Finite Group Databasebeta

← #10711990

Subgroups — #10711990

Order 768.

580 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 95 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#83174C211yesmaximal, characteristic
2C3×#1887C211yesmaximal, characteristic
3C2×#4545C211yesmaximal, characteristic
4#71657C212yesmaximal, complemented
5#67698C212yesmaximal, complemented
6#12966—31nomaximal, Sylow
7C2×C3×#127C2211yescharacteristic
8C3×#518C2211yescharacteristic
9C3×#418—21no—
10C2×C3×#100C2211yescharacteristic
11#5220—21no—
12#5220—22no—
13#5220—22no—
14#5220C2212yes—
15C2×#834C2212yes—
16#4946—22no—
17#4937—21no—
18C3×#339—21no—
19#4545C414yescomplemented
20#4526—22no—
21#2380—31no—
22C2×#353—31no—
23#1887D311yescharacteristic
24#1867—32no—
25#1390—32no—
26C22×C3×D4—22no—
27C22×C3×D4D411yescharacteristic
28C22×C3×D4C2311yescharacteristic
29C23×Dic3—21no—
30C23×Dic3D412yes—
31C2×C3×M16—21no—
32C3×#127—22no—
33C3×#127C2×C412yes—
34C3×#127Q812yes—
35C3×C4×D4—42no—
36C3×#117—21no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C42D411yescharacteristic
39C2×#217—22no—
40C2×#217—21no—
41C2×#217—22no—
42C2×#217—22no—
43C2×#217D412yes—
44C2×Dic3⋊C4—42no—
45C3×#102—42no—
46C3×#100—22no—
47#834—22no—
48#834C2×C414yescomplemented
49C3⋊#95—21no—
50C3⋊#95—22no—
51C3⋊#95—21no—
52C3⋊#95—22no—
53C3⋊#95—42no—
54C2×#127C2×D311yescharacteristic
55#518C2×D311yescharacteristic
56#418—21no—
57C2×#100C2×D311yescharacteristic
58C2×#99—32no—
59#386—32no—
60#386—62no—
61#386—62no—
62#386—61no—
63#385—62no—
64#383—61no—
65#353—34no—
66#342—62no—
67#339—21no—
68C24×C3—21no—
69C24×C3C2×D412yes—
70C2×C3×D4—24no—
71C2×C3×D4—24no—
72C2×C3×D4—21no—
73C2×C3×D4—41no—
74C2×C3×D4—41no—
75C2×C3×D4—41no—
76C2×C3×D4—41no—
77C2×C3×D4—82no—
78C2×C3×D4—82no—
79C2×C3×D4C4⋊C412yescomplemented
80C2×C3×D4C22×C411yescharacteristic
81C2×C3×D4C2×Q811yescharacteristic
82C2×C3×D4#4112yes—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4C2×D411yescharacteristic
87C22×Dic3—22no—
88C22×Dic3—22no—
89C22×Dic3—42no—
90C22×Dic3—42no—
91C22×Dic3—82no—
92C22×Dic3—82no—
93C22×Dic3—81no—
94C3×M16—41no—
95C3×M16—41no—
96C2×C3×C8—41no—
97C3×C4⋊C4—22no—
98C3×C22⋊C4—42no—
99C3×C22⋊C4—41no—
100C3×C42—21no—
101C3×C42—41no—
102C3×C42C4⋊C412yescomplemented
103#217—24no—
104#217—42no—
105#217—42no—
106#217—82no—
107#217—82no—
108#217—81no—
109Dic3⋊C4—84no—
110C22×D4—22no—
111C22×D4#6711yescharacteristic
112C22×D4C22×D311yescharacteristic
113C23×C4—32no—
114C23×C4—61no—
115C2×M16—21no—
116#127—22no—
117#127Dic612yes—
118#127C4×D312yes—
119C4×D4—42no—
120#117—21no—
121C2×C4⋊C4—122no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—32no—
124C2×C22⋊C4—62no—
125C2×C22⋊C4—62no—
126C2×C22⋊C4—61no—
127C2×C22⋊C4—62no—
128C2×C42#6711yescharacteristic
129#102—42no—
130#100—22no—
131#99—34no—
132#99—62no—
133#95—61no—
134#95—62no—
135#95—61no—
136#95—62no—
137#95—122no—
138C23×C3—22no—
139C23×C3—22no—
140C23×C3—22no—
141C23×C3—42no—
142C23×C3—42no—
143C23×C3—42no—
144C23×C3—42no—
145C23×C3—41no—
146C23×C3—41no—
147C23×C3C4×D412yes—
148C23×C3#12011yescharacteristic
149C23×C3C22⋊Q812yes—
150C3×D4—41no—
151C3×D4—41no—
152C3×D4—42no—
153C3×D4—42no—
154C3×D4—81no—
155C3×D4—84no—
156C3×D4—84no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—22no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—82no—
163C2×C3×C4—82no—
164C2×C3×C4C2×C4⋊C411yescharacteristic
165C2×C3×C4#12311yescharacteristic
166C2×Dic3—44no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169C2×Dic3—42no—
170C2×Dic3—42no—
171C2×Dic3—44no—
172C2×Dic3—42no—
173C2×Dic3—42no—
174C2×Dic3—82no—
175C2×Dic3—82no—
176C2×Dic3—84no—
177C2×Dic3—82no—
178C3×C8—81no—
179C24—21no—
180C24D3×D412yes—
181C2×D4—24no—
182C2×D4—24no—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—82no—
189C2×D4—82no—
190C2×D4Dic3⋊C412yescomplemented
191C2×D4C2×Dic611yescharacteristic
192C2×D4C2×C4×D311yescharacteristic
193C2×D4#23712yes—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—62no—
198C22×C4—62no—
199C22×C4—122no—
200C22×C4—122no—
201C22×C4—242no—
202C22×C4—242no—
203C22×C4—241no—
204C22×C4C2×#6711yescharacteristic
205M16—41no—
206M16—41no—
207C2×C8—41no—
208C4⋊C4—22no—
209C4⋊C4—244no—
210C22⋊C4—42no—
211C22⋊C4—41no—
212C22⋊C4—64no—
213C22⋊C4—122no—
214C22⋊C4—122no—
215C22⋊C4—242no—
216C22⋊C4—242no—
217C22⋊C4—241no—
218C42—21no—
219C42—41no—
220C42Dic3⋊C412yescomplemented
221C22×C3—22no—
222C22×C3—22no—
223C22×C3—22no—
224C22×C3—42no—
225C22×C3—42no—
226C22×C3—44no—
227C22×C3—42no—
228C22×C3—42no—
229C22×C3—42no—
230C22×C3—82no—
231C22×C3—81no—
232C22×C3—81no—
233C22×C3—82no—
234C22×C3#38511yescharacteristic
235C22×C3B(4,𝔽2)11yescharacteristic
236C22×C3#45811yescharacteristic
237C3×C4—21no—
238C3×C4—21no—
239C3×C4—42no—
240C3×C4—82no—
241Dic3—82no—
242Dic3—84no—
243Dic3—82no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—41no—
252C23—41no—
253C23#87812yes—
254C23#67⋊C412yes—
255C23#93711yescharacteristic
256D4—41no—
257D4—41no—
258D4—42no—
259D4—42no—
260D4—81no—
261D4—84no—
262D4—84no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—124no—
271C2×C4—122no—
272C2×C4—122no—
273C2×C4—122no—
274C2×C4—122no—
275C2×C4—124no—
276C2×C4—122no—
277C2×C4—122no—
278C2×C4—242no—
279C2×C4—242no—
280C2×C4—244no—
281C2×C4—242no—
282C2×C4C2×Dic3⋊C411yescharacteristic
283C2×C4#93511yescharacteristic
284C8—81no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—42no—
288C2×C3—42no—
289C2×C3—82no—
290C2×C3#186711yescharacteristic
291C2×C3#220012yes—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—42no—
296C22—42no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—81no—
303C22—81no—
304C22—82no—
305C22#494611yescharacteristic
306C22#503211yescharacteristic
307C22#505311yescharacteristic
308C4—21no—
309C4—21no—
310C4—42no—
311C4—82no—
312C4—242no—
313C4—244no—
314C4—242no—
315C3#1296611yescharacteristic, complemented, Sylow
316C2—21no—
317C2—21no—
318C2—42no—
319C2—42no—
320C2—82no—
321C2#6959711yescharacteristic
322C2#7098512yes—
subgroupG/H
centerC22#5032
commutatorC2×C3×D4C22×C4
FrattiniC22×D4C22×D3
FittingC3×#1887C2
socleC22×C3B(4,𝔽2)
prime radicalC2×C3×#127C22
cosocleC22×C3×D4C23
hypercenter#1887D3
hypocenterC3#12966
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1887

Complements

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

NH ≅ G/Ntype
4. #71657C2semidirect
5. #67698C2semidirect
19. #4545C4semidirect
48. #834C2×C4semidirect
79. C2×C3×D4C4⋊C4semidirect
102. C3×C42C4⋊C4semidirect
190. C2×D4Dic3⋊C4semidirect
220. C42Dic3⋊C4semidirect
315. C3#12966semidirect

Maximal subgroups

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

subgroupindex
#831742
C3×#18872
C2×#45452
#716572
#676982
#129663