Finite Group Databasebeta

← #10691771

Subgroups — #10691771

Order 768.

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

normal subgroups only

This group has 422 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#83359C211yesmaximal, characteristic
2#79338C211yesmaximal, characteristic
3C3×#1506C211yesmaximal, characteristic, complemented
4#71827C211yesmaximal, characteristic, complemented
5#69432C211yesmaximal, characteristic
6#67757C211yesmaximal, characteristic, complemented
7#67614C211yesmaximal, characteristic, complemented
8#9279—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#376C2211yescharacteristic
13#5088C411yescharacteristic, complemented
14#5088C411yescharacteristic, complemented
15#5080C411yescharacteristic, complemented
16#5078C411yescharacteristic, complemented
17#5066—22no—
18#5003—22no—
19#4926C2211yescharacteristic
20#4925C2211yescharacteristic
21#4925C2211yescharacteristic
22#4838—21no—
23#4834—21no—
24#4727—22no—
25#4706—22no—
26C3×#328—22no—
27#4535—21no—
28#4535—21no—
29#4451—22no—
30#3228—31no—
31#2981—31no—
32#1506D311yescharacteristic, complemented
33#1505—31no—
34#1497—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⋊Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×#117—21no—
46C2×C3×C42—21no—
47C2×C3×C42C2311yescharacteristic
48#930—21no—
49#930D412yes—
50C2×ΓL(1,𝔽25)—22no—
51C2×D3×C8—22no—
52#875—21no—
53#874C2×C411yescharacteristic
54C4×D12—21no—
55C4×D12D412yes—
56#869C2×C411yescharacteristic
57C3×C4⋊C8—21no—
58C3×C22⋊C8D412yes—
59C3×#97—21no—
60C3×#95—21no—
61C3×#95—21no—
62#830—21no—
63#830—21no—
64#830—21no—
65(C2×D3)⋊C8—22no—
66(C2×D3)⋊C8—22no—
67#809—22no—
68#809—22no—
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#533C2×D311yescharacteristic
80#532—31no—
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—22no—
97#327—62no—
98Q8⋊C8—61no—
99Q8⋊C8—61no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C22×C3×C4—41no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C2×#67—22no—
107#235—82no—
108C2×D12—21no—
109C2×D12—21no—
110C2×D12C22⋊C412yes—
111C2×C4×D3—22no—
112C2×Dic6—21no—
113C2×C3×C8—22no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C42—22no—
122C3×C42—41no—
123C3×C42—41no—
124C3×C42C22×C411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42#4111yescharacteristic
128(C2×D3)⋊C4—22no—
129(C2×D3)⋊C4—22no—
130C4⋊Dic3—21no—
131C4⋊Dic3C22⋊C412yes—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—41no—
137ΓL(1,𝔽25)—82no—
138D3×C8—82no—
139C2×#41—61no—
140C2×M16—62no—
141C22×C8—62no—
142C4⋊Q8—31no—
143C4⋊Q8C4×D311yescharacteristic
144C4⋊Q8C4×D311yescharacteristic
145#127—31no—
146#125—21no—
147#124—61no—
148C22⋊Q8—21no—
149C22⋊Q8—21no—
150#121—32no—
151#121—61no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×D4—32no—
155C4×D4—61no—
156#117—21no—
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—62no—
168C22⋊C8—62no—
169C22⋊C8—61no—
170C22⋊C8—61no—
171C22⋊C8—61no—
172C22⋊C8#6712yes—
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#12111yescharacteristic
182C22×D3—22no—
183C3×Q8—41no—
184C3×Q8—41no—
185C3×Q8—81no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—42no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—81no—
194C2×C3×C4—81no—
195C2×C3×C4C2×C22⋊C411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4C4×D411yescharacteristic
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12311yescharacteristic
201#67—82no—
202C2×Dic3—22no—
203D12—42no—
204D12—41no—
205D12—41no—
206D12—81no—
207C4×D3—42no—
208C4×D3—42no—
209Dic6—81no—
210C3×C8—42no—
211C3⋊C8—41no—
212C3⋊C8—41no—
213C3⋊C8—41no—
214C3⋊C8—41no—
215C3⋊C8—81no—
216#41—242no—
217C2×Q8—21no—
218C2×Q8—21no—
219C2×Q8—61no—
220C2×D4—32no—
221C2×D4—62no—
222C2×D4—61no—
223C2×D4—61no—
224C22×C4—41no—
225C22×C4—62no—
226C22×C4D3×D411yescharacteristic
227C22×C4#23711yescharacteristic
228C22×C4C2×#6711yescharacteristic
229M16—242no—
230C2×C8—22no—
231C2×C8—61no—
232C2×C8—61no—
233C2×C8—61no—
234C2×C8—61no—
235C2×C8—121no—
236C2×C8—242no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—32no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—61no—
245C22⋊C4—41no—
246C22⋊C4—62no—
247C22⋊C4—62no—
248C42—22no—
249C42—41no—
250C42—41no—
251C42C2×C4×D311yescharacteristic
252C42#23511yescharacteristic
253C42D3×D411yescharacteristic
254C42D3×D411yescharacteristic
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3#38611yescharacteristic
259C2×D3—42no—
260C2×D3—42no—
261C2×D3—42no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—82no—
268C3×C4—81no—
269C3×C4—81no—
270C3×C4#44311yescharacteristic
271C3×C4#44411yescharacteristic
272Dic3—42no—
273C23—62no—
274C23#93811yescharacteristic
275Q8—41no—
276Q8—41no—
277Q8—81no—
278Q8—241no—
279D4—122no—
280D4—121no—
281D4—121no—
282D4—241no—
283D4—242no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—62no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4D3×C22⋊C411yescharacteristic
297C2×C4#67⋊C411yescharacteristic
298C2×C4#88311yescharacteristic
299C2×C4#88411yescharacteristic
300C2×C4C4×#6711yescharacteristic
301C2×C4#93711yescharacteristic
302C8—42no—
303C8—121no—
304C8—121no—
305C8—121no—
306C8—121no—
307C8—241no—
308C2×C3—41no—
309C2×C3#187611yescharacteristic
310C2×C3#190411yescharacteristic
311C2×C3#190911yescharacteristic
312D3—82no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—122no—
317C22—122no—
318C22—122no—
319C22#495611yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—82no—
326C4—81no—
327C4—81no—
328C4—122no—
329C4#482311yescharacteristic
330C4#482411yescharacteristic
331C3#927911yescharacteristic, complemented, Sylow
332C2—41no—
333C2—242no—
334C2#7004911yescharacteristic
335C2#7035011yescharacteristic
336C2#7039011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1506C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1506D3
hypocenterC3#9279
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
3. C3×#1506C2semidirect
4. #71827C2semidirect
6. #67757C2semidirect
7. #67614C2semidirect
13. #5088C4semidirect
14. #5088C4semidirect
15. #5080C4semidirect
16. #5078C4semidirect
32. #1506D3semidirect
331. C3#9279semidirect

Maximal subgroups

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

subgroupindex
#833592
#793382
C3×#15062
#718272
#694322
#677572
#676142
#92793