Finite Group Databasebeta

← #10693282

Subgroups — #10693282

Order 768.

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

normal subgroups only

This group has 428 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#79344C211yesmaximal, characteristic
2#79316C211yesmaximal, characteristic, complemented
3#70827C211yesmaximal, characteristic
4#70510C211yesmaximal, characteristic, complemented
5#70496C211yesmaximal, characteristic
6C3×#1307C211yesmaximal, characteristic, complemented
7#67757C211yesmaximal, characteristic, complemented
8#9530—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#4926C2211yescharacteristic
15#4925C2211yescharacteristic
16#4846—21no—
17#4841—21no—
18#4831—21no—
19#4830—21no—
20#4731C412yes—
21#4722C412yes—
22#4720—22no—
23#4710—22no—
24#4701—22no—
25#4483—21no—
26#4483—21no—
27#4482—21no—
28#4481—21no—
29#4468—22no—
30#3110—31no—
31#2985—31no—
32#1559—31no—
33#1486—31no—
34#1466—31no—
35#1307D311yescharacteristic, complemented
36#1287—31no—
37C22×D12C2×C411yescharacteristic
38C22×C4×D3—21no—
39C2×C3×M16—21no—
40C2×C3×C42C2311yescharacteristic
41C2×(C2×D3)⋊C4—21no—
42#926—21no—
43C2×C4⋊Dic3C2×C411yescharacteristic
44#924—21no—
45#924C2×C412yes—
46C2×#204—22no—
47C4×D12—21no—
48C4×D12—21no—
49C4×D12—21no—
50C4×D12—21no—
51#870—21no—
52#869D411yescharacteristic
53#869D411yescharacteristic
54C4×Dic6—21no—
55C4×Dic6D412yes—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C22⋊C8C2×C412yes—
61C3×C22⋊C8D412yes—
62C3×#97—21no—
63C3×#97—21no—
64#830—21no—
65#830—21no—
66#821—22no—
67#818—22no—
68#816—22no—
69#815—22no—
70Dic3⋊C8—22no—
71Dic3⋊C8—22no—
72C8×Dic3—22no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75#803—21no—
76C4×C3⋊C8D411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78#533—31no—
79C2×C4×D4—31no—
80#484—32no—
81#474—61no—
82#461—61no—
83#450—32no—
84#446—61no—
85#446—62no—
86C8×Q8—61no—
87#440—62no—
88#433—62no—
89#432—31no—
90#432C2×D311yescharacteristic
91#432C2×D311yescharacteristic
92#431—31no—
93#423C2×D311yescharacteristic
94#333—61no—
95#332—61no—
96D4⋊C8—61no—
97D4⋊C8—61no—
98#323—62no—
99C23×D3—21no—
100C22×C3×C4C22×C411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C22×Dic3—21no—
104C2×D12—21no—
105C2×D12—21no—
106C2×D12—42no—
107C2×C4×D3—41no—
108C2×C4×D3—41no—
109C2×C4×D3—81no—
110C2×C4×D3—81no—
111C2×Dic6—21no—
112C2×Dic6C22⋊C412yes—
113C3×M16—81no—
114C2×C3×C8—22no—
115C2×C3×C8—22no—
116C2×C3×C8—41no—
117C3×C42C2×D411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42#4111yescharacteristic
121#217—22no—
122(C2×D3)⋊C4—41no—
123(C2×D3)⋊C4—41no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3C22⋊C412yes—
130Dic3⋊C4—22no—
131Dic3⋊C4—22no—
132C4×Dic3—22no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—41no—
136#204—44no—
137C22×D4—31no—
138C23×C4—61no—
139C2×SD16—62no—
140C2×M16—21no—
141C4⋊Q8—31no—
142C4⋊Q8—31no—
143#125—61no—
144C22⋊Q8—32no—
145C22⋊Q8—61no—
146C4×Q8—32no—
147C4×Q8—61no—
148C4×D4—61no—
149C4×D4—61no—
150C4×D4—61no—
151C4×D4—61no—
152#117—61no—
153C2×C4⋊C4—31no—
154C2×C22⋊C4—61no—
155C2×C42C22×D311yescharacteristic
156#107—62no—
157C4⋊C8—21no—
158C4⋊C8—21no—
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—61no—
168C22⋊C8—61no—
169C22⋊C8C4×D312yes—
170C22⋊C8D1212yes—
171#97—21no—
172#97—21no—
173#97—62no—
174#97—61no—
175C4×C8—31no—
176C4×C8—31no—
177C4×C8—62no—
178C23×C3C4×D411yescharacteristic
179C22×D3—22no—
180C22×D3—41no—
181C22×D3—41no—
182C22×D3—81no—
183C22×D3—81no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4C2×C22⋊C411yescharacteristic
189C2×C3×C4C4×D411yescharacteristic
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12311yescharacteristic
194C2×Dic3—22no—
195C2×Dic3—22no—
196C2×Dic3—41no—
197C2×Dic3—81no—
198D12—22no—
199D12—22no—
200D12—41no—
201D12—41no—
202C4×D3—82no—
203C4×D3—81no—
204C4×D3—81no—
205Dic6—41no—
206Dic6—41no—
207Dic6—42no—
208C3×C8—42no—
209C3×C8—42no—
210C3×C8—81no—
211C3⋊C8—41no—
212C3⋊C8—41no—
213C3⋊C8—81no—
214C24—61no—
215C2×Q8—32no—
216C2×Q8—61no—
217C2×D4—61no—
218C2×D4—61no—
219C2×D4—122no—
220C22×C4—61no—
221C22×C4—121no—
222C22×C4—121no—
223C22×C4—241no—
224C22×C4—241no—
225C22×C4C2×C4×D311yescharacteristic
226C22×C4C2×D1211yescharacteristic
227C22×C4#23511yescharacteristic
228SD16—124no—
229M16—81no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—41no—
233C2×C8—61no—
234C2×C8—61no—
235C2×C8—121no—
236C4⋊C4—32no—
237C4⋊C4—61no—
238C4⋊C4—62no—
239C4⋊C4—62no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C22⋊C4—62no—
245C22⋊C4—121no—
246C22⋊C4—121no—
247C42—62no—
248C42D3×D411yescharacteristic
249C42D3×D411yescharacteristic
250C42D3×D411yescharacteristic
251C42#23911yescharacteristic
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3#38611yescharacteristic
256C2×D3—42no—
257C2×D3—82no—
258C2×D3—81no—
259C2×D3—81no—
260C2×D3—82no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4#47111yescharacteristic
267C3×C4#47211yescharacteristic
268Dic3—42no—
269Dic3—42no—
270Dic3—81no—
271C23—62no—
272C23—121no—
273C23—121no—
274C23—241no—
275C23—241no—
276C23C4×D1211yescharacteristic
277Q8—122no—
278Q8—121no—
279Q8—121no—
280D4—62no—
281D4—62no—
282D4—121no—
283D4—121no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—62no—
289C2×C4—62no—
290C2×C4—121no—
291C2×C4—242no—
292C2×C4—241no—
293C2×C4—241no—
294C2×C4—241no—
295C2×C4D3×C22⋊C411yescharacteristic
296C2×C4#88211yescharacteristic
297C2×C4#88411yescharacteristic
298C2×C4#89311yescharacteristic
299C2×C4#89411yescharacteristic
300C2×C4#89511yescharacteristic
301C8—42no—
302C8—42no—
303C8—81no—
304C8—121no—
305C8—121no—
306C8—241no—
307C2×C3—41no—
308C2×C3#189111yescharacteristic
309C2×C3#190511yescharacteristic
310C2×C3#190711yescharacteristic
311D3—82no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—122no—
316C22—242no—
317C22—242no—
318C22—241no—
319C22—241no—
320C22#466511yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—122no—
327C4—122no—
328C4—241no—
329C4#474911yescharacteristic
330C4#475411yescharacteristic
331C3#953011yescharacteristic, complemented, Sylow
332C2—41no—
333C2—242no—
334C2#6900611yescharacteristic
335C2#6903011yescharacteristic
336C2#6903611yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1307C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1307D3
hypocenterC3#9530
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1307

Complements

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

NH ≅ G/Ntype
2. #79316C2semidirect
4. #70510C2semidirect
6. C3×#1307C2semidirect
7. #67757C2semidirect
35. #1307D3semidirect
331. C3#9530semidirect

Maximal subgroups

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

subgroupindex
#793442
#793162
#708272
#705102
#704962
C3×#13072
#677572
#95303