Finite Group Databasebeta

← #10836658

Subgroups — #10836658

Order 768.

706 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83212C211yesmaximal, characteristic, complemented
2C2×#5084C211yesmaximal, characteristic, complemented
3#79040C211yesmaximal, characteristic, complemented
4C4×#908C211yesmaximal, characteristic
5C3×#1997C211yesmaximal, characteristic, complemented
6#70527C211yesmaximal, characteristic, complemented
7#69875C211yesmaximal, characteristic, complemented
8#69011C212yesmaximal, complemented
9#68986C212yesmaximal
10#68984C212yesmaximal, complemented
11#68983C212yesmaximal, complemented
12#24721—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#908C2211yescharacteristic
15C4×#235C2211yescharacteristic
16C2×C4×D12C2211yescharacteristic
17C3×#423C2211yescharacteristic
18C3×C4×M16C2211yescharacteristic
19#5133C2211yescharacteristic
20#5131C2211yescharacteristic
21C2×#823C2212yes—
22#5129C2212yes—
23C2×#822C2212yes—
24#5122C2212yes—
25#5121C2211yescharacteristic
26#5094—21no—
27#5091—21no—
28#5087—21no—
29#5084—21no—
30#5084—21no—
31#5084—21no—
32#5084—22no—
33#4966C2212yes—
34C2×#809C2212yes—
35#4964C2212yes—
36C2×#808C2212yes—
37#4924C2211yescharacteristic
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4840—22no—
40#4835—21no—
41#4833—21no—
42#4711—21no—
43#4710—21no—
44C4×D24—21no—
45C4×#204—21no—
46C3×#343C2212yes—
47C3×#343C2212yes—
48#4558C2212yes—
49#4554C2212yes—
50#3079—31no—
51C2×#476—31no—
52C4×AGL(1,ℤ/8)—31no—
53#2925—31no—
54#2079—32no—
55#2051—32no—
56#1997D311yescharacteristic, complemented
57#1951—32no—
58#1950—31no—
59#1931—32no—
60#1800—31no—
61C22×C3×Q8—21no—
62C2×#235C2311yescharacteristic
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C2×C3×M16C2312yes—
66C2×C3×M16C2311yescharacteristic
67C2×C3×M16C2311yescharacteristic
68C3×C4⋊Q8—21no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8—21no—
71C3×C4⋊Q8—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C42C2311yescharacteristic
75C4×#67—21no—
76C4×#67—21no—
77C2×(C2×D3)⋊C4—21no—
78#926C2311yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×#208C2312yes—
81C22×C3⋊C8C2312yes—
82#908—22no—
83#908—22no—
84C2×D24—21no—
85C2×#204—21no—
86C4×D12—21no—
87C4×D12—21no—
88C4×D12—21no—
89C4×D12—21no—
90C4×D12D411yescharacteristic
91C4×D12D411yescharacteristic
92#872—21no—
93C42×D3—21no—
94C4×Dic6D411yescharacteristic
95C4×Dic6D411yescharacteristic
96C3×C4⋊C8—21no—
97C3×C4⋊C8—21no—
98C3×#97—21no—
99C3×C4×C8—21no—
100#823—24no—
101#822—24no—
102#821—21no—
103#821—21no—
104#821—21no—
105(C2×D3)⋊C8—22no—
106(C2×D3)⋊C8—22no—
107#818—21no—
108#817—21no—
109#816—21no—
110#815—22no—
111C8×Dic3—22no—
112#810—22no—
113#809—22no—
114#809—22no—
115#809—22no—
116#808—22no—
117#808—22no—
118#808—22no—
119#807—22no—
120C4⋊(C3⋊C8)—21no—
121C4⋊(C3⋊C8)—21no—
122C4⋊(C3⋊C8)D411yescharacteristic
123C4⋊(C3⋊C8)D411yescharacteristic
124C4⋊(C3⋊C8)D412yes—
125C2×AGL(1,ℤ/8)—31no—
126C2×C4⋊Q8C2×D311yescharacteristic
127C4×#41—31no—
128C2×C4×D4—31no—
129#477—61no—
130#476—61no—
131#476—61no—
132#476—62no—
133#476—61no—
134#475—61no—
135#474—61no—
136#442—61no—
137#440—61no—
138C4×SD16—61no—
139C4×D8—61no—
140#433—62no—
141#432—61no—
142#431—61no—
143C2×#108—32no—
144#428—31no—
145#428—32no—
146C2×#106—32no—
147#423—31no—
148#423C2×D311yescharacteristic
149C2×C4⋊C8—31no—
150#418—31no—
151#417—31no—
152#417—32no—
153C2×#102—32no—
154C2×#100—32no—
155#408—32no—
156#405—32no—
157C4×M16C2×D311yescharacteristic
158#343—32no—
159#343C2×D312yes—
160#343C2×D312yes—
161#341—32no—
162C23×D3—21no—
163C2×C3×Q8—41no—
164C2×C3×Q8—41no—
165C2×C3×Q8—82no—
166C22×C3×C4—22no—
167C22×C3×C4#4111yescharacteristic
168C22×C3×C4#4111yescharacteristic
169C22×C3×C4C2411yescharacteristic
170C2×#67—21no—
171C22×Dic3—21no—
172#235—21no—
173#235—21no—
174#235—21no—
175#235—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12C2×D411yescharacteristic
182C2×D12#4112yes—
183C2×C4×D3—21no—
184C2×C4×D3—21no—
185C2×C4×D3—21no—
186C2×C4×D3—41no—
187C2×C4×D3—41no—
188C2×C4×D3—82no—
189C2×Dic6C2×D411yescharacteristic
190C3×M16—21no—
191C3×M16—22no—
192C3×M16—21no—
193C3×M16—21no—
194C3×M16—22no—
195C3×M16#4112yes—
196C2×C3×C8—21no—
197C2×C3×C8—22no—
198C2×C3×C8—21no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—41no—
202C3×C4⋊C4—41no—
203C3×C42C2×D411yescharacteristic
204C3×C42C2×D411yescharacteristic
205C3×C42C2×D411yescharacteristic
206C3×C42C2×D411yescharacteristic
207#217—21no—
208(C2×D3)⋊C4—21no—
209(C2×D3)⋊C4—21no—
210(C2×D3)⋊C4—41no—
211(C2×D3)⋊C4—41no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3C2×D411yescharacteristic
215C4⋊Dic3C2×D411yescharacteristic
216Dic3⋊C4—21no—
217Dic3⋊C4—21no—
218C4×Dic3—21no—
219C4×Dic3—21no—
220C4×Dic3—21no—
221#208—22no—
222#208—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8C2×D412yes—
227C2×C3⋊C8C2×D412yes—
228D24—24no—
229#204—24no—
230C2×#41—31no—
231C22×Q8—21no—
232C22×D4—31no—
233C23×C4—61no—
234AGL(1,ℤ/8)—62no—
235AGL(1,ℤ/8)—62no—
236C2×SD16—61no—
237C2×D8—61no—
238C2×M16—32no—
239C2×M16C22×D311yescharacteristic
240C2×M16C22×D312yes—
241C2×M16C22×D311yescharacteristic
242C22×C8—32no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4×Q8—31no—
248C4×Q8—31no—
249C4×D4—31no—
250C4×D4—31no—
251C4×D4—61no—
252C4×D4—61no—
253C4×D4—61no—
254C4×D4—61no—
255C4×D4—61no—
256C4×D4—61no—
257#117—31no—
258#117—61no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4C22×D312yes—
262C2×C22⋊C4—61no—
263C2×C42—61no—
264C2×C42C22×D311yescharacteristic
265#108—64no—
266#107—62no—
267#107—61no—
268#106—62no—
269#106—62no—
270#106—62no—
271#106—61no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—31no—
275C4⋊C8—31no—
276C4⋊C8—32no—
277C4⋊C8—61no—
278C4⋊C8—61no—
279#103—62no—
280#103—61no—
281#102—62no—
282#102—62no—
283#102—62no—
284#102—61no—
285#102—61no—
286#102—61no—
287#100—64no—
288C22⋊C8—62no—
289C22⋊C8—62no—
290#97—21no—
291#97—62no—
292C4×C8—21no—
293C4×C8—62no—
294C23×C3C2×#4111yescharacteristic
295C22×D3—21no—
296C22×D3—22no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—81no—
300C22×D3—81no—
301C3×Q8—82no—
302C3×Q8—82no—
303C2×C3×C4—22no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—82no—
307C2×C3×C4#12111yescharacteristic
308C2×C3×C4#12111yescharacteristic
309C2×C3×C4#12111yescharacteristic
310C2×C3×C4#12111yescharacteristic
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C42−1+411yescharacteristic
316C2×C3×C42−1+411yescharacteristic
317#67—41no—
318#67—41no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—41no—
323C2×Dic3—81no—
324D12—22no—
325D12—21no—
326D12—21no—
327D12—22no—
328D12—41no—
329D12—41no—
330C4×D3—41no—
331C4×D3—41no—
332C4×D3—42no—
333C4×D3—81no—
334C4×D3—82no—
335C4×D3—82no—
336Dic6—21no—
337Dic6—21no—
338C3×C8—22no—
339C3×C8—41no—
340C3×C8—42no—
341C3⋊C8—22no—
342C3⋊C8—22no—
343C3⋊C8—42no—
344C24—61no—
345#41—61no—
346#41—61no—
347#41—61no—
348#41—61no—
349C2×Q8—31no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—82no—
353C2×D4—31no—
354C2×D4—32no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4—61no—
359C2×D4—61no—
360C2×D4—61no—
361C22×C4—22no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4—121no—
367C22×C4—121no—
368C22×C4—242no—
369C22×C4#23511yescharacteristic
370C22×C4#23511yescharacteristic
371C22×C4C23×D311yescharacteristic
372SD16—64no—
373D8—64no—
374M16—21no—
375M16—21no—
376M16—22no—
377M16—22no—
378M16—21no—
379M16—62no—
380M16—62no—
381M16#23912yes—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—22no—
385C2×C8—32no—
386C2×C8—32no—
387C2×C8—62no—
388C2×C8—62no—
389C2×C8—62no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—31no—
393C4⋊C4—31no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—61no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—61no—
400C22⋊C4—61no—
401C22⋊C4—61no—
402C22⋊C4—61no—
403C22⋊C4—121no—
404C22⋊C4—121no—
405C42—61no—
406C42—61no—
407C42—61no—
408C42D3×D411yescharacteristic
409C42D3×D411yescharacteristic
410C42D3×D411yescharacteristic
411C42D3×D411yescharacteristic
412C22×C3—21no—
413C22×C3—21no—
414C22×C3C2×#12111yescharacteristic
415C22×C3#55211yescharacteristic
416C22×C3#55611yescharacteristic
417C2×D3—41no—
418C2×D3—42no—
419C2×D3—41no—
420C2×D3—41no—
421C2×D3—82no—
422C2×D3—81no—
423C2×D3—81no—
424C2×D3—82no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—82no—
429C3×C4#47112yes—
430C3×C4#52511yescharacteristic
431C3×C4#53611yescharacteristic
432C3×C4#54711yescharacteristic
433C3×C4#54811yescharacteristic
434Dic3—41no—
435Dic3—41no—
436Dic3—41no—
437Dic3—81no—
438C23—61no—
439C23—62no—
440C23—121no—
441C23—121no—
442C23—241no—
443C23—241no—
444C23C2×#23511yescharacteristic
445Q8—61no—
446Q8—61no—
447Q8—82no—
448Q8—82no—
449D4—62no—
450D4—61no—
451D4—62no—
452D4—61no—
453D4—121no—
454D4—121no—
455D4—121no—
456D4—121no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—82no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—122no—
467C2×C4—121no—
468C2×C4—241no—
469C2×C4—242no—
470C2×C4—242no—
471C2×C4—241no—
472C2×C4#88411yescharacteristic
473C2×C4#88411yescharacteristic
474C2×C4#88411yescharacteristic
475C2×C4#88411yescharacteristic
476C2×C4C2×D3×D411yescharacteristic
477C2×C4C2×D3×D411yescharacteristic
478C2×C4C2×#23911yescharacteristic
479C2×C4#100711yescharacteristic
480C2×C4D3×#4111yescharacteristic
481C2×C4#101011yescharacteristic
482C8—22no—
483C8—42no—
484C8—41no—
485C8—62no—
486C8—62no—
487C8—122no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3#241511yescharacteristic
491C2×C3C2×#47111yescharacteristic
492C2×C3#306811yescharacteristic
493D3—81no—
494D3—82no—
495C22—21no—
496C22—21no—
497C22—121no—
498C22—122no—
499C22—121no—
500C22—121no—
501C22—242no—
502C22—242no—
503C22—241no—
504C22—241no—
505C22C2×#88411yescharacteristic
506C22#558111yescharacteristic
507C22#558411yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—82no—
512C4—121no—
513C4—121no—
514C4—121no—
515C4—241no—
516C4#475412yes—
517C4#562711yescharacteristic
518C4#562811yescharacteristic
519C4#567711yescharacteristic
520C4#572611yescharacteristic
521C3#2472111yescharacteristic, complemented, Sylow
522C2—21no—
523C2—21no—
524C2—241no—
525C2—242no—
526C2#7662011yescharacteristic
527C2#7870311yescharacteristic
528C2C2×#475411yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1997C2
socleC22×C3C2×#121
prime radicalC2×#908C22
hypercenter#1997D3
hypocenterC3#24721
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
1. #83212C2semidirect
2. C2×#5084C2semidirect
3. #79040C2semidirect
5. C3×#1997C2semidirect
6. #70527C2semidirect
7. #69875C2semidirect
8. #69011C2semidirect
10. #68984C2semidirect
11. #68983C2semidirect
56. #1997D3semidirect
521. C3#24721semidirect

Maximal subgroups

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

subgroupindex
#832122
C2×#50842
#790402
C4×#9082
C3×#19972
#705272
#698752
#690112
#689862
#689842
#689832
#247213