Finite Group Databasebeta

← #10691795

Subgroups — #10691795

Order 768.

416 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2985C211yesmaximal, characteristic, complemented
2#83356C211yesmaximal, characteristic
3#71836C211yesmaximal, characteristic, complemented
4#70832C211yesmaximal, characteristic
5#70765C211yesmaximal, characteristic
6#67757C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#9282—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#446—22no—
12C3×#446—21no—
13C3×C8×Q8—21no—
14C3×#433—22no—
15C3×#432C2211yescharacteristic
16C3×#431C2211yescharacteristic
17#5093C411yescharacteristic, complemented
18#5092C411yescharacteristic, complemented
19#5088C411yescharacteristic, complemented
20#5085C411yescharacteristic, complemented
21#5047—22no—
22#5028—22no—
23#4932C2211yescharacteristic
24#4926C2211yescharacteristic
25#4925C2211yescharacteristic
26#4538—22no—
27#4470—22no—
28#4459—21no—
29#4459—21no—
30#3210—31no—
31#2985D311yescharacteristic, complemented
32#1506—31no—
33#1505—31no—
34#1496—31no—
35#1287—31no—
36#1274—31no—
37C2×#235—21no—
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×#125—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8D412yes—
43C3×C4×Q8—21no—
44C3×C4×Q8D412yes—
45C3×#117—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#216—22no—
48C2×#215—22no—
49#930—21no—
50#930—21no—
51C2×C4×Dic3—21no—
52#875—21no—
53#874C2×C411yescharacteristic
54C4×D12—21no—
55C4×D12—21no—
56#869C2×C411yescharacteristic
57C3×C4⋊C8—22no—
58C3×C4⋊C8—22no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×C22⋊C8D412yes—
63C3×#97—22no—
64C3×#97—21no—
65C3×C4×C8—22no—
66C3×C4×C8D411yescharacteristic
67C3×C4×C8D411yescharacteristic
68C3⋊#95—21no—
69C3⋊#95—21no—
70#830—21no—
71#830—21no—
72#810—22no—
73#809—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76#803—21no—
77C4×C3⋊C8C2×C411yescharacteristic
78C4×C3⋊C8C2×C411yescharacteristic
79#533C2×D311yescharacteristic
80#532—31no—
81#494—31no—
82#492—31no—
83#487—31no—
84#486—31no—
85#464—62no—
86#452—62no—
87#446—22no—
88#446—21no—
89C8×Q8—21no—
90#433—22no—
91#432—31no—
92#432C2×D311yescharacteristic
93#431—31no—
94#431C2×D311yescharacteristic
95#376—31no—
96#328—62no—
97#327—62no—
98D4⋊C8—61no—
99D4⋊C8—61no—
100C2×C3×Q8—21no—
101C2×C3×Q8C22⋊C412yes—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C2×#67—41no—
106C22×Dic3—41no—
107#235—82no—
108C2×D12—21no—
109C2×D12—21no—
110C2×D12—21no—
111C2×C4×D3—41no—
112C2×Dic6—21no—
113C2×C3×C8—21no—
114C2×C3×C8—22no—
115C2×C3×C8—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4C22⋊C412yes—
122C3×C22⋊C4—22no—
123C3×C42—22no—
124C3×C42C22×C411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42#4111yescharacteristic
128#216—82no—
129#215—82no—
130(C2×D3)⋊C4—41no—
131(C2×D3)⋊C4—41no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4×Dic3—22no—
135C4×Dic3—41no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—21no—
138C2×C3⋊C8—41no—
139C2×#41—61no—
140C2×Q16—62no—
141C2×SD16—62no—
142C4⋊Q8—31no—
143C4⋊Q8C4×D311yescharacteristic
144C4⋊Q8C4×D311yescharacteristic
145#127—31no—
146#125—21no—
147#124—61no—
148C22⋊Q8—21no—
149C22⋊Q8D1212yes—
150#121—61no—
151#121—61no—
152C4×Q8—21no—
153C4×Q8#6712yes—
154C4×D4—61no—
155C4×D4—61no—
156#117—21no—
157C2×C42—61no—
158C2×C42C22×D311yescharacteristic
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—21no—
162C4⋊C8—21no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165#103—62no—
166#102—62no—
167C22⋊C8—21no—
168C22⋊C8—61no—
169C22⋊C8—61no—
170C22⋊C8#6712yes—
171#97—22no—
172#97—21no—
173#97—61no—
174C4×C8—22no—
175C4×C8—31no—
176C4×C8—31no—
177C4×C8#6711yescharacteristic
178C4×C8#6711yescharacteristic
179#95—61no—
180#95—61no—
181C23×C3#12111yescharacteristic
182C22×D3—41no—
183C3×Q8—42no—
184C3×Q8—41no—
185C3×Q8—41no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4C2×C22⋊C411yescharacteristic
193C2×C3×C4C4×D411yescharacteristic
194C2×C3×C4C4×D411yescharacteristic
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12311yescharacteristic
198#67—82no—
199C2×Dic3—42no—
200C2×Dic3—41no—
201C2×Dic3—81no—
202C2×Dic3—81no—
203D12—42no—
204D12—41no—
205D12—41no—
206D12—81no—
207C4×D3—82no—
208Dic6—42no—
209C3×C8—41no—
210C3×C8—42no—
211C3×C8—41no—
212C3⋊C8—41no—
213C3⋊C8—41no—
214C3⋊C8—81no—
215#41—242no—
216C2×Q8—21no—
217C2×Q8—61no—
218C2×Q8(C2×D3)⋊C412yes—
219C2×D4—61no—
220C2×D4—61no—
221C2×D4—61no—
222C2×D4—121no—
223C22×C4—121no—
224C22×C4—121no—
225C22×C4C2×D1211yescharacteristic
226C22×C4#23511yescharacteristic
227C22×C4C2×#6711yescharacteristic
228Q16—242no—
229SD16—242no—
230C2×C8—21no—
231C2×C8—22no—
232C2×C8—21no—
233C2×C8—61no—
234C2×C8—61no—
235C2×C8—121no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4(C2×D3)⋊C412yes—
244C22⋊C4—22no—
245C22⋊C4—121no—
246C22⋊C4—121no—
247C42—22no—
248C42—62no—
249C42—121no—
250C42C2×C4×D311yescharacteristic
251C42#23511yescharacteristic
252C42C2×#6711yescharacteristic
253C42C2×#6711yescharacteristic
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3#38611yescharacteristic
258C2×D3—81no—
259C2×D3—81no—
260C2×D3—81no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—42no—
267C3×C4—42no—
268C3×C4#44311yescharacteristic
269C3×C4#44411yescharacteristic
270Dic3—82no—
271Dic3—81no—
272C23—121no—
273C23#93011yescharacteristic
274Q8—42no—
275Q8—41no—
276Q8—41no—
277Q8—122no—
278D4—122no—
279D4—121no—
280D4—121no—
281D4—241no—
282D4—242no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—122no—
290C2×C4—121no—
291C2×C4—242no—
292C2×C4—241no—
293C2×C4—241no—
294C2×C4C2×(C2×D3)⋊C411yescharacteristic
295C2×C4C4×#6711yescharacteristic
296C2×C4C4×#6711yescharacteristic
297C2×C4#92911yescharacteristic
298C2×C4#93011yescharacteristic
299C2×C4#95311yescharacteristic
300C8—41no—
301C8—42no—
302C8—41no—
303C8—121no—
304C8—121no—
305C8—241no—
306C2×C3—41no—
307C2×C3#187611yescharacteristic
308C2×C3#190511yescharacteristic
309C2×C3#191011yescharacteristic
310D3—161no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—241no—
315C22—241no—
316C22—241no—
317C22#521311yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—42no—
324C4—42no—
325C4—242no—
326C4—241no—
327C4#514411yescharacteristic
328C4#514511yescharacteristic
329C3#928211yescharacteristic, complemented, Sylow
330C2—41no—
331C2—481no—
332C2#7166211yescharacteristic
333C2#7201011yescharacteristic
334C2#7202511yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2985C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#2985D3
hypocenterC3#9282
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2985

Complements

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

NH ≅ G/Ntype
1. C3×#2985C2semidirect
3. #71836C2semidirect
6. #67757C2semidirect
7. #67737C2semidirect
17. #5093C4semidirect
18. #5092C4semidirect
19. #5088C4semidirect
20. #5085C4semidirect
31. #2985D3semidirect
329. C3#9282semidirect

Maximal subgroups

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

subgroupindex
C3×#29852
#833562
#718362
#708322
#707652
#677572
#677372
#92823