Finite Group Databasebeta

← #10711986

Subgroups — #10711986

Order 768.

536 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2372C211yesmaximal, characteristic
2C2×#4541C211yesmaximal, characteristic
3#71723C211yesmaximal, characteristic
4#69597C212yesmaximal
5#67637C212yesmaximal
6#12965—31nomaximal, Sylow
7C2×C3×#124C2211yescharacteristic
8#5798C2211yescharacteristic
9C3×#391—21no—
10C3×#388—22no—
11C3×#386C2212yes—
12C3×C4×C22⋊C4—22no—
13C3×#376—21no—
14#5225—21no—
15C2×#834C2212yes—
16C2×#833C2211yescharacteristic
17#5210—22no—
18#4541C414yescomplemented
19#4527—22no—
20#4523—21no—
21#2372D311yescharacteristic
22C2×#354—31no—
23#1885—31no—
24#1867—32no—
25#1390—32no—
26C22×C3×Q8D411yescharacteristic
27C22×C3×D4C2311yescharacteristic
28C23×C3×C4D412yes—
29C3×#124—22no—
30C3×#124C2×C412yes—
31C3×#124Q812yes—
32C2×C3×C4⋊C4—21no—
33C2×C3×C22⋊C4—22no—
34C2×C3×C22⋊C4—22no—
35C2×C3×C22⋊C4D412yes—
36C2×C3×C42—22no—
37C2×C3×C42—21no—
38C2×C3×C42D411yescharacteristic
39C2×#217—22no—
40C2×#217—22no—
41D4×Dic3—42no—
42#926—21no—
43C2×Dic3⋊C4—42no—
44C2×#208—21no—
45C3×#95—21no—
46C3×#95—21no—
47C3×#95—22no—
48C3×#95—22no—
49#835—42no—
50#834—22no—
51#834C2×C414yes—
52#833—22no—
53C3⋊#95—42no—
54C2×#124C2×D311yescharacteristic
55#518—31no—
56#419—61no—
57C2×#100—31no—
58C2×#99—32no—
59#391—21no—
60#388—22no—
61#386C2×D312yes—
62#385—62no—
63C4×C22⋊C4—22no—
64#376—21no—
65#354—34no—
66#342—62no—
67#339—61no—
68C24×C3C2×D412yes—
69C2×C3×Q8—21no—
70C2×C3×Q8—41no—
71C2×C3×Q8—41no—
72C2×C3×Q8C4⋊C412yescomplemented
73C2×C3×D4—41no—
74C2×C3×D4—41no—
75C2×C3×D4C22×C411yescharacteristic
76C2×C3×D4C2×Q811yescharacteristic
77C2×C3×D4#4112yes—
78C22×C3×C4—22no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—22no—
82C22×C3×C4—44no—
83C22×C3×C4—42no—
84C22×C3×C4C2×D411yescharacteristic
85C22×Dic3—22no—
86C22×Dic3—22no—
87C22×Dic3—42no—
88C3×C4⋊C4—81no—
89C3×C22⋊C4—24no—
90C3×C22⋊C4—24no—
91C3×C22⋊C4—44no—
92C3×C22⋊C4—42no—
93C3×C42—21no—
94C3×C42—44no—
95C3×C42—81no—
96C3×C42C4⋊C412yescomplemented
97#217—24no—
98#217—42no—
99#217—42no—
100#217—41no—
101C4⋊Dic3—22no—
102Dic3⋊C4—84no—
103C4×Dic3—41no—
104#208—41no—
105#208—41no—
106C2×C3⋊C8—41no—
107C22×Q8#6711yescharacteristic
108C22×D4C22×D311yescharacteristic
109C23×C4#6712yes—
110C2×M16—61no—
111#124—22no—
112#124Dic612yes—
113#124C4×D312yes—
114C4×D4—122no—
115#117—61no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—122no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—62no—
121C2×C22⋊C4—62no—
122C2×C22⋊C4#6712yes—
123C2×C42—22no—
124C2×C42—21no—
125C2×C42#6711yescharacteristic
126#103—122no—
127#100—62no—
128#99—34no—
129#99—62no—
130#95—21no—
131#95—21no—
132#95—22no—
133#95—22no—
134#95—122no—
135C23×C3—22no—
136C23×C3—22no—
137C23×C3—42no—
138C23×C3—42no—
139C23×C3C4×D412yes—
140C23×C3#12011yescharacteristic
141C23×C3C22⋊Q812yes—
142C3×Q8—42no—
143C3×Q8—81no—
144C3×D4—41no—
145C3×D4—41no—
146C3×D4—42no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—24no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—44no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—82no—
162C2×C3×C4C2×C4⋊C411yescharacteristic
163C2×C3×C4#12311yescharacteristic
164C2×Dic3—44no—
165C2×Dic3—42no—
166C2×Dic3—82no—
167C2×Dic3—82no—
168C2×Dic3—82no—
169C2×Dic3—82no—
170C3⋊C8—81no—
171C24C2×#6712yes—
172C2×Q8—21no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8Dic3⋊C412yescomplemented
176C2×D4—41no—
177C2×D4—41no—
178C2×D4C2×Dic611yescharacteristic
179C2×D4C2×C4×D311yescharacteristic
180C2×D4#23512yes—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—62no—
188C22×C4—62no—
189C22×C4—122no—
190C22×C4C2×#6711yescharacteristic
191M16—121no—
192M16—121no—
193C2×C8—121no—
194C4⋊C4—62no—
195C4⋊C4—81no—
196C4⋊C4—244no—
197C22⋊C4—24no—
198C22⋊C4—24no—
199C22⋊C4—44no—
200C22⋊C4—42no—
201C22⋊C4—64no—
202C22⋊C4—122no—
203C22⋊C4—122no—
204C22⋊C4—121no—
205C42—21no—
206C42—44no—
207C42—81no—
208C42—121no—
209C42Dic3⋊C412yescomplemented
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—42no—
214C22×C3—42no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3—82no—
218C22×C3#38511yescharacteristic
219C22×C3B(4,𝔽2)11yescharacteristic
220C22×C3#45811yescharacteristic
221C3×C4—21no—
222C3×C4—21no—
223C3×C4—42no—
224C3×C4—42no—
225C3×C4—44no—
226C3×C4—82no—
227Dic3—84no—
228Dic3—82no—
229C23—22no—
230C23—22no—
231C23—42no—
232C23—42no—
233C23#92412yes—
234C23C4×#6712yes—
235C23#95311yescharacteristic
236Q8—42no—
237Q8—81no—
238D4—41no—
239D4—41no—
240D4—42no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—82no—
256C2×C4—124no—
257C2×C4—122no—
258C2×C4—242no—
259C2×C4—242no—
260C2×C4—242no—
261C2×C4—242no—
262C2×C4C2×Dic3⋊C411yescharacteristic
263C2×C4#92911yescharacteristic
264C8—241no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—42no—
268C2×C3—42no—
269C2×C3#186711yescharacteristic
270C2×C3#219712yes—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—82no—
279C22#521011yescharacteristic
280C22#524311yescharacteristic
281C22#524411yescharacteristic
282C4—21no—
283C4—21no—
284C4—42no—
285C4—42no—
286C4—44no—
287C4—82no—
288C4—244no—
289C4—242no—
290C3#1296511yescharacteristic, complemented, Sylow
291C2—21no—
292C2—21no—
293C2—42no—
294C2—42no—
295C2#7165711yescharacteristic
296C2#7192212yes—
subgroupG/H
centerC22#5244
commutatorC2×C3×D4C22×C4
FrattiniC22×D4C22×D3
FittingC3×#2372C2
socleC22×C3B(4,𝔽2)
prime radical, cosocleC22×C3×D4C23
hypercenter#2372D3
hypocenterC3#12965
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2372

Complements

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

NH ≅ G/Ntype
18. #4541C4semidirect
72. C2×C3×Q8C4⋊C4semidirect
96. C3×C42C4⋊C4semidirect
175. C2×Q8Dic3⋊C4semidirect
209. C42Dic3⋊C4semidirect
290. C3#12965semidirect

Maximal subgroups

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

subgroupindex
C3×#23722
C2×#45412
#717232
#695972
#676372
#129653