Finite Group Databasebeta

← #10692429

Subgroups — #10692429

Order 768.

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

normal subgroups only

This group has 420 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#83358C211yesmaximal, characteristic
2#79338C211yesmaximal, characteristic
3C3×#1537C211yesmaximal, characteristic, complemented
4#71824C211yesmaximal, characteristic, complemented
5#70816C211yesmaximal, characteristic
6#67757C211yesmaximal, characteristic, complemented
7#67755C211yesmaximal, characteristic, complemented
8#9373—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13#5089C412yes—
14#5086C412yes—
15#5047—22no—
16#5031—21no—
17#5027—21no—
18#4926C2211yescharacteristic
19#4925C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21#4838—21no—
22#4834—21no—
23#4727—22no—
24#4706—22no—
25C3×#330—22no—
26#4535—21no—
27#4535—21no—
28#4485—21no—
29#4481—21no—
30#3216—31no—
31#2981—31no—
32#1551—31no—
33#1537D311yescharacteristic, complemented
34#1481—31no—
35#1290—31no—
36#1287—31no—
37C2×#235D412yes—
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×C42C2311yescharacteristic
47C2×#216—22no—
48C2×#215—22no—
49#930—21no—
50#930—21no—
51C22×C3⋊C8—21no—
52C2×ΓL(1,𝔽25)—22no—
53C2×D3×C8—22no—
54#875C2×C412yes—
55#874D411yescharacteristic
56C4×D12—21no—
57C4×D12—21no—
58#869D411yescharacteristic
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×#97—21no—
64C3×#97D412yes—
65#830—21no—
66#830—21no—
67(C2×D3)⋊C8—22no—
68(C2×D3)⋊C8—22no—
69#810—22no—
70#809—22no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75#803C2×C412yes—
76C4×C3⋊C8D411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78#533C2×D311yescharacteristic
79#532—31no—
80#497—32no—
81#489—32no—
82#462—61no—
83#460—61no—
84#452—62no—
85#436—61no—
86#435—62no—
87C8×D4—61no—
88#432—31no—
89#432C2×D311yescharacteristic
90#432C2×D311yescharacteristic
91#431—31no—
92C2×C4⋊C8—31no—
93#408—62no—
94#333—61no—
95#331—61no—
96#330—22no—
97Q8⋊C8—61no—
98Q8⋊C8—61no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C2×#67—22no—
105#235—82no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12C22⋊C412yes—
109C2×C4×D3—22no—
110C2×Dic6C22⋊C412yes—
111C2×C3×C8—22no—
112C2×C3×C8—41no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119C3×C22⋊C4—41no—
120C3×C42—41no—
121C3×C42C22×C411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125#216—82no—
126#215—82no—
127(C2×D3)⋊C4—22no—
128(C2×D3)⋊C4—22no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C2×C3⋊C8—22no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—81no—
136ΓL(1,𝔽25)—82no—
137D3×C8—82no—
138C2×#41—32no—
139C2×Q16—62no—
140C2×SD16—62no—
141C2×M16—62no—
142C22×C8—62no—
143C22×C8—61no—
144C4⋊Q8—31no—
145C4⋊Q8C4×D311yescharacteristic
146C4⋊Q8C4×D311yescharacteristic
147#127—31no—
148#125—21no—
149#124—32no—
150C22⋊Q8—21no—
151C22⋊Q8—21no—
152#121—61no—
153#121—61no—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×D4—61no—
157C4×D4—61no—
158#117—21no—
159C2×C42C22×D311yescharacteristic
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166#103—62no—
167#102—62no—
168C22⋊C8—21no—
169C22⋊C8—21no—
170C22⋊C8—62no—
171C22⋊C8—62no—
172C22⋊C8—61no—
173C22⋊C8—61no—
174#97—21no—
175#97—32no—
176#97#6712yes—
177C4×C8—31no—
178C4×C8—31no—
179C23×C3#12311yescharacteristic
180C22×D3—22no—
181C3×Q8—41no—
182C3×Q8—41no—
183C3×Q8—42no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4C2×C22⋊C411yescharacteristic
191C2×C3×C4C4×D411yescharacteristic
192C2×C3×C4C4×D411yescharacteristic
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12111yescharacteristic
196#67—82no—
197C2×Dic3—22no—
198D12—42no—
199D12—41no—
200D12—41no—
201D12—81no—
202C4×D3—42no—
203C4×D3—42no—
204Dic6—42no—
205C3×C8—42no—
206C3×C8—81no—
207C3⋊C8—41no—
208C3⋊C8—41no—
209C3⋊C8—42no—
210C3⋊C8—81no—
211#41—242no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—32no—
215C2×D4—32no—
216C2×D4—62no—
217C2×D4—61no—
218C2×D4—61no—
219C22×C4—62no—
220C22×C4#23511yescharacteristic
221C22×C4D3×D411yescharacteristic
222C22×C4#23711yescharacteristic
223Q16—242no—
224SD16—242no—
225M16—242no—
226C2×C8—22no—
227C2×C8—41no—
228C2×C8—62no—
229C2×C8—61no—
230C2×C8—61no—
231C2×C8—121no—
232C2×C8—242no—
233C2×C8—241no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C22⋊C4—41no—
243C22⋊C4—62no—
244C22⋊C4—62no—
245C42—41no—
246C42C2×C4×D311yescharacteristic
247C42D3×D411yescharacteristic
248C42D3×D411yescharacteristic
249C42C2×#6711yescharacteristic
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3#38611yescharacteristic
254C2×D3—42no—
255C2×D3—42no—
256C2×D3—42no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—81no—
263C3×C4—81no—
264C3×C4#47111yescharacteristic
265C3×C4#47311yescharacteristic
266Dic3—42no—
267C23—62no—
268C23#88311yescharacteristic
269Q8—41no—
270Q8—41no—
271Q8—42no—
272Q8—122no—
273D4—122no—
274D4—121no—
275D4—121no—
276D4—241no—
277D4—242no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—62no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4D3×C22⋊C411yescharacteristic
288C2×C4#67⋊C411yescharacteristic
289C2×C4#88411yescharacteristic
290C2×C4C4×#6711yescharacteristic
291C2×C4#93711yescharacteristic
292C2×C4#93811yescharacteristic
293C8—42no—
294C8—81no—
295C8—121no—
296C8—121no—
297C8—122no—
298C8—241no—
299C2×C3—41no—
300C2×C3#188111yescharacteristic
301C2×C3#190811yescharacteristic
302C2×C3#191111yescharacteristic
303D3—82no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—122no—
308C22—122no—
309C22—122no—
310C22#495611yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—81no—
317C4—81no—
318C4—122no—
319C4#475411yescharacteristic
320C4#475511yescharacteristic
321C3#937311yescharacteristic, complemented, Sylow
322C2—41no—
323C2—242no—
324C2#7012311yescharacteristic
325C2#7040811yescharacteristic
326C2#7041611yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1537C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1537D3
hypocenterC3#9373
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
3. C3×#1537C2semidirect
4. #71824C2semidirect
6. #67757C2semidirect
7. #67755C2semidirect
33. #1537D3semidirect
321. C3#9373semidirect

Maximal subgroups

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

subgroupindex
#833582
#793382
C3×#15372
#718242
#708162
#677572
#677552
#93733