Finite Group Databasebeta

← #10695831

Subgroups — #10695831

Order 768.

722 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 81 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×#2817C211yesmaximal, characteristic, complemented
2#82735C212yesmaximal
3#71689C212yesmaximal, complemented
4#69537C211yesmaximal, characteristic
5#67629C211yesmaximal, characteristic, complemented
6#9982—31nomaximal, Sylow
7C22×#235C2211yescharacteristic
8C2×C3×C22⋊Q8C2212yes—
9C2×#942—22no—
10C3×#400—22no—
11C3×#397—22no—
12C3×#395—22no—
13C3×#381C2211yescharacteristic
14#5211C2211yescharacteristic
15#5207C2212yes—
16#5050—22no—
17#5047—24no—
18#5046—22no—
19#4966—22no—
20C2×#806—22no—
21#4538—24no—
22#3007—32no—
23#2817D311yescharacteristic, complemented
24#1782—31no—
25#1511—32no—
26#1306—31no—
27C22×C3×Q8D412yes—
28C23×C3×C4C2311yescharacteristic
29C22×#67—21no—
30C2×#235—22no—
31C2×#235—22no—
32C2×#235—44no—
33C22×D12D411yescharacteristic
34C22×C4×D3—21no—
35C22×Dic6D411yescharacteristic
36C3×C22⋊Q8—22no—
37C3×C22⋊Q8—22no—
38C3×C22⋊Q8—22no—
39C3×C22⋊Q8—22no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4—22no—
42C2×C3×C4⋊C4D412yes—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4D412yes—
45C2×#217—21no—
46C2×#216—24no—
47#942—48no—
48C2×#215—24no—
49C2×#208—22no—
50C3×#95—24no—
51C3×#95—22no—
52C3×#95—22no—
53C3⋊#95—22no—
54#830—22no—
55#830D414yes—
56#810—24no—
57#809—24no—
58#806—44no—
59C22×#41—31no—
60C2×#137—62no—
61C2×C22⋊Q8C2×D312yes—
62#452—64no—
63#451—62no—
64#450—62no—
65#417—62no—
66C2×#99—62no—
67#407—32no—
68#400—22no—
69#397—22no—
70#395—22no—
71#381—31no—
72#381C2×D311yescharacteristic
73#328—64no—
74C24×C3C2×D411yescharacteristic
75C23×D3—21no—
76C2×C3×Q8—22no—
77C2×C3×Q8—22no—
78C2×C3×Q8—44no—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4—22no—
82C22×C3×C4—42no—
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4C2×D412yes—
85C22×C3×C4C2×D412yes—
86C22×C3×C4#4111yescharacteristic
87C2×#67—24no—
88C2×#67—42no—
89C2×#67—41no—
90C2×#67—41no—
91C22×Dic3—21no—
92C22×Dic3—41no—
93#235—28no—
94#235—44no—
95#235—44no—
96#235—44no—
97C2×D12—22no—
98C2×D12—42no—
99C2×C4×D3—42no—
100C2×C4×D3—42no—
101C2×C4×D3—42no—
102C2×Dic6—22no—
103C2×Dic6—42no—
104C3×C4⋊C4—22no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—42no—
107C3×C4⋊C4—42no—
108C3×C4⋊C4—82no—
109C3×C22⋊C4—42no—
110C3×C22⋊C4—42no—
111C3×C22⋊C4—42no—
112#217—42no—
113#216—48no—
114#215—48no—
115#208—44no—
116C2×C3⋊C8—24no—
117C2×#41—62no—
118C2×#41—62no—
119C2×#41—124no—
120C22×Q8—31no—
121C22×Q8#6712yes—
122C22×D4—31no—
123C22×D4—61no—
124C23×C4—61no—
125C23×C4C22×D311yescharacteristic
126#137—128no—
127C2×Q16—64no—
128C2×SD16—64no—
129C2×M16—62no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4#6712yes—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—61no—
139C2×C22⋊C4#6712yes—
140#103—64no—
141#102—64no—
142#99—124no—
143C22⋊C8—34no—
144C22⋊C8—62no—
145#95—24no—
146#95—22no—
147#95—22no—
148#95—62no—
149C23×C3—22no—
150C23×C3—42no—
151C23×C3#12011yescharacteristic
152C23×C3#12111yescharacteristic
153C23×C3#12711yescharacteristic
154C22×D3—22no—
155C22×D3—41no—
156C22×D3—41no—
157C22×D3—82no—
158C3×Q8—44no—
159C3×Q8—82no—
160C2×C3×C4—24no—
161C2×C3×C4—41no—
162C2×C3×C4—41no—
163C2×C3×C4—42no—
164C2×C3×C4—44no—
165C2×C3×C4—44no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—44no—
170C2×C3×C4#12012yes—
171C2×C3×C4#12112yes—
172#67—44no—
173#67—44no—
174C2×Dic3—22no—
175C2×Dic3—41no—
176C2×Dic3—41no—
177C2×Dic3—81no—
178C2×Dic3—82no—
179D12—24no—
180D12—42no—
181C4×D3—48no—
182Dic6—24no—
183Dic6—42no—
184C3⋊C8—44no—
185C24—61no—
186C24C2×#6711yescharacteristic
187#41—68no—
188#41—124no—
189#41—124no—
190#41—124no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—44no—
194C2×Q8—62no—
195C2×Q8—122no—
196C2×D4—64no—
197C2×D4—62no—
198C2×D4—122no—
199C2×D4—122no—
200C2×D4—121no—
201C2×D4—121no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—42no—
206C22×C4—61no—
207C22×C4—122no—
208C22×C4—122no—
209C22×C4—122no—
210C22×C4—121no—
211C22×C4D3×D411yescharacteristic
212C22×C4D3×D412yes—
213C22×C4#23711yescharacteristic
214C22×C4C2×#6712yes—
215Q16—128no—
216SD16—128no—
217M16—124no—
218C2×C8—64no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—82no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—122no—
228C22×C3—22no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—82no—
234C22×C3#39211yescharacteristic
235C22×C3B(4,𝔽2)12yes—
236C2×D3—42no—
237C2×D3—82no—
238C2×D3—82no—
239C2×D3—82no—
240C3×C4—22no—
241C3×C4—22no—
242C3×C4—82no—
243C3×C4—82no—
244C3×C4—82no—
245Dic3—42no—
246Dic3—161no—
247C23—22no—
248C23—42no—
249C23—62no—
250C23—121no—
251C23—121no—
252C23—242no—
253C23#93811yescharacteristic
254C23#94011yescharacteristic
255C23#95311yescharacteristic
256Q8—44no—
257Q8—64no—
258Q8—82no—
259Q8—122no—
260D4—64no—
261D4—122no—
262D4—124no—
263D4—124no—
264C2×C4—24no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—62no—
275C2×C4—128no—
276C2×C4—121no—
277C2×C4—121no—
278C2×C4—241no—
279C2×C4—242no—
280C2×C4#93712yes—
281C2×C4#93912yes—
282C8—124no—
283C2×C3—22no—
284C2×C3—41no—
285C2×C3—41no—
286C2×C3#200912yes—
287C2×C3#202111yescharacteristic
288D3—82no—
289C22—22no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—82no—
295C22—122no—
296C22—242no—
297C22—242no—
298C22—242no—
299C22#522311yescharacteristic
300C22#524412yes—
301C4—22no—
302C4—22no—
303C4—82no—
304C4—82no—
305C4—82no—
306C4—122no—
307C4—481no—
308C3#998211yescharacteristic, complemented, Sylow
309C2—22no—
310C2—41no—
311C2—41no—
312C2—242no—
313C2#7185111yescharacteristic
314C2#7186912yes—
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2817C2
socleC22×C3#392
prime radicalC22×#235C22
hypercenter#2817D3
hypocenterC3#9982
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2817

Complements

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

NH ≅ G/Ntype
1. C3×#2817C2semidirect
3. #71689C2semidirect
5. #67629C2semidirect
23. #2817D3semidirect
308. C3#9982semidirect

Maximal subgroups

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

subgroupindex
C3×#28172
#827352
#716892
#695372
#676292
#99823