Finite Group Databasebeta

← #10699216

Subgroups — #10699216

Order 768.

374 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 99 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#83224C211yesmaximal, characteristic
2C3×#1997C211yesmaximal, characteristic
3#71779C211yesmaximal, characteristic
4#70509C211yesmaximal, characteristic
5#69868C211yesmaximal, characteristic
6#69850C211yesmaximal, characteristic
7#67715C211yesmaximal, characteristic
8#10506—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5093—21no—
14#5091—21no—
15#5091—21no—
16#5078—21no—
17#4970—22no—
18C4×C4⋊Dic3C2211yescharacteristic
19#4925C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×#343—21no—
22C3×#343—21no—
23#4555—21no—
24#4555—21no—
25#4529—22no—
26#4486—21no—
27#4481—21no—
28#4458—21no—
29#4457—21no—
30#3230—31no—
31#1997D311yescharacteristic
32#1987—31no—
33#1986—31no—
34#1565—31no—
35#1550—31no—
36#1303—31no—
37C22×C3×Q8—21no—
38C2×C3×M16—21no—
39C2×C3×M16D411yescharacteristic
40C2×C3×M16D411yescharacteristic
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D412yes—
47C2×C3×C42C2311yescharacteristic
48#926—21no—
49C2×C4⋊Dic3—21no—
50#924—21no—
51#924—21no—
52C2×C4×Dic3—21no—
53C22×C3⋊C8—21no—
54#870—21no—
55#869D411yescharacteristic
56#869D411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×#97—21no—
62C3×C4×C8—21no—
63C3⋊#95—21no—
64#830—21no—
65#810—22no—
66#810—22no—
67#807—22no—
68#807—22no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)D411yescharacteristic
72C4⋊(C3⋊C8)D411yescharacteristic
73C4×C3⋊C8—21no—
74#533—31no—
75C2×C4⋊Q8C2×D311yescharacteristic
76#500—61no—
77#487—61no—
78#484—62no—
79#477—61no—
80#477—61no—
81#431—31no—
82#423C2×D311yescharacteristic
83C2×C4⋊C8—31no—
84C4×M16C2×D311yescharacteristic
85C4×C4⋊C4—31no—
86#343—21no—
87#343—21no—
88#340—62no—
89#339—61no—
90#339—61no—
91#335—61no—
92#334—61no—
93#333—61no—
94#332—61no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8—82no—
98C22×C3×C4—22no—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4#4111yescharacteristic
102C22×Dic3—41no—
103C2×Dic6—21no—
104C2×Dic6—21no—
105C3×M16—21no—
106C3×M16—21no—
107C3×M16—21no—
108C3×M16—41no—
109C3×M16—41no—
110C3×M16Q1612yes—
111C2×C3×C8—21no—
112C2×C3×C8—21no—
113C2×C3×C8—41no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42#4111yescharacteristic
121C3×C42#4111yescharacteristic
122#217—41no—
123C4⋊Dic3—22no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—41no—
129Dic3⋊C4—41no—
130Dic3⋊C4—41no—
131C4×Dic3—41no—
132C4×Dic3—41no—
133C4×Dic3—41no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—42no—
137C2×C3⋊C8—41no—
138C22×Q8—21no—
139C2×M16—21no—
140C2×M16#6711yescharacteristic
141C2×M16#6711yescharacteristic
142C22×C8—61no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—31no—
148C4⋊Q8—31no—
149#125—61no—
150C22⋊Q8—61no—
151C22⋊Q8—61no—
152C4×Q8—61no—
153C4×Q8—61no—
154#117—61no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4D1212yes—
158C2×C42—61no—
159C2×C42C22×D311yescharacteristic
160#107—62no—
161#107—62no—
162C4⋊C8—21no—
163C4⋊C8—21no—
164C4⋊C8—31no—
165C4⋊C8—31no—
166C4⋊C8—61no—
167C4⋊C8—61no—
168#103—62no—
169#103—62no—
170C22⋊C8—61no—
171#97—21no—
172C4×C8—21no—
173C4×C8—61no—
174#95—61no—
175C23×C3#12111yescharacteristic
176C3×Q8—82no—
177C3×Q8—82no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—41no—
181C2×C3×C4—42no—
182C2×C3×C4—82no—
183C2×C3×C4#12011yescharacteristic
184C2×C3×C4#12111yescharacteristic
185C2×C3×C4#12111yescharacteristic
186C2×C3×C4#12311yescharacteristic
187C2×C3×C4#12311yescharacteristic
188C2×C3×C4#12411yescharacteristic
189C2×C3×C4C2×Q1611yescharacteristic
190C2×C3×C4#13511yescharacteristic
191C2×C3×C4#13711yescharacteristic
192C2×C3×C4#13711yescharacteristic
193C2×Dic3—42no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—81no—
197C2×Dic3—81no—
198Dic6—41no—
199Dic6—41no—
200Dic6—81no—
201C3×C8—22no—
202C3×C8—41no—
203C3×C8—81no—
204C3⋊C8—41no—
205C3⋊C8—41no—
206C3⋊C8—81no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—61no—
210C2×Q8—61no—
211C2×Q8—82no—
212C22×C4—22no—
213C22×C4—121no—
214C22×C4C2×D1211yescharacteristic
215C22×C4#23511yescharacteristic
216C22×C4C2×#6711yescharacteristic
217M16—21no—
218M16—21no—
219M16—21no—
220M16—41no—
221M16—41no—
222M16#21612yes—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—41no—
226C2×C8—61no—
227C2×C8—61no—
228C2×C8—122no—
229C2×C8—121no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—62no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C22⋊C4—121no—
243C42—121no—
244C42—121no—
245C42—121no—
246C42D3×D411yescharacteristic
247C42D3×D411yescharacteristic
248C42#23711yescharacteristic
249C42#23711yescharacteristic
250C22×C3—21no—
251C22×C3—21no—
252C22×C3#39411yescharacteristic
253C22×C3#46211yescharacteristic
254C22×C3#46411yescharacteristic
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—82no—
259C3×C4#45411yescharacteristic
260C3×C4#45611yescharacteristic
261C3×C4#46211yescharacteristic
262C3×C4#46411yescharacteristic
263C3×C4#46711yescharacteristic
264C3×C4#47011yescharacteristic
265Dic3—82no—
266Dic3—81no—
267Dic3—81no—
268C23#93011yescharacteristic
269Q8—82no—
270Q8—82no—
271Q8—121no—
272Q8—121no—
273Q8—241no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—122no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—241no—
283C2×C4—241no—
284C2×C4#88211yescharacteristic
285C2×C4#88311yescharacteristic
286C2×C4#88511yescharacteristic
287C2×C4#88611yescharacteristic
288C2×C4#93811yescharacteristic
289C2×C4#93911yescharacteristic
290C2×C4#94211yescharacteristic
291C2×C4C2×#21611yescharacteristic
292C2×C4#95011yescharacteristic
293C2×C4#95111yescharacteristic
294C8—22no—
295C8—41no—
296C8—81no—
297C8—121no—
298C8—121no—
299C8—241no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3#203811yescharacteristic
303C2×C3#204811yescharacteristic
304C2×C3#205611yescharacteristic
305C22—21no—
306C22—21no—
307C22#496111yescharacteristic
308C22#502811yescharacteristic
309C22#503111yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—82no—
314C4—242no—
315C4—241no—
316C4—241no—
317C4#475211yescharacteristic
318C4#475311yescharacteristic
319C4#505111yescharacteristic
320C4#505211yescharacteristic
321C4#505411yescharacteristic
322C4#509011yescharacteristic
323C3#1050611yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2#6968111yescharacteristic
327C2#7007611yescharacteristic
328C2#7010011yescharacteristic
subgroupG/H
centerC22#4961
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1997C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#1997D3
hypocenterC3#10506
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
323. C3#10506semidirect

Maximal subgroups

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

subgroupindex
#832242
C3×#19972
#717792
#705092
#698682
#698502
#677152
#105063