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—34no—
254C4×D4—62no—
255#117—68no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4#6711yescharacteristic
264C2×C4⋊C4#6712yes—
265C2×C4⋊C4C22×D311yescharacteristic
266C2×C4⋊C4C22×D312yes—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—62no—
273C2×C42—62no—
274C2×C42C22×D312yes—
275C2×C42C22×D311yescharacteristic
276#107—62no—
277C4⋊C8—62no—
278#103—64no—
279#103—64no—
280#103—122no—
281#102—62no—
282#102—62no—
283#102—122no—
284C4×C8—62no—
285#95—22no—
286C23×C3—22no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3C22×D411yescharacteristic
294C22×D3—22no—
295C22×D3—41no—
296C22×D3—41no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—41no—
300C22×D3—41no—
301C3×Q8—44no—
302C3×D4—22no—
303C3×D4—22no—
304C3×D4—41no—
305C3×D4—41no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—41no—
315C2×C3×C4—42no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—84no—
321C2×C3×C4#12012yes—
322C2×C3×C4#12012yes—
323C2×C3×C4#13512yes—
324C2×C3×C4AGL(1,ℤ/8)12yes—
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4H(5,𝔽2)11yescharacteristic
330C2×C3×C42−1+411yescharacteristic
331C2×Dic3—22no—
332C2×Dic3—24no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337D12—22no—
338D12—22no—
339D12—41no—
340D12—41no—
341C4×D3—48no—
342Dic6—24no—
343Dic6—24no—
344Dic6—42no—
345Dic6—42no—
346C3⋊C8—22no—
347C3⋊C8—41no—
348C3⋊C8—42no—
349C24—21no—
350C24—61no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—34no—
355C2×Q8—62no—
356C2×Q8—62no—
357C2×Q8—62no—
358C2×Q8—62no—
359C2×Q8—62no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—32no—
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4#23912yes—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—61no—
378C22×C4—62no—
379C22×C4—64no—
380C22×C4—122no—
381C22×C4—122no—
382C22×C4C2×#6712yes—
383C22×C4C2×#6712yes—
384C22×C4C2×#6711yescharacteristic
385C22×C4C2×#6711yescharacteristic
386C22×C4C23×D311yescharacteristic
387SD16—128no—
388D8—68no—
389C2×C8—32no—
390C2×C8—61no—
391C2×C8—61no—
392C2×C8—61no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—61no—
396C2×C8—62no—
397C2×C8—61no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—32no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C4⋊C4—68no—
408C4⋊C4—64no—
409C4⋊C4—64no—
410C4⋊C4—61no—
411C4⋊C4—81no—
412C22⋊C4—44no—
413C22⋊C4—42no—
414C22⋊C4—68no—
415C22⋊C4—68no—
416C22⋊C4—64no—
417C22⋊C4—82no—
418C42—22no—
419C42—22no—
420C42—68no—
421C42D3×D412yes—
422C42D3×D412yes—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—81no—
427C22×C3—81no—
428C22×C3—81no—
429C22×C3—81no—
430C22×C3C2×#12011yescharacteristic
431C22×C3#54811yescharacteristic
432C22×C3#55011yescharacteristic
433C22×C3C2×#13511yescharacteristic
434C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
435C22×C3#57711yescharacteristic
436C22×C3#57811yescharacteristic
437C2×D3—42no—
438C2×D3—42no—
439C2×D3—42no—
440C2×D3—81no—
441C2×D3—81no—
442C2×D3—81no—
443C2×D3—81no—
444C3×C4—22no—
445C3×C4—42no—
446C3×C4—42no—
447C3×C4—82no—
448C3×C4—81no—
449C3×C4#44914yes—
450C3×C4#52511yescharacteristic
451C3×C4#53711yescharacteristic
452C3×C4#54611yescharacteristic
453C3×C4#54711yescharacteristic
454Dic3—44no—
455Dic3—42no—
456C23—22no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—62no—
464C23—121no—
465C23—121no—
466C23—121no—
467C23—121no—
468C23—121no—
469C23—121no—
470C23C22×#6711yescharacteristic
471Q8—44no—
472Q8—64no—
473Q8—64no—
474Q8—122no—
475Q8—122no—
476D4—22no—
477D4—22no—
478D4—41no—
479D4—41no—
480D4—62no—
481D4—62no—
482D4—121no—
483D4—121no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—62no—
499C2×C4—64no—
500C2×C4—84no—
501C2×C4—121no—
502C2×C4—122no—
503C2×C4—121no—
504C2×C4—122no—
505C2×C4—128no—
506C2×C4#93212yes—
507C2×C4#93712yes—
508C2×C4#93712yes—
509C2×C4#95012yes—
510C2×C4C2×D3×D411yescharacteristic
511C2×C4C2×D3×D411yescharacteristic
512C2×C4C2×#23911yescharacteristic
513C2×C4#100711yescharacteristic
514C2×C4D3×#4111yescharacteristic
515C2×C4#100911yescharacteristic
516C8—62no—
517C8—121no—
518C8—122no—
519C2×C3—82no—
520C2×C3#240711yescharacteristic
521C2×C3C2×#44911yescharacteristic
522C2×C3#301311yescharacteristic
523C2×C3#336011yescharacteristic
524C2×C3#336711yescharacteristic
525C2×C3#339311yescharacteristic
526C2×C3#339711yescharacteristic
527D3—82no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—122no—
536C22—122no—
537C22—122no—
538C22—241no—
539C22—241no—
540C22—241no—
541C22—241no—
542C22C2×#93211yescharacteristic
543C22C2×#93711yescharacteristic
544C22#581711yescharacteristic
545C22C2×#95011yescharacteristic
546C22#583411yescharacteristic
547C22#583611yescharacteristic
548C22#583811yescharacteristic
549C4—22no—
550C4—42no—
551C4—42no—
552C4—82no—
553C4—81no—
554C4—124no—
555C4—122no—
556C4#503714yes—
557C4#562711yescharacteristic
558C4#562911yescharacteristic
559C4#563011yescharacteristic
560C4#563111yescharacteristic
561C3#2418511yescharacteristic, complemented, Sylow
562C2—82no—
563C2—242no—
564C2C2×#503711yescharacteristic
565C2#8281211yescharacteristic
566C2#8342711yescharacteristic
567C2#8356311yescharacteristic
568C2#8357611yescharacteristic
569C2#8363711yescharacteristic
570C2#8367011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2538C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2538D3
hypocenterC3#24185
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C2×#5584C2semidirect
2. C3×#2538C2semidirect
3. C2×#5019C2semidirect
5. #71634C2semidirect
6. #69660C2semidirect
8. #69631C2semidirect
9. #69628C2semidirect
10. #69584C2semidirect
11. #69579C2semidirect
49. #2538D3semidirect
561. C3#24185semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
C3×#25382
C2×#50192
C2×C4×#2132
#716342
#696602
#696592
#696312
#696282
#695842
#695792
#241853