Finite Group Databasebeta

← #10693269

Subgroups — #10693269

Order 768.

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

normal subgroups only

This group has 396 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
1C3×#2985C211yesmaximal, characteristic
2#83369C211yesmaximal, characteristic
3#71833C211yesmaximal, characteristic
4#70830C211yesmaximal, characteristic
5#70770C211yesmaximal, characteristic
6#67758C211yesmaximal, characteristic
7#67747C211yesmaximal, characteristic
8#9528—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#446—22no—
12C3×#446—21no—
13C3×C8×Q8—21no—
14C3×#433—22no—
15C3×#432C2211yescharacteristic
16C3×#431C2211yescharacteristic
17#5094—21no—
18#5090—21no—
19#5050C412yes—
20#5043C412yes—
21#5030—22no—
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24#4924C2211yescharacteristic
25#4547—21no—
26#4546—21no—
27#4469—22no—
28#4456—21no—
29#4456—21no—
30#3113—31no—
31#2985D311yescharacteristic
32#1559—31no—
33#1487—31no—
34#1467—31no—
35#1307—31no—
36#1287—31no—
37C22×Dic6C2×C411yescharacteristic
38C3×C4⋊Q8D411yescharacteristic
39C3×C4⋊Q8D411yescharacteristic
40C3×#125—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8C2×C412yes—
43C3×C4×Q8—21no—
44C3×C4×Q8D412yes—
45C3×#117—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#216—22no—
48C2×C4⋊Dic3C2×C411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×C4×Dic3—21no—
51C2×#208—21no—
52C4×Dic6—21no—
53C4×Dic6—21no—
54C4×Dic6—21no—
55C4×Dic6—21no—
56C3×C4⋊C8—22no—
57C3×C4⋊C8—22no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×C22⋊C8D412yes—
62C3×#97—22no—
63C3×#97—21no—
64C3×C4×C8—22no—
65C3×C4×C8D411yescharacteristic
66C3×C4×C8D411yescharacteristic
67#835—22no—
68#830—21no—
69#830C2×C412yes—
70#810—22no—
71#808—22no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76#803—21no—
77#803—21no—
78#533C2×D311yescharacteristic
79C2×C4×Q8—31no—
80#483—32no—
81#475—61no—
82#462—61no—
83#451—32no—
84#446—22no—
85#446—21no—
86C8×Q8—21no—
87#441—62no—
88#433—22no—
89#432—31no—
90#432—31no—
91#432C2×D311yescharacteristic
92#431C2×D311yescharacteristic
93#423—31no—
94#333—61no—
95#332—61no—
96Q8⋊C8—61no—
97Q8⋊C8—61no—
98#324—62no—
99C2×C3×Q8—21no—
100C2×C3×Q8C22⋊C412yes—
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×Dic3—21no—
105C22×Dic3—21no—
106C2×Dic6—21no—
107C2×Dic6—21no—
108C2×Dic6—42no—
109C2×C3×C8—21no—
110C2×C3×C8—22no—
111C2×C3×C8—21no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4C22⋊C412yes—
118C3×C22⋊C4—22no—
119C3×C42—22no—
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42#4111yescharacteristic
124#216—44no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127Dic3⋊C4—41no—
128Dic3⋊C4—41no—
129C4×Dic3—41no—
130C4×Dic3—41no—
131#208—81no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—41no—
134C2×C3⋊C8—41no—
135C22×Q8—31no—
136C2×Q16—62no—
137C2×M16—61no—
138C4⋊Q8D1211yescharacteristic
139C4⋊Q8D1211yescharacteristic
140#125—21no—
141C22⋊Q8—21no—
142C22⋊Q8C4×D312yes—
143C4×Q8—21no—
144C4×Q8—61no—
145C4×Q8—61no—
146C4×Q8—61no—
147C4×Q8—61no—
148C4×Q8#6712yes—
149#117—21no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—61no—
152C2×C42—61no—
153C2×C42C22×D311yescharacteristic
154#106—62no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163#103—62no—
164#103—62no—
165C22⋊C8—21no—
166C22⋊C8—32no—
167C22⋊C8—61no—
168C22⋊C8#6712yes—
169#97—22no—
170#97—21no—
171#97—61no—
172#97—61no—
173C4×C8—22no—
174C4×C8#6711yescharacteristic
175C4×C8#6711yescharacteristic
176C23×C3C4×D411yescharacteristic
177C3×Q8—42no—
178C3×Q8—41no—
179C3×Q8—41no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4C2×C22⋊C411yescharacteristic
187C2×C3×C4C4×D411yescharacteristic
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12311yescharacteristic
192C2×Dic3—22no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—81no—
197Dic6—22no—
198Dic6—22no—
199Dic6—41no—
200Dic6—41no—
201C3×C8—41no—
202C3×C8—42no—
203C3×C8—41no—
204C3⋊C8—42no—
205C3⋊C8—81no—
206C3⋊C8—81no—
207C2×Q8—21no—
208C2×Q8—61no—
209C2×Q8—61no—
210C2×Q8—122no—
211C2×Q8(C2×D3)⋊C412yes—
212C22×C4—61no—
213C22×C4—61no—
214C22×C4C2×C4×D311yescharacteristic
215C22×C4#23511yescharacteristic
216C22×C4C2×#6711yescharacteristic
217Q16—124no—
218M16—241no—
219C2×C8—21no—
220C2×C8—22no—
221C2×C8—21no—
222C2×C8—62no—
223C2×C8—121no—
224C2×C8—121no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—121no—
233C4⋊C4—121no—
234C4⋊C4(C2×D3)⋊C412yes—
235C22⋊C4—22no—
236C42—22no—
237C42—121no—
238C42—121no—
239C42C2×D1211yescharacteristic
240C42#23511yescharacteristic
241C42C2×#6711yescharacteristic
242C42C2×#6711yescharacteristic
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3#38611yescharacteristic
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—42no—
253C3×C4—42no—
254C3×C4#47111yescharacteristic
255C3×C4#47311yescharacteristic
256Dic3—42no—
257Dic3—81no—
258C23C4×#6711yescharacteristic
259Q8—42no—
260Q8—41no—
261Q8—41no—
262Q8—62no—
263Q8—62no—
264Q8—121no—
265Q8—121no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—62no—
273C2×C4—121no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—241no—
277C2×C4C2×(C2×D3)⋊C411yescharacteristic
278C2×C4C4×#6711yescharacteristic
279C2×C4#92911yescharacteristic
280C2×C4#93011yescharacteristic
281C2×C4#93011yescharacteristic
282C2×C4#95311yescharacteristic
283C8—41no—
284C8—42no—
285C8—41no—
286C8—122no—
287C8—241no—
288C8—241no—
289C2×C3—41no—
290C2×C3#189111yescharacteristic
291C2×C3#190511yescharacteristic
292C2×C3#190811yescharacteristic
293C22—21no—
294C22—21no—
295C22—21no—
296C22#521311yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—122no—
305C4—241no—
306C4#514111yescharacteristic
307C4#514311yescharacteristic
308C3#952811yescharacteristic, complemented, Sylow
309C2—41no—
310C2#7166211yescharacteristic
311C2#7201911yescharacteristic
312C2#7202911yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2985C2
socleC22×C3#386
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenter#2985D3
hypocenterC3#9528
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2985

Complements

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

NH ≅ G/Ntype
308. C3#9528semidirect

Maximal subgroups

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

subgroupindex
C3×#29852
#833692
#718332
#708302
#707702
#677582
#677472
#95283