Finite Group Databasebeta

← #10693270

Subgroups — #10693270

Order 768.

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

normal subgroups only

This group has 396 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#79356C211yesmaximal, characteristic
2#79266C211yesmaximal, characteristic
3#70830C211yesmaximal, characteristic
4#70825C211yesmaximal, characteristic
5#70770C211yesmaximal, characteristic
6C3×#1287C211yesmaximal, characteristic
7#67758C211yesmaximal, characteristic
8#9528—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#4926C2211yescharacteristic
15#4924C2211yescharacteristic
16#4851—21no—
17#4849—21no—
18#4828—22no—
19#4739C412yes—
20#4721C412yes—
21#4720—22no—
22#4716—22no—
23#4679—21no—
24C8×Dic6—21no—
25#4486—21no—
26#4481—21no—
27#4469—22no—
28#4456—21no—
29#4456—21no—
30#3113—31no—
31#2985—31no—
32#1559—31no—
33#1487—31no—
34#1467—31no—
35#1307—31no—
36#1287D311yescharacteristic
37C22×Dic6C2×C411yescharacteristic
38C2×C3×C42C2311yescharacteristic
39#926—21no—
40C2×C4⋊Dic3C2×C411yescharacteristic
41#924—21no—
42#924C2×C412yes—
43C2×Dic3⋊C4—21no—
44C2×C4×Dic3—21no—
45C2×#208—21no—
46C2×Dic12—22no—
47#870—21no—
48#869D411yescharacteristic
49#869D411yescharacteristic
50C4×Dic6—21no—
51C4×Dic6—21no—
52C4×Dic6—21no—
53C4×Dic6—21no—
54C4×Dic6—21no—
55C4×Dic6D412yes—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×C22⋊C8C2×C412yes—
62C3×#97—21no—
63C3×#97—21no—
64C3×C4×C8D411yescharacteristic
65C3×C4×C8D411yescharacteristic
66#830D412yes—
67#817—22no—
68#816—22no—
69#816—22no—
70#815—22no—
71Dic3⋊C8—22no—
72Dic3⋊C8—22no—
73C8×Dic3—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77#803—21no—
78#533—31no—
79C2×C4×Q8—31no—
80#483—32no—
81#475—61no—
82#462—61no—
83#451—32no—
84#446—62no—
85#446—61no—
86C8×Q8—61no—
87#441—62no—
88#433—62no—
89#432—31no—
90#432C2×D311yescharacteristic
91#432C2×D311yescharacteristic
92#431C2×D311yescharacteristic
93#423—31no—
94#333—61no—
95#332—61no—
96Q8⋊C8—61no—
97Q8⋊C8—61no—
98#324—62no—
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4#4111yescharacteristic
102C22×Dic3—21no—
103C22×Dic3—21no—
104C2×Dic6—21no—
105C2×Dic6—21no—
106C2×Dic6—21no—
107C2×Dic6—42no—
108C2×Dic6C22⋊C412yes—
109C2×C3×C8—22no—
110C2×C3×C8—21no—
111C2×C3×C8—21no—
112C2×C3×C8—41no—
113C3×C42C2×D411yescharacteristic
114C3×C42C2×D411yescharacteristic
115C3×C42C2×D411yescharacteristic
116C3×C42#4111yescharacteristic
117#217—22no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3C22⋊C412yes—
124Dic3⋊C4—22no—
125Dic3⋊C4—22no—
126Dic3⋊C4—41no—
127Dic3⋊C4—41no—
128C4×Dic3—22no—
129C4×Dic3—41no—
130C4×Dic3—41no—
131#208—81no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—41no—
134Dic12—44no—
135C22×Q8—31no—
136C2×Q16—62no—
137C2×M16—61no—
138C4⋊Q8—31no—
139C4⋊Q8—31no—
140#125—61no—
141C22⋊Q8—32no—
142C22⋊Q8—61no—
143C4×Q8—32no—
144C4×Q8—61no—
145C4×Q8—61no—
146C4×Q8—61no—
147C4×Q8—61no—
148C4×Q8—61no—
149#117—61no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—61no—
152C2×C42—61no—
153C2×C42C22×D311yescharacteristic
154#106—62no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—62no—
159C4⋊C8—62no—
160C4⋊C8—61no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163#103—62no—
164#103—62no—
165C22⋊C8—21no—
166C22⋊C8—21no—
167C22⋊C8—32no—
168C22⋊C8C4×D312yes—
169#97—21no—
170#97—21no—
171#97—62no—
172#97—61no—
173C4×C8—62no—
174C4×C8D1211yescharacteristic
175C4×C8D1211yescharacteristic
176C23×C3C4×D411yescharacteristic
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4C2×C22⋊C411yescharacteristic
182C2×C3×C4C4×D411yescharacteristic
183C2×C3×C4#12011yescharacteristic
184C2×C3×C4#12111yescharacteristic
185C2×C3×C4#12111yescharacteristic
186C2×C3×C4#12311yescharacteristic
187C2×Dic3—22no—
188C2×Dic3—22no—
189C2×Dic3—22no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—81no—
194Dic6—22no—
195Dic6—22no—
196Dic6—41no—
197Dic6—41no—
198Dic6—41no—
199Dic6—41no—
200Dic6—42no—
201C3×C8—42no—
202C3×C8—41no—
203C3×C8—41no—
204C3×C8—81no—
205C3⋊C8—42no—
206C3⋊C8—81no—
207C2×Q8—32no—
208C2×Q8—61no—
209C2×Q8—61no—
210C2×Q8—61no—
211C2×Q8—122no—
212C22×C4—61no—
213C22×C4—61no—
214C22×C4C2×C4×D311yescharacteristic
215C22×C4D3×D411yescharacteristic
216C22×C4#23911yescharacteristic
217Q16—124no—
218M16—241no—
219C2×C8—22no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—41no—
223C2×C8—62no—
224C2×C8—121no—
225C4⋊C4—32no—
226C4⋊C4—62no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—62no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—121no—
234C4⋊C4—121no—
235C22⋊C4—62no—
236C42—62no—
237C42—121no—
238C42—121no—
239C42C2×D1211yescharacteristic
240C42#23511yescharacteristic
241C42D3×D411yescharacteristic
242C42D3×D411yescharacteristic
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#47111yescharacteristic
253C3×C4#47311yescharacteristic
254Dic3—42no—
255Dic3—42no—
256Dic3—42no—
257Dic3—81no—
258C23#89311yescharacteristic
259Q8—62no—
260Q8—62no—
261Q8—122no—
262Q8—121no—
263Q8—121no—
264Q8—121no—
265Q8—121no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—62no—
273C2×C4—121no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—241no—
277C2×C4C4×D1211yescharacteristic
278C2×C4D3×C22⋊C411yescharacteristic
279C2×C4#88211yescharacteristic
280C2×C4#88411yescharacteristic
281C2×C4#89411yescharacteristic
282C2×C4#89511yescharacteristic
283C8—42no—
284C8—41no—
285C8—41no—
286C8—81no—
287C8—122no—
288C8—241no—
289C2×C3—41no—
290C2×C3#189111yescharacteristic
291C2×C3#190511yescharacteristic
292C2×C3#190811yescharacteristic
293C22—21no—
294C22—21no—
295C22—21no—
296C22#466511yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—122no—
303C4—122no—
304C4—122no—
305C4—241no—
306C4#489611yescharacteristic
307C4#489811yescharacteristic
308C3#952811yescharacteristic, complemented, Sylow
309C2—41no—
310C2#6870711yescharacteristic
311C2#6894911yescharacteristic
312C2#6903611yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1287C2
socleC22×C3#386
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenter#1287D3
hypocenterC3#9528
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1287

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
#793562
#792662
#708302
#708252
#707702
C3×#12872
#677582
#95283