Finite Group Databasebeta

← #10695797

Subgroups — #10695797

Order 768.

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

normal subgroups only

This group has 716 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
1C3×#3189C211yesmaximal, characteristic, complemented
2#82735C212yesmaximal
3#71691C212yesmaximal, complemented
4#69956C211yesmaximal, characteristic
5#67679C211yesmaximal, characteristic, complemented
6#9976—31nomaximal, Sylow
7C22×#235C2211yescharacteristic
8C2×C3×C22⋊Q8C2212yes—
9C2×#942—22no—
10C3×#484—22no—
11C3×#467—22no—
12C3×#451—22no—
13C2×C3×C22⋊C8C2211yescharacteristic
14#5207C2212yes—
15C2×#830C2211yescharacteristic
16C2×(C2×D3)⋊C8—22no—
17#5050—22no—
18#5047—24no—
19#5046—22no—
20#4966—22no—
21#4528—24no—
22#3189D311yescharacteristic, complemented
23#3007—32no—
24#1780—31no—
25#1472—32no—
26#1293—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
36C2×C3×Q16—22no—
37C22×C3×C8D412yes—
38C3×C22⋊Q8—22no—
39C3×C22⋊Q8—22no—
40C3×C22⋊Q8—22no—
41C3×C22⋊Q8—22no—
42C2×C3×C4⋊C4—22no—
43C2×C3×C4⋊C4D412yes—
44C2×C3×C22⋊C4—22no—
45C2×#216—24no—
46#942—48no—
47C2×#215—24no—
48C2×#208—22no—
49C22×C3⋊C8—21no—
50C3×#107—22no—
51C3×#103—24no—
52C3×C22⋊C8—22no—
53#830—22no—
54#830—22no—
55#830D414yes—
56(C2×D3)⋊C8—44no—
57#810—24no—
58#809—24no—
59C22×#41—31no—
60C2×#137—62no—
61C2×C22⋊Q8C2×D312yes—
62#484—22no—
63#467—22no—
64#452—64no—
65#451—22no—
66#451—62no—
67#450—62no—
68#417—62no—
69#407—32no—
70C2×C22⋊C8—31no—
71C2×C22⋊C8—62no—
72C2×C22⋊C8C2×D311yescharacteristic
73#324—64no—
74C24×C3#4111yescharacteristic
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—42no—
82C22×C3×C4C2×D411yescharacteristic
83C22×C3×C4C2×D412yes—
84C22×C3×C4C2×D412yes—
85C22×C3×C4C2×D411yescharacteristic
86C2×#67—42no—
87C2×#67—42no—
88C2×#67—42no—
89C22×Dic3—21no—
90#235—28no—
91#235—44no—
92#235—44no—
93#235—44no—
94C2×D12—22no—
95C2×D12—42no—
96C2×C4×D3—24no—
97C2×C4×D3—42no—
98C2×C4×D3—41no—
99C2×C4×D3—41no—
100C2×Dic6—22no—
101C2×Dic6—42no—
102C3×Q16—82no—
103C2×C3×C8—22no—
104C2×C3×C8—42no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—42no—
108C3×C4⋊C4—42no—
109C3×C22⋊C4—42no—
110C3×C22⋊C4—42no—
111#216—48no—
112#215—48no—
113#208—44no—
114C2×C3⋊C8—24no—
115C2×C3⋊C8—42no—
116C2×C3⋊C8—41no—
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—22no—
128C2×Q16—64no—
129C2×SD16—64no—
130C2×M16—62no—
131C22×C8—61no—
132C22×C8#6712yes—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4#6712yes—
139C2×C22⋊C4—22no—
140#107—22no—
141#103—24no—
142#103—64no—
143#102—64no—
144C22⋊C8—22no—
145C22⋊C8—34no—
146C22⋊C8—62no—
147C22⋊C8—62no—
148C22⋊C8—124no—
149C23×C3—22no—
150C23×C3—41no—
151C23×C3—41no—
152C23×C3#12111yescharacteristic
153C23×C3#12112yes—
154C22×D3—22no—
155C22×D3—41no—
156C22×D3—42no—
157C22×D3—41no—
158C22×D3—42no—
159C3×Q8—44no—
160C3×Q8—44no—
161C2×C3×C4—22no—
162C2×C3×C4—24no—
163C2×C3×C4—41no—
164C2×C3×C4—42no—
165C2×C3×C4—44no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—82no—
169C2×C3×C4#12012yes—
170C2×C3×C4#12011yescharacteristic
171C2×C3×C4#12711yescharacteristic
172#67—48no—
173C2×Dic3—22no—
174C2×Dic3—41no—
175C2×Dic3—41no—
176D12—24no—
177D12—42no—
178C4×D3—44no—
179C4×D3—44no—
180Dic6—24no—
181Dic6—42no—
182C3×C8—42no—
183C3⋊C8—44no—
184C3⋊C8—81no—
185C24—61no—
186C24#23711yescharacteristic
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—62no—
197C2×D4—122no—
198C2×D4—122no—
199C2×D4—122no—
200C2×D4—122no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—42no—
204C22×C4—64no—
205C22×C4—61no—
206C22×C4—122no—
207C22×C4—121no—
208C22×C4—121no—
209C22×C4D3×D411yescharacteristic
210C22×C4D3×D412yes—
211C22×C4C2×#6712yes—
212C22×C4C2×#6711yescharacteristic
213Q16—82no—
214Q16—128no—
215SD16—128no—
216M16—124no—
217C2×C8—22no—
218C2×C8—42no—
219C2×C8—64no—
220C2×C8—122no—
221C2×C8—121no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—82no—
234C22×C3#39211yescharacteristic
235C22×C3#47312yes—
236C2×D3—44no—
237C2×D3—42no—
238C2×D3—82no—
239C2×D3—81no—
240C2×D3—81no—
241C3×C4—22no—
242C3×C4—21no—
243C3×C4—82no—
244C3×C4—82no—
245C3×C4B(4,𝔽2)12yes—
246Dic3—42no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—62no—
251C23—121no—
252C23—122no—
253C23—122no—
254C23—121no—
255C23#93811yescharacteristic
256C23#93912yes—
257Q8—44no—
258Q8—44no—
259Q8—64no—
260Q8—122no—
261D4—64no—
262D4—122no—
263D4—128no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—62no—
272C2×C4—82no—
273C2×C4—124no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—124no—
277C2×C4#93712yes—
278C2×C4#94011yescharacteristic
279C2×C4#95311yescharacteristic
280C8—42no—
281C8—124no—
282C8—241no—
283C2×C3—22no—
284C2×C3—42no—
285C2×C3#201711yescharacteristic
286C2×C3#202011yescharacteristic
287C2×C3#202111yescharacteristic
288D3—82no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—82no—
295C22—124no—
296C22—122no—
297C22—242no—
298C22—241no—
299C22—241no—
300C22#519212yes—
301C22#522311yescharacteristic
302C4—22no—
303C4—21no—
304C4—82no—
305C4—82no—
306C4—122no—
307C4#524412yes—
308C3#997611yescharacteristic, complemented, Sylow
309C2—22no—
310C2—42no—
311C2—242no—
312C2#7185111yescharacteristic
313C2#7207411yescharacteristic
314C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3189C2
socleC22×C3#392
prime radicalC22×#235C22
hypercenter#3189D3
hypocenterC3#9976
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

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

NH ≅ G/Ntype
1. C3×#3189C2semidirect
3. #71691C2semidirect
5. #67679C2semidirect
22. #3189D3semidirect
308. C3#9976semidirect

Maximal subgroups

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

subgroupindex
C3×#31892
#827352
#716912
#699562
#676792
#99763