Finite Group Databasebeta

← #10692162

Subgroups — #10692162

Order 768.

398 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 85 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#82949C211yesmaximal, characteristic
2#79317C211yesmaximal, characteristic, complemented
3C3×#1469C211yesmaximal, characteristic
4#71719C211yesmaximal, characteristic
5#70781C211yesmaximal, characteristic, complemented
6C3⋊#1308C211yesmaximal, characteristic, complemented
7#67747C211yesmaximal, characteristic, complemented
8#9334—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#431C2211yescharacteristic
13#5228C412yes—
14#5218C412yes—
15#5060—21no—
16#5055—21no—
17#5054—22no—
18#4926C2211yescharacteristic
19#4926C2211yescharacteristic
20#4926C2211yescharacteristic
21#4828—22no—
22#4720—22no—
23#4701—22no—
24C3×#323—22no—
25#4540—22no—
26#4486—21no—
27#4481—21no—
28#3215—31no—
29#2987—31no—
30#1554—31no—
31#1533—31no—
32#1469D311yescharacteristic
33#1308—31no—
34#1287—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C22×C3×D4C2×C411yescharacteristic
37C3×#124—21no—
38C3×#124—21no—
39C3×#124—21no—
40C3×#124—21no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C42C2311yescharacteristic
44C2×#216—22no—
45C2×#214—22no—
46#926—21no—
47#924—21no—
48#924C2×C412yes—
49#870—21no—
50#869D411yescharacteristic
51#869D411yescharacteristic
52C4×Dic6—21no—
53C4×Dic6D412yes—
54C3×C4⋊C8—21no—
55C3×C4⋊C8—21no—
56C3×C22⋊C8—21no—
57C3×C22⋊C8D412yes—
58C3×#97—21no—
59C3×C4×C8—21no—
60#835—22no—
61#832—22no—
62#830—21no—
63#830—21no—
64#830C2×C412yes—
65#815—22no—
66Dic3⋊C8—24no—
67C8×Dic3—22no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71#803—21no—
72#803—21no—
73#803D412yes—
74#533—31no—
75C2×#124C2×D311yescharacteristic
76#497—61no—
77#489—61no—
78#464—62no—
79#451—32no—
80#450—32no—
81#446—62no—
82#446—62no—
83#433—62no—
84#432—31no—
85#432—31no—
86#432—31no—
87#432C2×D311yescharacteristic
88#431C2×D311yescharacteristic
89#333—61no—
90#332—61no—
91#329—62no—
92#323—22no—
93C24×C3—21no—
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C2×C3×Q8—81no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4—81no—
100C22×C3×C4—21no—
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C2×Dic6—21no—
105C2×Dic6C22⋊C412yes—
106C2×C3×C8—22no—
107C2×C3×C8—41no—
108C3×C22⋊C4—41no—
109C3×C22⋊C4—41no—
110C3×C22⋊C4—41no—
111C3×C22⋊C4—41no—
112C3×C42C2×D411yescharacteristic
113C3×C42C2×D411yescharacteristic
114C3×C42C2×D411yescharacteristic
115C3×C42#4111yescharacteristic
116#217—22no—
117#216—82no—
118#214—82no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3C22⋊C412yes—
123Dic3⋊C4—22no—
124Dic3⋊C4—22no—
125C4×Dic3—22no—
126C2×C3⋊C8—22no—
127C2×C3⋊C8—22no—
128C2×C3⋊C8—41no—
129C22×Q8C4×D311yescharacteristic
130C22×D4C4×D311yescharacteristic
131C2×Q16—62no—
132C2×SD16—62no—
133C4⋊Q8—31no—
134C4⋊Q8—31no—
135#125—61no—
136#124—21no—
137#124—21no—
138#124—21no—
139#124—21no—
140C22⋊Q8—32no—
141C22⋊Q8—61no—
142C4×Q8—32no—
143C4×Q8—61no—
144#117—61no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C42C22×D311yescharacteristic
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—64no—
151C4⋊C8—61no—
152C4⋊C8—61no—
153C4⋊C8—61no—
154#103—62no—
155#102—62no—
156C22⋊C8—21no—
157C22⋊C8—32no—
158C22⋊C8—61no—
159C22⋊C8—61no—
160C22⋊C8#6712yes—
161#97—21no—
162#97—32no—
163#97—62no—
164#97—61no—
165#97—61no—
166C4×C8—21no—
167C4×C8—62no—
168C23×C3—42no—
169C23×C3—41no—
170C23×C3—81no—
171C23×C3—81no—
172C23×C3C4×D411yescharacteristic
173C3×Q8—42no—
174C3×Q8—81no—
175C3×D4—42no—
176C3×D4—81no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—81no—
182C2×C3×C4C2×C22⋊C411yescharacteristic
183C2×C3×C4C4×D411yescharacteristic
184C2×C3×C4#12011yescharacteristic
185C2×C3×C4#12111yescharacteristic
186C2×C3×C4#12111yescharacteristic
187C2×C3×C4#12311yescharacteristic
188C2×Dic3—22no—
189C2×Dic3—22no—
190Dic6—42no—
191Dic6—81no—
192C3×C8—42no—
193C3×C8—81no—
194C3⋊C8—42no—
195C3⋊C8—42no—
196C3⋊C8—81no—
197C24—21no—
198C2×Q8—21no—
199C2×Q8—21no—
200C2×Q8—32no—
201C2×Q8—61no—
202C2×Q8—81no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—81no—
206C22×C4—21no—
207C22×C4C2×C4×D311yescharacteristic
208C22×C4#23511yescharacteristic
209C22×C4C2×#6711yescharacteristic
210Q16—242no—
211SD16—242no—
212C2×C8—22no—
213C2×C8—41no—
214C2×C8—62no—
215C2×C8—62no—
216C2×C8—121no—
217C4⋊C4—32no—
218C4⋊C4—62no—
219C4⋊C4—62no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—62no—
228C42—62no—
229C42D3×D411yescharacteristic
230C42D3×D411yescharacteristic
231C42D3×D411yescharacteristic
232C42#23711yescharacteristic
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—81no—
237C22×C3—81no—
238C22×C3—81no—
239C22×C3—162no—
240C22×C3#38611yescharacteristic
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—21no—
246C3×C4—21no—
247C3×C4—81no—
248Dic3—42no—
249Dic3—42no—
250C23—42no—
251C23—41no—
252C23—81no—
253C23—81no—
254C23C4×#6711yescharacteristic
255Q8—42no—
256Q8—81no—
257Q8—122no—
258Q8—241no—
259D4—42no—
260D4—81no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—81no—
268C2×C4D3×C22⋊C411yescharacteristic
269C2×C4#67⋊C411yescharacteristic
270C2×C4#88311yescharacteristic
271C2×C4#88411yescharacteristic
272C2×C4#93711yescharacteristic
273C2×C4#93811yescharacteristic
274C8—42no—
275C8—81no—
276C8—122no—
277C8—122no—
278C8—241no—
279C2×C3—41no—
280C2×C3—161no—
281C2×C3#188111yescharacteristic
282C2×C3#190511yescharacteristic
283C2×C3#191411yescharacteristic
284C22—21no—
285C22—21no—
286C22—21no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—162no—
291C22#495611yescharacteristic
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—81no—
299C4—122no—
300C4—122no—
301C3#933411yescharacteristic, complemented, Sylow
302C2—41no—
303C2—161no—
304C2#7001211yescharacteristic
305C2#7005211yescharacteristic
306C2#7008911yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1469C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#1469D3
hypocenterC3#9334
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1469

Complements

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

NH ≅ G/Ntype
2. #79317C2semidirect
5. #70781C2semidirect
6. C3⋊#1308C2semidirect
7. #67747C2semidirect
301. C3#9334semidirect

Maximal subgroups

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

subgroupindex
#829492
#793172
C3×#14692
#717192
#707812
C3⋊#13082
#677472
#93343