Finite Group Databasebeta

← #10690179

Subgroups — #10690179

Order 768.

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

normal subgroups only

This group has 756 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×#4979C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic
3C2×C3×#324C211yesmaximal, characteristic
4#69963C212yesmaximal
5#67675C212yesmaximal
6#9014—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C22×Dic3⋊C4—21no—
10C2×C3×C22⋊C8C2212yes—
11#5212C2212yes—
12#5211—21no—
13C4×#217—21no—
14C2×C3⋊#95—21no—
15C2×C3⋊#95C2211yescharacteristic
16C2×#816—24no—
17#4980—24no—
18#4979—22no—
19#4979C414yescomplemented
20C3×#324—22no—
21C2×#395—31no—
22#2313—31no—
23#1787—32no—
24C2×#324D311yescharacteristic
25#1294—32no—
26C23×C3×C4D411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×C3×C4C2311yescharacteristic
29C23×Dic3—21no—
30C23×Dic3—21no—
31C22×Dic6—22no—
32C22×Dic6—22no—
33C22×Dic6—21no—
34C22×Dic6—44no—
35C22×Dic6C2×C412yes—
36C22×C3×C8—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4C2×C412yes—
40C2×C3×C4⋊C4D412yes—
41C2×#217—22no—
42C2×C4⋊Dic3D414yes—
43C2×Dic3⋊C4—24no—
44C2×Dic3⋊C4—44no—
45C2×C4×Dic3—22no—
46C3×C22⋊C8—22no—
47C3×C22⋊C8—22no—
48C3⋊#95—22no—
49C3⋊#95—24no—
50C3⋊#95—24no—
51C3⋊#95—21no—
52C3⋊#95—22no—
53C3⋊#95C2×C412yes—
54#816—48no—
55C23×Q8—31no—
56C22×C4⋊C4—61no—
57C22×C4⋊C4C2×D311yescharacteristic
58C2×#103—64no—
59C2×C22⋊C8C2×D312yes—
60#395—34no—
61#395—62no—
62#387—64no—
63#381—61no—
64#380—32no—
65C4×C22⋊C4—61no—
66C2×#95—31no—
67C2×#95—61no—
68#324—22no—
69C24×C3C2×D411yescharacteristic
70C22×C3×C4—21no—
71C22×C3×C4—21no—
72C22×C3×C4—21no—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4—21no—
76C22×C3×C4—42no—
77C22×C3×C4—42no—
78C22×C3×C4C22⋊C412yes—
79C22×C3×C4C22⋊C412yes—
80C22×C3×C4C22×C411yescharacteristic
81C22×C3×C4C2×D412yes—
82C22×C3×C4C2×D411yescharacteristic
83C22×C3×C4#4112yes—
84C22×Dic3—24no—
85C22×Dic3—24no—
86C22×Dic3—22no—
87C22×Dic3—22no—
88C22×Dic3—24no—
89C22×Dic3—42no—
90C22×Dic3—42no—
91C22×Dic3—41no—
92C22×Dic3—41no—
93C22×Dic3—42no—
94C2×Dic6—28no—
95C2×Dic6—24no—
96C2×Dic6—44no—
97C2×Dic6—48no—
98C2×Dic6—44no—
99C2×Dic6—44no—
100C2×Dic6—42no—
101C2×C3×C8—44no—
102C2×C3×C8—42no—
103C3×C4⋊C4—24no—
104C3×C4⋊C4—24no—
105#217—42no—
106#217—42no—
107C4⋊Dic3—44no—
108Dic3⋊C4—48no—
109C4×Dic3—82no—
110C22×Q8—32no—
111C22×Q8—61no—
112C22×Q8—62no—
113C22×Q8—62no—
114C22×Q8—124no—
115C23×C4—61no—
116C23×C4—61no—
117C23×C4D1211yescharacteristic
118C23×C4D1211yescharacteristic
119C23×C4C22×D311yescharacteristic
120C22×C8—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—34no—
124C2×C4⋊C4—64no—
125C2×C4⋊C4—124no—
126C2×C4⋊C4C4×D312yes—
127C2×C4⋊C4#6712yes—
128C2×C22⋊C4—62no—
129C2×C42—62no—
130#103—128no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133#95—32no—
134#95—62no—
135#95—64no—
136#95—64no—
137#95—62no—
138#95—61no—
139C23×C3—22no—
140C23×C3—22no—
141C23×C3C2×C22⋊C411yescharacteristic
142C23×C3#12011yescharacteristic
143C23×C3#12311yescharacteristic
144C23×C3#13512yes—
145C23×C3#13712yes—
146C2×C3×C4—24no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—42no—
158C2×C3×C4—81no—
159C2×C3×C4C4×D412yes—
160C2×C3×C4#12112yes—
161C2×Dic3—24no—
162C2×Dic3—24no—
163C2×Dic3—42no—
164C2×Dic3—42no—
165C2×Dic3—44no—
166C2×Dic3—48no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169C2×Dic3—44no—
170C2×Dic3—44no—
171C2×Dic3—44no—
172C2×Dic3—44no—
173Dic6—28no—
174Dic6—48no—
175Dic6—44no—
176C3×C8—82no—
177C24C2×D1211yescharacteristic
178C2×Q8—68no—
179C2×Q8—64no—
180C2×Q8—124no—
181C2×Q8—124no—
182C2×Q8—124no—
183C2×Q8—128no—
184C2×Q8—122no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—64no—
194C22×C4—64no—
195C22×C4—62no—
196C22×C4—62no—
197C22×C4—64no—
198C22×C4—122no—
199C22×C4—122no—
200C22×C4—121no—
201C22×C4—121no—
202C22×C4—122no—
203C22×C4(C2×D3)⋊C412yes—
204C22×C4(C2×D3)⋊C412yes—
205C22×C4C2×C4×D311yescharacteristic
206C22×C4D3×D412yes—
207C22×C4#23712yes—
208C22×C4C2×#6711yescharacteristic
209C2×C8—44no—
210C2×C8—42no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—124no—
214C4⋊C4—128no—
215C22⋊C4—122no—
216C22⋊C4—122no—
217C42—242no—
218C22×C3—24no—
219C22×C3—24no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3#38611yescharacteristic
225C22×C3#41611yescharacteristic
226C22×C3#41911yescharacteristic
227C22×C3#45212yes—
228C22×C3#45612yes—
229C22×C3B(4,𝔽2)12yes—
230C22×C3#48512yes—
231C3×C4—22no—
232C3×C4—22no—
233C3×C4—42no—
234C3×C4—42no—
235Dic3—44no—
236Dic3—44no—
237Dic3—84no—
238C23—22no—
239C23—22no—
240C23#88211yescharacteristic
241C23#88611yescharacteristic
242C23#90412yes—
243C23#90912yes—
244C23C2×(C2×D3)⋊C411yescharacteristic
245Q8—68no—
246Q8—128no—
247Q8—124no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—64no—
261C2×C4—64no—
262C2×C4—81no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4—124no—
266C2×C4—128no—
267C2×C4—122no—
268C2×C4—124no—
269C2×C4—122no—
270C2×C4—124no—
271C2×C4—124no—
272C2×C4—124no—
273C2×C4#67⋊C412yes—
274C2×C4#93912yes—
275C8—82no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—22no—
279C2×C3#160314yes—
280C2×C3#187311yescharacteristic
281C2×C3#188611yescharacteristic
282C2×C3#188911yescharacteristic
283C22—24no—
284C22—24no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22#472312yes—
290C22#473312yes—
291C22#474112yes—
292C22#475212yes—
293C22#495811yescharacteristic
294C22#511311yescharacteristic
295C22#512011yescharacteristic
296C4—22no—
297C4—22no—
298C4—42no—
299C4—42no—
300C4—124no—
301C4—124no—
302C4—244no—
303C3#901411yescharacteristic, complemented, Sylow
304C2—21no—
305C2—21no—
306C2—22no—
307C2#6813214yes—
308C2#6993111yescharacteristic
309C2#7002211yescharacteristic
310C2#7010311yescharacteristic
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#9014
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
19. #4979C4semidirect
303. C3#9014semidirect

Maximal subgroups

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

subgroupindex
C2×#49792
#821552
C2×C3×#3242
#699632
#676752
#90143