Finite Group Databasebeta

← #10695833

Subgroups — #10695833

Order 768.

722 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 79 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#84084C212yesmaximal
2#74114C211yesmaximal, characteristic, complemented
3C3×#1782C211yesmaximal, characteristic
4#69533C212yesmaximal, complemented
5#67629C211yesmaximal, characteristic, complemented
6#9982—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×#951—22no—
9C2×#924C2212yes—
10C2×C3×#99—22no—
11C3×#381C2211yescharacteristic
12C3×#381C2211yescharacteristic
13#5239—24no—
14#5232—22no—
15#5228—22no—
16#5218—22no—
17#5207C2212yes—
18#4953—22no—
19#4659—22no—
20#4656—22no—
21#4452—24no—
22#3007—32no—
23#2817—31no—
24#1782D311yescharacteristic
25#1511—32no—
26#1306—31no—
27C2×C3×#41—22no—
28C2×C3×#41—22no—
29C2×C3×#41—44no—
30C22×C3×Q8D411yescharacteristic
31C22×C3×D4—21no—
32C22×C3×D4D411yescharacteristic
33C23×C3×C4—21no—
34C23×C3×C4C2311yescharacteristic
35C22×Dic6D412yes—
36C2×C3×C22⋊C4—21no—
37C2×C3×C22⋊C4D412yes—
38C2×#217—22no—
39#951—48no—
40C2×#216—24no—
41C2×#214—24no—
42C2×C4⋊Dic3—22no—
43C2×C4⋊Dic3D412yes—
44#924—22no—
45#924—22no—
46#924—22no—
47#924—22no—
48C2×Dic3⋊C4—22no—
49C2×#208—22no—
50C3×#99—44no—
51C3×#95—22no—
52C3×#95—22no—
53#835—24no—
54#832—24no—
55C3⋊#95—24no—
56C3⋊#95—22no—
57#830—22no—
58#830D414yes—
59C22×#41C2×D311yescharacteristic
60C2×#137—62no—
61C2×C22⋊Q8—32no—
62#452—64no—
63#451—62no—
64#450—62no—
65#417—62no—
66C2×#99—22no—
67#407—32no—
68#400—62no—
69#397—62no—
70#395—62no—
71#381C2×D311yescharacteristic
72#381C2×D311yescharacteristic
73#328—64no—
74C24×C3—21no—
75C24×C3C2×D411yescharacteristic
76C3×#41—28no—
77C3×#41—44no—
78C3×#41—44no—
79C3×#41—44no—
80C2×C3×Q8—22no—
81C2×C3×Q8—42no—
82C2×C3×D4—24no—
83C2×C3×D4—22no—
84C2×C3×D4—42no—
85C2×C3×D4—42no—
86C2×C3×D4—41no—
87C2×C3×D4—41no—
88C22×C3×C4—21no—
89C22×C3×C4—22no—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4—42no—
93C22×C3×C4—42no—
94C22×C3×C4—41no—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D412yes—
97C22×C3×C4C2×D412yes—
98C22×C3×C4#4111yescharacteristic
99C22×Dic3—22no—
100C22×Dic3—22no—
101C2×Dic6—22no—
102C2×Dic6—22no—
103C2×Dic6—44no—
104C3×C22⋊C4—42no—
105C3×C22⋊C4—42no—
106#217—42no—
107#217—42no—
108#216—48no—
109#214—48no—
110C4⋊Dic3—22no—
111C4⋊Dic3—22no—
112C4⋊Dic3—82no—
113Dic3⋊C4—42no—
114Dic3⋊C4—42no—
115#208—44no—
116C2×C3⋊C8—24no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—44no—
120C22×Q8—32no—
121C22×Q8D1211yescharacteristic
122C22×D4—21no—
123C22×D4D1211yescharacteristic
124C23×C4—21no—
125C23×C4C22×D311yescharacteristic
126#137—128no—
127C2×Q16—64no—
128C2×SD16—64no—
129C2×M16—62no—
130C22⋊Q8—62no—
131C22⋊Q8—62no—
132C22⋊Q8—62no—
133C22⋊Q8—62no—
134C2×C4⋊C4—32no—
135C2×C4⋊C4—62no—
136C2×C4⋊C4—62no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—62no—
139C2×C22⋊C4#6712yes—
140#103—64no—
141#102—64no—
142#99—44no—
143C22⋊C8—34no—
144C22⋊C8—62no—
145#95—22no—
146#95—22no—
147#95—64no—
148#95—62no—
149C23×C3—22no—
150C23×C3—22no—
151C23×C3—42no—
152C23×C3—41no—
153C23×C3—41no—
154C23×C3—82no—
155C23×C3#12011yescharacteristic
156C23×C3#12111yescharacteristic
157C23×C3#12711yescharacteristic
158C3×Q8—24no—
159C3×Q8—42no—
160C3×D4—24no—
161C3×D4—42no—
162C3×D4—44no—
163C3×D4—44no—
164C2×C3×C4—24no—
165C2×C3×C4—22no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—42no—
169C2×C3×C4—48no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—44no—
173C2×C3×C4—42no—
174C2×C3×C4—81no—
175C2×C3×C4—82no—
176C2×C3×C4#12012yes—
177C2×C3×C4#12112yes—
178C2×Dic3—44no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181C2×Dic3—44no—
182Dic6—44no—
183Dic6—82no—
184C3⋊C8—44no—
185C24—21no—
186C24C2×#6711yescharacteristic
187#41—28no—
188#41—44no—
189#41—44no—
190#41—44no—
191C2×Q8—22no—
192C2×Q8—42no—
193C2×Q8—62no—
194C2×Q8—62no—
195C2×Q8—124no—
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—41no—
201C2×D4—41no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—62no—
210C22×C4—62no—
211C22×C4C2×D1211yescharacteristic
212C22×C4#23511yescharacteristic
213C22×C4D3×D412yes—
214C22×C4D3×D412yes—
215Q16—128no—
216SD16—128no—
217M16—124no—
218C2×C8—64no—
219C4⋊C4—62no—
220C4⋊C4—62no—
221C4⋊C4—122no—
222C4⋊C4—122no—
223C4⋊C4—242no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—122no—
227C22⋊C4—122no—
228C22×C3—22no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—42no—
234C22×C3—82no—
235C22×C3—82no—
236C22×C3—82no—
237C22×C3—82no—
238C22×C3#39211yescharacteristic
239C22×C3B(4,𝔽2)12yes—
240C3×C4—22no—
241C3×C4—22no—
242C3×C4—42no—
243C3×C4—82no—
244C3×C4—161no—
245Dic3—82no—
246Dic3—82no—
247C23—22no—
248C23—22no—
249C23—42no—
250C23—41no—
251C23—41no—
252C23—82no—
253C23#93011yescharacteristic
254C23#93711yescharacteristic
255C23#94011yescharacteristic
256Q8—24no—
257Q8—42no—
258Q8—124no—
259Q8—242no—
260D4—24no—
261D4—42no—
262D4—44no—
263D4—44no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—48no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—124no—
277C2×C4—122no—
278C2×C4—122no—
279C2×C4—124no—
280C2×C4#88212yes—
281C2×C4#88412yes—
282C8—124no—
283C2×C3—22no—
284C2×C3—41no—
285C2×C3—41no—
286C2×C3—82no—
287C2×C3#200912yes—
288C2×C3#202111yescharacteristic
289C22—22no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22#496011yescharacteristic
300C22#503212yes—
301C4—22no—
302C4—22no—
303C4—42no—
304C4—82no—
305C4—161no—
306C4—242no—
307C4—242no—
308C3#998211yescharacteristic, complemented, Sylow
309C2—22no—
310C2—41no—
311C2—41no—
312C2—82no—
313C2#6950011yescharacteristic
314C2#7005312yes—
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1782C2
socleC22×C3#392
prime radicalC22×C3×#41C22
hypercenter#1782D3
hypocenterC3#9982
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. #74114C2semidirect
4. #69533C2semidirect
5. #67629C2semidirect
308. C3#9982semidirect

Maximal subgroups

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

subgroupindex
#840842
#741142
C3×#17822
#695332
#676292
#99823