Finite Group Databasebeta

← #10691772

Subgroups — #10691772

Order 768.

422 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 95 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
1#83280C211yesmaximal, characteristic
2#83272C211yesmaximal, characteristic
3C3×#1505C211yesmaximal, characteristic
4#70803C211yesmaximal, characteristic, complemented
5#69433C211yesmaximal, characteristic, complemented
6#67756C211yesmaximal, characteristic, complemented
7#67614C211yesmaximal, characteristic, complemented
8#9279—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#376C2211yescharacteristic
13#5234—22no—
14#5080C411yescharacteristic, complemented
15#5078C411yescharacteristic, complemented
16#5070C411yescharacteristic, complemented
17#5070C411yescharacteristic, complemented
18#5035—22no—
19#5012—21no—
20#5011—22no—
21D4×C3⋊C8—21no—
22#5009—22no—
23#4926C2211yescharacteristic
24#4925C2211yescharacteristic
25#4925C2211yescharacteristic
26C3×#327—22no—
27#4484—21no—
28#4484—21no—
29#4452—22no—
30#3228—31no—
31#2981—31no—
32#1506—31no—
33#1505D311yescharacteristic
34#1497—31no—
35#1287—31no—
36#1274—31no—
37C2×C3×#41—21no—
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×#127C2×C411yescharacteristic
40C3×#124—21no—
41C3×#121—21no—
42C3×#121D412yes—
43C3×C4×D4—21no—
44C3×C4×D4D412yes—
45C2×C3×C42—21no—
46C2×C3×C42C2311yescharacteristic
47#926—21no—
48#924—21no—
49#924—21no—
50C2×#208—22no—
51C22×C3⋊C8—22no—
52#870—21no—
53#869C2×C411yescharacteristic
54#869C2×C411yescharacteristic
55C4×Dic6—21no—
56C4×Dic6—21no—
57C3×C4⋊C8—21no—
58C3×C22⋊C8—21no—
59C3×#97—21no—
60C3×#95—21no—
61C3×#95—21no—
62#832—22no—
63#832—22no—
64#830—22no—
65#830—22no—
66#830—21no—
67#830—21no—
68#830D412yes—
69#808—22no—
70#807—22no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74#803—21no—
75C4×C3⋊C8C2×C411yescharacteristic
76C4×C3⋊C8C2×C411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78C4×C3⋊C8D411yescharacteristic
79#533—31no—
80#532C2×D311yescharacteristic
81#500—31no—
82#498—31no—
83#486—31no—
84#486—31no—
85#482—62no—
86#478—62no—
87#436—61no—
88#435—62no—
89C8×D4—61no—
90#432—31no—
91#432C2×D311yescharacteristic
92#431—31no—
93#431—31no—
94#408—62no—
95#376C2×D311yescharacteristic
96#328—62no—
97#327—22no—
98Q8⋊C8—61no—
99Q8⋊C8—61no—
100C3×#41—82no—
101C2×C3×Q8—21no—
102C2×C3×D4—22no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4C22⋊C412yes—
106C22×C3×C4—22no—
107C22×C3×C4—41no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C2×Dic6—21no—
112C2×Dic6—21no—
113C2×C3×C8—41no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4C22⋊C412yes—
116C3×C22⋊C4—22no—
117C3×C22⋊C4—22no—
118C3×C42—22no—
119C3×C42—41no—
120C3×C42C22×C411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42#4111yescharacteristic
124#217—41no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129Dic3⋊C4—41no—
130Dic3⋊C4—41no—
131C4×Dic3—41no—
132#208—82no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—82no—
139C2×#41—21no—
140C2×M16—62no—
141C22×C8—62no—
142C4⋊Q8—31no—
143C4⋊Q8—31no—
144C4⋊Q8C4×D311yescharacteristic
145#127C4×D311yescharacteristic
146#125—61no—
147#124—21no—
148C22⋊Q8—61no—
149C22⋊Q8—61no—
150#121—21no—
151#121#6712yes—
152C4×Q8—61no—
153C4×Q8—61no—
154C4×D4—21no—
155C4×D4D1212yes—
156#117—61no—
157C2×C42—21no—
158C2×C42C22×D311yescharacteristic
159#107—62no—
160#106—62no—
161C4⋊C8—21no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165#102—62no—
166#102—62no—
167C22⋊C8—21no—
168C22⋊C8—32no—
169C22⋊C8—62no—
170C22⋊C8—62no—
171C22⋊C8—61no—
172C22⋊C8—61no—
173#97—21no—
174#97—61no—
175C4×C8—31no—
176C4×C8—31no—
177C4×C8—31no—
178C4×C8—31no—
179#95—21no—
180#95—21no—
181C23×C3—22no—
182C23×C3#12111yescharacteristic
183C3×Q8—81no—
184C3×D4—42no—
185C3×D4—41no—
186C3×D4—41no—
187C3×D4—81no—
188C3×D4—82no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—22no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—81no—
198C2×C3×C4—81no—
199C2×C3×C4C2×C22⋊C411yescharacteristic
200C2×C3×C4C4×D411yescharacteristic
201C2×C3×C4C4×D411yescharacteristic
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12311yescharacteristic
205C2×Dic3—41no—
206C2×Dic3—41no—
207Dic6—41no—
208Dic6—41no—
209Dic6—81no—
210C3×C8—81no—
211C3⋊C8—41no—
212C3⋊C8—41no—
213C3⋊C8—41no—
214C3⋊C8—42no—
215C3⋊C8—41no—
216#41—82no—
217C2×Q8—21no—
218C2×Q8—61no—
219C2×Q8—61no—
220C2×D4—22no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4(C2×D3)⋊C412yes—
224C22×C4—22no—
225C22×C4—41no—
226C22×C4D3×D411yescharacteristic
227C22×C4#23711yescharacteristic
228C22×C4C2×#6711yescharacteristic
229M16—242no—
230C2×C8—41no—
231C2×C8—61no—
232C2×C8—61no—
233C2×C8—62no—
234C2×C8—61no—
235C2×C8—61no—
236C2×C8—242no—
237C4⋊C4—21no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C4⋊C4(C2×D3)⋊C412yes—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—121no—
248C42—22no—
249C42—41no—
250C42—121no—
251C42C2×C4×D311yescharacteristic
252C42C2×D1211yescharacteristic
253C42D3×D411yescharacteristic
254C42#23711yescharacteristic
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—42no—
259C22×C3—42no—
260C22×C3—42no—
261C22×C3#38611yescharacteristic
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—42no—
268C3×C4—82no—
269C3×C4#44311yescharacteristic
270C3×C4#44411yescharacteristic
271Dic3—81no—
272Dic3—81no—
273C23—22no—
274C23#93911yescharacteristic
275Q8—81no—
276Q8—121no—
277Q8—121no—
278Q8—241no—
279D4—42no—
280D4—41no—
281D4—41no—
282D4—81no—
283D4—82no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4#67⋊C411yescharacteristic
297C2×C4#67⋊C411yescharacteristic
298C2×C4#88211yescharacteristic
299C2×C4#88611yescharacteristic
300C2×C4C2×(C2×D3)⋊C411yescharacteristic
301C2×C4#93911yescharacteristic
302C8—81no—
303C8—121no—
304C8—122no—
305C8—121no—
306C8—121no—
307C8—121no—
308C2×C3—41no—
309C2×C3—82no—
310C2×C3#187611yescharacteristic
311C2×C3#190411yescharacteristic
312C2×C3#190911yescharacteristic
313C22—21no—
314C22—21no—
315C22—21no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22#495811yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—42no—
326C4—82no—
327C4—241no—
328C4—241no—
329C4#482311yescharacteristic
330C4#482411yescharacteristic
331C3#927911yescharacteristic, complemented, Sylow
332C2—41no—
333C2—82no—
334C2#7004611yescharacteristic
335C2#7031411yescharacteristic
336C2#7037711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1505C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1505D3
hypocenterC3#9279
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1505

Complements

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

NH ≅ G/Ntype
4. #70803C2semidirect
5. #69433C2semidirect
6. #67756C2semidirect
7. #67614C2semidirect
14. #5080C4semidirect
15. #5078C4semidirect
16. #5070C4semidirect
17. #5070C4semidirect
331. C3#9279semidirect

Maximal subgroups

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

subgroupindex
#832802
#832722
C3×#15052
#708032
#694332
#677562
#676142
#92793