Finite Group Databasebeta

← #10691794

Subgroups — #10691794

Order 768.

416 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 89 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#79306C211yesmaximal, characteristic
2#79266C211yesmaximal, characteristic, complemented
3#70765C211yesmaximal, characteristic
4#69433C211yesmaximal, characteristic, complemented
5#69432C211yesmaximal, characteristic
6C3×#1274C211yesmaximal, characteristic, complemented
7#67747C211yesmaximal, characteristic, complemented
8#9282—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#376C2211yescharacteristic
14#4926C2211yescharacteristic
15#4926C2211yescharacteristic
16#4845—22no—
17#4828—22no—
18#4733—22no—
19#4720—22no—
20#4699C411yescharacteristic, complemented
21#4694C411yescharacteristic, complemented
22#4693C411yescharacteristic, complemented
23#4681C411yescharacteristic, complemented
24#4679—21no—
25C8×Dic6—21no—
26#4459—21no—
27#4459—21no—
28#4452—22no—
29#4451—22no—
30#3210—31no—
31#2985—31no—
32#1506—31no—
33#1505—31no—
34#1496—31no—
35#1287—31no—
36#1274D311yescharacteristic, complemented
37C2×#235—21no—
38C2×C3×C42—21no—
39C2×C3×C42C2311yescharacteristic
40#930—21no—
41#930—21no—
42#926—21no—
43#924—21no—
44#924D412yes—
45C2×Dic12—22no—
46C2×#204—22no—
47#875—21no—
48#874C2×C411yescharacteristic
49C4×D12—21no—
50C4×D12—21no—
51#870—21no—
52#869C2×C411yescharacteristic
53#869C2×C411yescharacteristic
54#869C2×C411yescharacteristic
55C4×Dic6—21no—
56C4×Dic6D412yes—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×C4×C8C2×C411yescharacteristic
62C3×C4×C8C2×C411yescharacteristic
63C3×C4×C8D411yescharacteristic
64C3×C4×C8D411yescharacteristic
65C3×#95—21no—
66C3×#95—21no—
67#830—21no—
68#830D412yes—
69#821—22no—
70#816—22no—
71#815—22no—
72Dic3⋊C8—22no—
73Dic3⋊C8—22no—
74C8×Dic3—22no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77#803—21no—
78#803—21no—
79#533—31no—
80#532—31no—
81#494—31no—
82#492—31no—
83#487—31no—
84#486—31no—
85#464—62no—
86#452—62no—
87#446—62no—
88#446—61no—
89C8×Q8—61no—
90#433—62no—
91#432—31no—
92#432—31no—
93#431C2×D311yescharacteristic
94#431C2×D311yescharacteristic
95#376C2×D311yescharacteristic
96#328—62no—
97#327—62no—
98D4⋊C8—61no—
99D4⋊C8—61no—
100C22×C3×C4—41no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C2×#67—41no—
105#235—82no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12—21no—
109C2×C4×D3—41no—
110C2×Dic6—21no—
111C2×Dic6—21no—
112C2×Dic6C22⋊C412yes—
113C2×C3×C8—21no—
114C2×C3×C8—21no—
115C2×C3×C8—21no—
116C2×C3×C8—21no—
117C3×C42—22no—
118C3×C42—41no—
119C3×C42C22×C411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123#217—22no—
124(C2×D3)⋊C4—41no—
125(C2×D3)⋊C4—41no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3C22⋊C412yes—
132Dic3⋊C4—22no—
133Dic3⋊C4—22no—
134C4×Dic3—22no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—41no—
137Dic12—82no—
138#204—82no—
139C2×#41—61no—
140C2×Q16—62no—
141C2×SD16—62no—
142C4⋊Q8—31no—
143C4⋊Q8—31no—
144C4⋊Q8—31no—
145#127—31no—
146#125—61no—
147#124—61no—
148C22⋊Q8—32no—
149C22⋊Q8—61no—
150#121—61no—
151#121—61no—
152C4×Q8—32no—
153C4×Q8—61no—
154C4×D4—61no—
155C4×D4—61no—
156#117—61no—
157C2×C42—21no—
158C2×C42C22×D311yescharacteristic
159C4⋊C8—21no—
160C4⋊C8—21no—
161C4⋊C8—62no—
162C4⋊C8—62no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165#103—62no—
166#102—62no—
167C22⋊C8—21no—
168C22⋊C8—21no—
169C22⋊C8—32no—
170C22⋊C8—61no—
171#97—62no—
172#97—61no—
173#97—61no—
174C4×C8—62no—
175C4×C8C4×D311yescharacteristic
176C4×C8C4×D311yescharacteristic
177C4×C8D1211yescharacteristic
178C4×C8D1211yescharacteristic
179#95—21no—
180#95—21no—
181C23×C3#12111yescharacteristic
182C22×D3—41no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—42no—
188C2×C3×C4—81no—
189C2×C3×C4—81no—
190C2×C3×C4C2×C22⋊C411yescharacteristic
191C2×C3×C4C4×D411yescharacteristic
192C2×C3×C4C4×D411yescharacteristic
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12311yescharacteristic
196#67—82no—
197C2×Dic3—22no—
198C2×Dic3—22no—
199C2×Dic3—41no—
200D12—42no—
201D12—41no—
202D12—41no—
203D12—81no—
204C4×D3—82no—
205Dic6—41no—
206Dic6—41no—
207Dic6—42no—
208Dic6—42no—
209C3×C8—41no—
210C3×C8—41no—
211C3×C8—41no—
212C3×C8—41no—
213C3⋊C8—42no—
214C3⋊C8—81no—
215#41—242no—
216C2×Q8—32no—
217C2×Q8—61no—
218C2×Q8—61no—
219C2×D4—61no—
220C2×D4—61no—
221C2×D4—61no—
222C2×D4—121no—
223C22×C4—41no—
224C22×C4—121no—
225C22×C4#23511yescharacteristic
226C22×C4D3×D411yescharacteristic
227C22×C4D3×D411yescharacteristic
228Q16—242no—
229SD16—242no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—62no—
235C2×C8—121no—
236C4⋊C4—32no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—62no—
241C4⋊C4—62no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C22⋊C4—62no—
245C22⋊C4—121no—
246C22⋊C4—121no—
247C42—22no—
248C42—41no—
249C42—62no—
250C42C2×C4×D311yescharacteristic
251C42C2×D1211yescharacteristic
252C42D3×D411yescharacteristic
253C42#23911yescharacteristic
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—82no—
267C3×C4#44311yescharacteristic
268C3×C4#44411yescharacteristic
269Dic3—42no—
270Dic3—42no—
271Dic3—81no—
272C23—121no—
273C23#88411yescharacteristic
274Q8—122no—
275Q8—122no—
276Q8—121no—
277Q8—121no—
278D4—122no—
279D4—121no—
280D4—121no—
281D4—241no—
282D4—242no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—62no—
289C2×C4—62no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—121no—
293C2×C4—242no—
294C2×C4C4×D1211yescharacteristic
295C2×C4D3×C22⋊C411yescharacteristic
296C2×C4#88211yescharacteristic
297C2×C4#89311yescharacteristic
298C2×C4#89411yescharacteristic
299C2×C4#89511yescharacteristic
300C8—41no—
301C8—41no—
302C8—41no—
303C8—41no—
304C8—122no—
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#466511yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—82no—
324C4—122no—
325C4—122no—
326C4—241no—
327C4#470011yescharacteristic
328C4#471711yescharacteristic
329C3#928211yescharacteristic, complemented, Sylow
330C2—41no—
331C2—481no—
332C2#6869511yescharacteristic
333C2#6870711yescharacteristic
334C2#6871311yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1274C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1274D3
hypocenterC3#9282
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
2. #79266C2semidirect
4. #69433C2semidirect
6. C3×#1274C2semidirect
7. #67747C2semidirect
20. #4699C4semidirect
21. #4694C4semidirect
22. #4693C4semidirect
23. #4681C4semidirect
36. #1274D3semidirect
329. C3#9282semidirect

Maximal subgroups

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

subgroupindex
#793062
#792662
#707652
#694332
#694322
C3×#12742
#677472
#92823