Finite Group Databasebeta

← #10695145

Subgroups — #10695145

Order 768.

390 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 113 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#83339C211yesmaximal, characteristic
2C3×#1493C211yesmaximal, characteristic
3#71826C211yesmaximal, characteristic
4#70516C211yesmaximal, characteristic
5#70501C211yesmaximal, characteristic
6#69738C211yesmaximal, characteristic
7C2×#4455C211yesmaximal, characteristic
8#9859—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13#5080C411yescharacteristic
14#5080C411yescharacteristic
15#5078C411yescharacteristic
16#5078C411yescharacteristic
17#5065—22no—
18#4998—22no—
19#4924C2211yescharacteristic
20C2×C4×C3⋊C8C2211yescharacteristic
21C2×C4×C3⋊C8C2211yescharacteristic
22C3×Q8⋊C8—22no—
23#4558—22no—
24#4535—22no—
25#4484—22no—
26#4456—22no—
27#4455—22no—
28#3160—31no—
29#1837—31no—
30#1493—31no—
31#1493—31no—
32#1493—31no—
33#1493D311yescharacteristic
34C2×#322—31no—
35C2×C3×M16—21no—
36C22×C3×C8—21no—
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×#125—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×#117—21no—
45C2×C3×C42C2311yescharacteristic
46#926—21no—
47#924—21no—
48#924—21no—
49C2×#208—21no—
50C22×C3⋊C8—21no—
51C22×C3⋊C8D412yes—
52#870—21no—
53#869C2×C411yescharacteristic
54#869C2×C411yescharacteristic
55C4×Dic6—21no—
56C4×Dic6—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C4×C8—21no—
60C3×C4×C8D412yes—
61#808—22no—
62#808—22no—
63#807—22no—
64#807—22no—
65C4⋊(C3⋊C8)—21no—
66C4⋊(C3⋊C8)—21no—
67C4×C3⋊C8—21no—
68C4×C3⋊C8C2×C411yescharacteristic
69C4×C3⋊C8C2×C411yescharacteristic
70C4×C3⋊C8D412yes—
71C4×C3⋊C8D412yes—
72#533—31no—
73#533C2×D311yescharacteristic
74#500—31no—
75#500—31no—
76#499—62no—
77#498—31no—
78#498—31no—
79#427—62no—
80#423—31no—
81#423C2×D311yescharacteristic
82C2×C4×C8—31no—
83C2×C4×C8—31no—
84C2×C4×C8C2×D311yescharacteristic
85#341—62no—
86Q8⋊C8—22no—
87Q8⋊C8—62no—
88Q8⋊C8—62no—
89Q8⋊C8—62no—
90#322—62no—
91C2×C3×Q8—21no—
92C2×C3×Q8—21no—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4#4111yescharacteristic
95C22×C3×C4#4111yescharacteristic
96C2×Dic6—21no—
97C2×Dic6—21no—
98C3×M16—42no—
99C2×C3×C8—24no—
100C2×C3×C8—22no—
101C2×C3×C8—41no—
102C3×C4⋊C4—21no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C22⋊C4—41no—
109C3×C42—41no—
110C3×C42C22×C411yescharacteristic
111C3×C42C2×D411yescharacteristic
112C3×C42C2×D411yescharacteristic
113C3×C42C2×D411yescharacteristic
114#217—41no—
115C4⋊Dic3—21no—
116C4⋊Dic3—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119Dic3⋊C4—41no—
120Dic3⋊C4—41no—
121C4×Dic3—41no—
122#208—42no—
123C2×C3⋊C8—22no—
124C2×C3⋊C8—22no—
125C2×C3⋊C8—22no—
126C2×C3⋊C8—42no—
127C2×C3⋊C8—41no—
128C2×C3⋊C8C22⋊C412yes—
129C2×C3⋊C8C22⋊C412yes—
130C2×M16—21no—
131C2×M16—61no—
132C22×C8—21no—
133C22×C8—32no—
134C22×C8—61no—
135C4⋊Q8—31no—
136C4⋊Q8—31no—
137C4⋊Q8C4×D311yescharacteristic
138C4⋊Q8C4×D311yescharacteristic
139#125—21no—
140#125—61no—
141C22⋊Q8—21no—
142C22⋊Q8—21no—
143C22⋊Q8—61no—
144C22⋊Q8—61no—
145C4×Q8—21no—
146C4×Q8—21no—
147C4×Q8—61no—
148C4×Q8—61no—
149#117—21no—
150#117—61no—
151C2×C42C22×D311yescharacteristic
152#107—62no—
153#107—62no—
154#106—62no—
155#106—62no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—61no—
159C4⋊C8—61no—
160C4×C8—21no—
161C4×C8—31no—
162C4×C8—31no—
163C4×C8—32no—
164C4×C8—32no—
165C4×C8—61no—
166C4×C8#6712yes—
167C23×C3#12411yescharacteristic
168C3×Q8—42no—
169C3×Q8—42no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4C2×C22⋊C411yescharacteristic
175C2×C3×C4C4×D411yescharacteristic
176C2×C3×C4C4×D411yescharacteristic
177C2×C3×C4#12111yescharacteristic
178C2×C3×C4#12111yescharacteristic
179C2×C3×C4#12711yescharacteristic
180C2×C3×C4#13712yes—
181C2×C3×C4#13712yes—
182C2×Dic3—41no—
183C2×Dic3—41no—
184Dic6—42no—
185Dic6—42no—
186C3×C8—24no—
187C3×C8—81no—
188C3⋊C8—22no—
189C3⋊C8—22no—
190C3⋊C8—42no—
191C3⋊C8—81no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8—61no—
195C2×Q8—61no—
196C22×C4#23511yescharacteristic
197C22×C4D3×D411yescharacteristic
198C22×C4#23711yescharacteristic
199M16—42no—
200M16—122no—
201C2×C8—24no—
202C2×C8—22no—
203C2×C8—32no—
204C2×C8—32no—
205C2×C8—41no—
206C2×C8—62no—
207C2×C8—62no—
208C2×C8—62no—
209C2×C8—122no—
210C2×C8—121no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—61no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4—121no—
222C4⋊C4—121no—
223C22⋊C4—41no—
224C22⋊C4—121no—
225C42—41no—
226C42—121no—
227C42C2×C4×D311yescharacteristic
228C42D3×D411yescharacteristic
229C42D3×D411yescharacteristic
230C42C2×#6711yescharacteristic
231C22×C3—21no—
232C22×C3—21no—
233C22×C3#39011yescharacteristic
234C22×C3#49112yes—
235C3×C4—21no—
236C3×C4—21no—
237C3×C4—81no—
238C3×C4—81no—
239C3×C4#41912yes—
240C3×C4#44312yes—
241C3×C4#44412yes—
242C3×C4#49711yescharacteristic
243C3×C4#49711yescharacteristic
244Dic3—81no—
245Dic3—81no—
246C23#88511yescharacteristic
247Q8—42no—
248Q8—42no—
249Q8—122no—
250Q8—122no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—121no—
256C2×C4—121no—
257C2×C4D3×C22⋊C411yescharacteristic
258C2×C4#67⋊C411yescharacteristic
259C2×C4#88411yescharacteristic
260C2×C4#91512yes—
261C2×C4#91812yes—
262C2×C4C4×#6711yescharacteristic
263C2×C4#93911yescharacteristic
264C2×C4#94011yescharacteristic
265C8—24no—
266C8—62no—
267C8—62no—
268C8—81no—
269C8—122no—
270C8—241no—
271C2×C3—22no—
272C2×C3#196011yescharacteristic
273C2×C3#197411yescharacteristic
274C2×C3#197511yescharacteristic
275C22—21no—
276C22—21no—
277C22#479712yes—
278C22#495711yescharacteristic
279C4—21no—
280C4—21no—
281C4—81no—
282C4—81no—
283C4—241no—
284C4—241no—
285C4#480212yes—
286C4#482412yes—
287C4#514512yes—
288C4#516711yescharacteristic
289C4#519011yescharacteristic
290C3#985911yescharacteristic, complemented, Sylow
291C2—22no—
292C2#7000011yescharacteristic
293C2#7022811yescharacteristic
294C2#7037311yescharacteristic
subgroupG/H
centerC2×C4#940
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1493C2
socleC22×C3#390
prime radicalC23×C3#124
cosocleC2×C3×C42C23
hypercenter#1493D3
hypocenterC3#9859
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
290. C3#9859semidirect

Maximal subgroups

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

subgroupindex
#833392
C3×#14932
#718262
#705162
#705012
#697382
C2×#44552
#98593