Finite Group Databasebeta

← #10708402

Subgroups — #10708402

Order 768.

348 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 61 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#80765C211yesmaximal, characteristic
2#71078C211yesmaximal, characteristic, complemented
3#69095C211yesmaximal, characteristic, complemented
4#68830C211yesmaximal, characteristic, complemented
5C3×#1348C211yesmaximal, characteristic, complemented
6#67522C211yesmaximal, characteristic, complemented
7#67485C211yesmaximal, characteristic, complemented
8#12280—31nomaximal, Sylow
9#5627—21no—
10C3×#467C2211yescharacteristic
11#5187—21no—
12#5115C2211yescharacteristic
13C2×(C2×D3)⋊C8C2211yescharacteristic
14C2×#818C2211yescharacteristic
15#5048—21no—
16#4987—21no—
17#4885—21no—
18#4814—21no—
19#4810—21no—
20C3×C22⋊C16C2211yescharacteristic
21C3×#340C2211yescharacteristic
22#4564—21no—
23#4549C2211yescharacteristic
24#4529—21no—
25#4509—21no—
26#4504—21no—
27#4490—21no—
28#3191—31no—
29#2235—31no—
30#2232—31no—
31#2075—31no—
32#1348D311yescharacteristic, complemented
33#1347—31no—
34#1314—31no—
35C2×#239—21no—
36C22×C4×D3D411yescharacteristic
37C2×C3×Q16D411yescharacteristic
38C22×C3×C8C2311yescharacteristic
39C3×C22⋊Q8—21no—
40C2×C3×C4⋊C4—21no—
41#946—21no—
42C2×#215—21no—
43#930—21no—
44C2×(C2×D3)⋊C4—41no—
45C2×C4⋊Dic3—21no—
46C22×C3⋊C8D411yescharacteristic
47C2×D24D411yescharacteristic
48#895—21no—
49#894—41no—
50#893—41no—
51#892—21no—
52#882—41no—
53#67⋊C4—41no—
54C2×C3×C16—21no—
55C3×#107D411yescharacteristic
56C3×#103—21no—
57#835—21no—
58C3⋊#95—41no—
59#821—21no—
60(C2×D3)⋊C8—21no—
61(C2×D3)⋊C8—21no—
62#818—21no—
63#818—21no—
64#818D411yescharacteristic
65C2×C3⋊C16—21no—
66#809—21no—
67#808—21no—
68#525—61no—
69#485—61no—
70#482—61no—
71#467C2×D311yescharacteristic
72#466—31no—
73#465—61no—
74#452—61no—
75#449—61no—
76C2×#107—31no—
77C2×C22⋊C8—31no—
78#380—61no—
79#367—61no—
80#366—61no—
81#358—61no—
82#357—61no—
83C22⋊C16—31no—
84C22⋊C16C2×D311yescharacteristic
85#340—61no—
86#340C2×D311yescharacteristic
87C23×D3D811yescharacteristic
88C2×C3×Q8—21no—
89C22×C3×C4—41no—
90C22×C3×C4C2×D411yescharacteristic
91C2×#67—41no—
92C22×Dic3—41no—
93C22×Dic3D811yescharacteristic
94#239—81no—
95#239—81no—
96C2×D12—21no—
97C2×D12—41no—
98C2×D12—41no—
99C2×C4×D3—21no—
100C2×C4×D3—21no—
101C2×C4×D3—41no—
102C2×C4×D3—41no—
103C2×C4×D3—41no—
104C3×Q16—41no—
105C2×C3×C8—41no—
106C2×C3×C8C2×D411yescharacteristic
107C2×C3×C8C2×D411yescharacteristic
108C3×C4⋊C4—21no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C22⋊C4—41no—
113#215—81no—
114(C2×D3)⋊C4—41no—
115(C2×D3)⋊C4—41no—
116(C2×D3)⋊C4—41no—
117(C2×D3)⋊C4—41no—
118(C2×D3)⋊C4—82no—
119C4⋊Dic3—21no—
120C4⋊Dic3—41no—
121C4⋊Dic3—41no—
122Dic3⋊C4—41no—
123C4×Dic3—41no—
124C2×C3⋊C8—21no—
125C2×C3⋊C8—41no—
126D24—41no—
127C3×C16—41no—
128C3⋊C16—41no—
129C2×#41—61no—
130C23×C4—31no—
131C2×Q16D1211yescharacteristic
132C2×SD16—61no—
133C2×D8—31no—
134C22×C8—31no—
135C22×C8C22×D311yescharacteristic
136#123—121no—
137C22⋊Q8—21no—
138C22⋊Q8—61no—
139#121—61no—
140#121—61no—
141#120—121no—
142C4×D4—121no—
143C4×D4—121no—
144#117—61no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—61no—
147C2×C22⋊C4—121no—
148C2×C16—21no—
149C2×C16—61no—
150#107—31no—
151#107—61no—
152#107—61no—
153#107D1211yescharacteristic
154#106—61no—
155#103—21no—
156#103—61no—
157#102—61no—
158#102—61no—
159C22⋊C8—61no—
160C22⋊C8—61no—
161#95—121no—
162C23×C3C2×D811yescharacteristic
163C22×D3—21no—
164C22×D3—21no—
165C22×D3—21no—
166C22×D3—41no—
167C22×D3—41no—
168C22×D3—41no—
169C3×Q8—41no—
170C3×Q8—41no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—81no—
176C2×C3×C4—81no—
177C2×C3×C4—81no—
178C2×C3×C4#12011yescharacteristic
179C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
180#67—81no—
181#67—81no—
182C2×Dic3—21no—
183C2×Dic3—41no—
184C2×Dic3—41no—
185C2×Dic3—81no—
186C2×Dic3—81no—
187C2×Dic3—81no—
188D12—41no—
189D12—41no—
190D12—82no—
191D12—82no—
192C4×D3—41no—
193C4×D3—41no—
194C4×D3—81no—
195C4×D3—82no—
196C3×C8—21no—
197C3×C8—21no—
198C3⋊C8—41no—
199C24—31no—
200#41—241no—
201#41—241no—
202C2×Q8—21no—
203C2×D4—61no—
204C2×D4—121no—
205C2×D4—121no—
206C2×D4—121no—
207C22×C4—31no—
208C22×C4—41no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—121no—
212C22×C4—121no—
213C22×C4—121no—
214C22×C4—121no—
215C22×C4D3×D411yescharacteristic
216Q16—41no—
217SD16—241no—
218D8—121no—
219C2×C8—41no—
220C2×C8—61no—
221C2×C8—121no—
222C2×C8C2×D1211yescharacteristic
223C2×C8D3×D411yescharacteristic
224C4⋊C4—21no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—61no—
229C4⋊C4—121no—
230C4⋊C4—121no—
231C4⋊C4—121no—
232C22⋊C4—41no—
233C22⋊C4—121no—
234C22⋊C4—121no—
235C22⋊C4—121no—
236C22⋊C4—121no—
237C22⋊C4—242no—
238C42—121no—
239C16—41no—
240C16—121no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3#44711yescharacteristic
244C22×C3#50811yescharacteristic
245C22×C3#50911yescharacteristic
246C2×D3—42no—
247C2×D3—41no—
248C2×D3—41no—
249C2×D3—41no—
250C2×D3—81no—
251C2×D3—81no—
252C2×D3—81no—
253C2×D3—81no—
254C3×C4—21no—
255C3×C4—81no—
256C3×C4—81no—
257C3×C4—161no—
258C3×C4#45511yescharacteristic
259C3×C4#45811yescharacteristic
260Dic3—41no—
261Dic3—81no—
262Dic3—161no—
263C23—61no—
264C23—61no—
265C23—61no—
266C23—121no—
267C23—121no—
268C23—121no—
269C23D3×D811yescharacteristic
270Q8—41no—
271Q8—41no—
272D4—121no—
273D4—121no—
274D4—242no—
275D4—242no—
276D4—241no—
277D4—241no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—61no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—241no—
291C2×C4—242no—
292C2×C4—241no—
293C2×C4—241no—
294C2×C4—241no—
295C2×C4#88211yescharacteristic
296C2×C4#91411yescharacteristic
297C8—21no—
298C8—21no—
299C8—121no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3#160711yescharacteristic
303C2×C3#218611yescharacteristic
304C2×C3#219111yescharacteristic
305D3—81no—
306D3—161no—
307C22—21no—
308C22—21no—
309C22—122no—
310C22—121no—
311C22—121no—
312C22—121no—
313C22—241no—
314C22—241no—
315C22—241no—
316C22—241no—
317C22#477311yescharacteristic
318C22#491411yescharacteristic
319C22#491911yescharacteristic
320C4—21no—
321C4—81no—
322C4—81no—
323C4—121no—
324C4—161no—
325C4—241no—
326C4—481no—
327C4#474211yescharacteristic
328C4#482511yescharacteristic
329C3#1228011yescharacteristic, complemented, Sylow
330C2—21no—
331C2—21no—
332C2—241no—
333C2—481no—
334C2#6828811yescharacteristic
335C2#6922411yescharacteristic
336C2#6925311yescharacteristic
subgroupG/H
centerC22#4773
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC3×#1348C2
socleC22×C3#447
prime radical#80765C2
hypercenter#1348D3
hypocenterC3#12280
LCSG ▷ C22×C3×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1348

Complements

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

NH ≅ G/Ntype
2. #71078C2semidirect
3. #69095C2semidirect
4. #68830C2semidirect
5. C3×#1348C2semidirect
6. #67522C2semidirect
7. #67485C2semidirect
32. #1348D3semidirect
329. C3#12280semidirect

Maximal subgroups

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

subgroupindex
#807652
#710782
#690952
#688302
C3×#13482
#675222
#674852
#122803