Finite Group Databasebeta

← #10690219

Subgroups — #10690219

Order 768.

572 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#4986C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic
3C2×C3×#324C211yesmaximal, characteristic
4#69925C212yesmaximal
5#67675C212yesmaximal
6#9022—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C22×C4⋊Dic3C2211yescharacteristic
9C2×C3×C22⋊C8C2212yes—
10#5212C2212yes—
11#5211—21no—
12C4×#217—21no—
13C2×C3⋊#95—21no—
14C2×C3⋊#95—21no—
15C2×C3⋊#95C2211yescharacteristic
16#4986—22no—
17#4986C414yescomplemented
18#4980—24no—
19C3×#324—22no—
20#4551—24no—
21C2×#400—31no—
22#2313—31no—
23#1824—32no—
24C2×#324D311yescharacteristic
25#1294—32no—
26C23×C3×C4D411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×C3×C4C2311yescharacteristic
29C23×Dic3—21no—
30C23×Dic3—21no—
31C22×C3×C8—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4C2×C412yes—
35C2×C3×C4⋊C4D412yes—
36C2×#217—22no—
37C2×C4⋊Dic3—22no—
38C2×C4⋊Dic3—22no—
39C2×C4⋊Dic3—21no—
40C2×C4⋊Dic3C2×C412yes—
41C2×C4⋊Dic3D414yes—
42C2×C4×Dic3—22no—
43C3×C22⋊C8—22no—
44C3×C22⋊C8—22no—
45C3⋊#95—22no—
46C3⋊#95—24no—
47C3⋊#95—24no—
48C3⋊#95—24no—
49C3⋊#95—21no—
50C3⋊#95—22no—
51C3⋊#95C2×C412yes—
52C22×C4⋊C4—31no—
53C22×C4⋊C4C2×D311yescharacteristic
54C2×C22⋊C8C2×D312yes—
55#400—34no—
56#400—62no—
57#387—64no—
58#381—61no—
59#380—32no—
60C4×C22⋊C4—61no—
61C2×#95—31no—
62C2×#95—61no—
63C2×#95—61no—
64#340—64no—
65#324—22no—
66C24×C3C2×D411yescharacteristic
67C22×C3×C4—21no—
68C22×C3×C4—21no—
69C22×C3×C4—21no—
70C22×C3×C4—22no—
71C22×C3×C4—22no—
72C22×C3×C4—21no—
73C22×C3×C4—42no—
74C22×C3×C4—42no—
75C22×C3×C4C22⋊C412yes—
76C22×C3×C4C22⋊C412yes—
77C22×C3×C4C22×C411yescharacteristic
78C22×C3×C4C2×D412yes—
79C22×C3×C4C2×D411yescharacteristic
80C22×C3×C4#4112yes—
81C22×Dic3—22no—
82C22×Dic3—22no—
83C22×Dic3—24no—
84C22×Dic3—42no—
85C22×Dic3—42no—
86C22×Dic3—44no—
87C22×Dic3—41no—
88C22×Dic3—41no—
89C22×Dic3—42no—
90C2×C3×C8—44no—
91C2×C3×C8—42no—
92C3×C4⋊C4—24no—
93C3×C4⋊C4—24no—
94#217—42no—
95#217—42no—
96C4⋊Dic3—24no—
97C4⋊Dic3—42no—
98C4⋊Dic3—44no—
99C4×Dic3—82no—
100C23×C4—61no—
101C23×C4—61no—
102C23×C4D1211yescharacteristic
103C23×C4D1211yescharacteristic
104C23×C4C22×D311yescharacteristic
105C22×C8—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—32no—
109C2×C4⋊C4—34no—
110C2×C4⋊C4—62no—
111C2×C4⋊C4—62no—
112C2×C4⋊C4—61no—
113C2×C4⋊C4C4×D312yes—
114C2×C4⋊C4#6712yes—
115C2×C22⋊C4—62no—
116C2×C42—62no—
117C22⋊C8—22no—
118C22⋊C8—22no—
119#95—32no—
120#95—62no—
121#95—64no—
122#95—64no—
123#95—64no—
124#95—62no—
125#95—61no—
126C23×C3—22no—
127C23×C3—22no—
128C23×C3C2×C22⋊C411yescharacteristic
129C23×C3#12011yescharacteristic
130C23×C3#12311yescharacteristic
131C23×C3#13512yes—
132C23×C3#13712yes—
133C2×C3×C4—24no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—41no—
139C2×C3×C4—41no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4—42no—
143C2×C3×C4—42no—
144C2×C3×C4—42no—
145C2×C3×C4—81no—
146C2×C3×C4C4×D412yes—
147C2×C3×C4#12112yes—
148C2×Dic3—42no—
149C2×Dic3—42no—
150C2×Dic3—44no—
151C2×Dic3—42no—
152C2×Dic3—42no—
153C2×Dic3—44no—
154C2×Dic3—44no—
155C2×Dic3—44no—
156C2×Dic3—84no—
157C2×Dic3—82no—
158C2×Dic3—82no—
159C3×C8—82no—
160C24C2×D1211yescharacteristic
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—64no—
170C22×C4—62no—
171C22×C4—62no—
172C22×C4—122no—
173C22×C4—122no—
174C22×C4—124no—
175C22×C4—121no—
176C22×C4—121no—
177C22×C4—122no—
178C22×C4(C2×D3)⋊C412yes—
179C22×C4(C2×D3)⋊C412yes—
180C22×C4C2×C4×D311yescharacteristic
181C22×C4D3×D412yes—
182C22×C4#23712yes—
183C22×C4C2×#6711yescharacteristic
184C2×C8—44no—
185C2×C8—42no—
186C4⋊C4—24no—
187C4⋊C4—24no—
188C4⋊C4—64no—
189C4⋊C4—124no—
190C4⋊C4—122no—
191C22⋊C4—122no—
192C22⋊C4—122no—
193C42—242no—
194C22×C3—24no—
195C22×C3—24no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3#38611yescharacteristic
201C22×C3#41611yescharacteristic
202C22×C3#41911yescharacteristic
203C22×C3#45212yes—
204C22×C3#45412yes—
205C22×C3#45812yes—
206C22×C3#48512yes—
207C3×C4—22no—
208C3×C4—22no—
209C3×C4—42no—
210C3×C4—42no—
211Dic3—82no—
212Dic3—84no—
213Dic3—82no—
214C23—22no—
215C23—22no—
216C23#88211yescharacteristic
217C23#88611yescharacteristic
218C23#90412yes—
219C23#90912yes—
220C23C2×(C2×D3)⋊C411yescharacteristic
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—81no—
234C2×C4—122no—
235C2×C4—122no—
236C2×C4—124no—
237C2×C4—122no—
238C2×C4—124no—
239C2×C4—124no—
240C2×C4—124no—
241C2×C4—122no—
242C2×C4—244no—
243C2×C4—242no—
244C2×C4—242no—
245C2×C4#67⋊C412yes—
246C2×C4#93912yes—
247C8—82no—
248C2×C3—21no—
249C2×C3—21no—
250C2×C3—22no—
251C2×C3#161514yes—
252C2×C3#187311yescharacteristic
253C2×C3#188811yescharacteristic
254C2×C3#189011yescharacteristic
255C22—24no—
256C22—24no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22#472312yes—
262C22#473312yes—
263C22#474212yes—
264C22#475312yes—
265C22#495811yescharacteristic
266C22#511311yescharacteristic
267C22#512011yescharacteristic
268C4—22no—
269C4—22no—
270C4—42no—
271C4—42no—
272C4—242no—
273C4—244no—
274C4—242no—
275C3#902211yescharacteristic, complemented, Sylow
276C2—21no—
277C2—21no—
278C2—22no—
279C2#6813614yes—
280C2#6993111yescharacteristic
281C2#7002311yescharacteristic
282C2#7010411yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#324C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#324D3
hypocenterC3#9022
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
17. #4986C4semidirect
275. C3#9022semidirect

Maximal subgroups

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

subgroupindex
C2×#49862
#821552
C2×C3×#3242
#699252
#676752
#90223