Finite Group Databasebeta

← #10710163

Subgroups — #10710163

Order 768.

826 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2311C211yesmaximal, characteristic
2C2×#5240C212yesmaximal
3#67641C214yesmaximal, complemented
4#12583—31nomaximal, Sylow
5C22×C3×#41C2211yescharacteristic
6C2×C3×#117C411yescharacteristic, complemented
7C22×C3×C4⋊C4C411yescharacteristic, complemented
8C2×#926—22no—
9C3×#391C412yescomplemented
10C3×#390C2212yes—
11C3×#386—22no—
12C3×#380—22no—
13#5240—24no—
14#5240C418yescomplemented
15C2×#834—22no—
16C2×#834C2214yes—
17C2×#410—32no—
18#2311D311yescharacteristic
19#1393—34no—
20C2×C3×#41—42no—
21C2×C3×#41—42no—
22C2×C3×#41C2×C414yes—
23C2×C3×#41D412yes—
24C2×C3×#41Q812yes—
25C22×C3×Q8C2×C411yescharacteristic
26C22×C3×D4D412yes—
27C22×C3×D4C2311yescharacteristic
28C23×C3×C4—21no—
29C23×C3×C4C2×C411yescharacteristic
30C23×C3×C4D412yes—
31C3×#117—42no—
32C2×C3×C4⋊C4—21no—
33C2×C3×C4⋊C4—21no—
34C2×C3×C4⋊C4—42no—
35C2×C3×C4⋊C4C2×C412yes—
36C2×C3×C22⋊C4—21no—
37C2×C3×C22⋊C4—24no—
38C2×C3×C42C2×C412yes—
39C2×#217—24no—
40#926—28no—
41#926—44no—
42C2×C4⋊Dic3—42no—
43C2×C4×Dic3—42no—
44C3×#95—24no—
45#834—28no—
46#834—24no—
47#834C2×C418yescomplemented
48C22×#41C2×D311yescharacteristic
49C2×#117—62no—
50C2×#117Dic311yescharacteristic, complemented
51C22×C4⋊C4Dic311yescharacteristic, complemented
52#410—38no—
53#410—64no—
54C2×#99—34no—
55C2×#99—62no—
56#391Dic312yescomplemented
57#390C2×D312yes—
58#386—22no—
59#380—22no—
60C24×C3C22⋊C411yescharacteristic
61C24×C3C2×D412yes—
62C3×#41—44no—
63C3×#41—44no—
64C3×#41—44no—
65C3×#41—44no—
66C2×C3×Q8—41no—
67C2×C3×Q8—41no—
68C2×C3×Q8C4212yes—
69C2×C3×Q8C4⋊C412yes—
70C2×C3×D4—42no—
71C2×C3×D4—42no—
72C2×C3×D4—41no—
73C2×C3×D4—41no—
74C2×C3×D4C22⋊C414yes—
75C2×C3×D4C4⋊C414yes—
76C2×C3×D4C22×C412yes—
77C2×C3×D4C2×D411yescharacteristic
78C2×C3×D4C2×Q811yescharacteristic
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4—24no—
82C22×C3×C4—22no—
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—42no—
87C22×C3×C4—82no—
88C22×C3×C4C4212yes—
89C22×C3×C4C22⋊C414yes—
90C22×C3×C4C22⋊C412yes—
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C4⋊C414yes—
93C22×C3×C4C4⋊C412yes—
94C22×C3×C4C22×C411yescharacteristic
95C22×Dic3—44no—
96C3×C4⋊C4—42no—
97C3×C4⋊C4—42no—
98C3×C4⋊C4—42no—
99C3×C22⋊C4—42no—
100C3×C22⋊C4—84no—
101C3×C42—42no—
102#217—28no—
103#217—44no—
104C4⋊Dic3—48no—
105C4×Dic3—48no—
106C2×#41—42no—
107C2×#41—42no—
108C2×#41Dic612yes—
109C2×#41C4×D314yes—
110C2×#41D1212yes—
111C22×Q8C2×Dic311yescharacteristic
112C22×D4#6712yes—
113C22×D4C22×D311yescharacteristic
114C23×C4—21no—
115C23×C4C2×Dic311yescharacteristic
116C23×C4#6712yes—
117#117—42no—
118#117—68no—
119#117—124no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—122no—
124C2×C4⋊C4C2×Dic312yes—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—64no—
128C2×C42—122no—
129C2×C42C2×Dic312yes—
130#99—38no—
131#99—68no—
132#99—64no—
133#95—24no—
134C23×C3—21no—
135C23×C3—22no—
136C23×C3—21no—
137C23×C3—22no—
138C23×C3—41no—
139C23×C3—41no—
140C23×C3—42no—
141C23×C3—42no—
142C23×C3#9512yes—
143C23×C3C2×C22⋊C412yes—
144C23×C3C2×C22⋊C411yescharacteristic
145C23×C3C2×C4⋊C412yes—
146C3×Q8—42no—
147C3×Q8—42no—
148C3×D4—44no—
149C3×D4—44no—
150C3×D4—42no—
151C3×D4—41no—
152C3×D4—41no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—24no—
158C2×C3×C4—24no—
159C2×C3×C4—24no—
160C2×C3×C4—24no—
161C2×C3×C4—24no—
162C2×C3×C4—24no—
163C2×C3×C4—42no—
164C2×C3×C4—44no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—84no—
169C2×C3×C4#9514yes—
170C2×C3×C4#9512yes—
171C2×C3×C4#9912yes—
172C2×C3×C4#9912yes—
173C2×C3×C4C2×C4211yescharacteristic
174C2×C3×C4C2×C4⋊C411yescharacteristic
175C2×Dic3—48no—
176C2×Dic3—84no—
177C2×Dic3—84no—
178C24#21711yescharacteristic
179C24C2×#6712yes—
180#41—44no—
181#41—44no—
182#41—44no—
183#41—44no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8C4×Dic312yes—
187C2×Q8C4⋊Dic312yes—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4Dic3⋊C414yes—
193C2×D4(C2×D3)⋊C414yes—
194C2×D4C2×Dic611yescharacteristic
195C2×D4C2×C4×D312yes—
196C2×D4C2×D1211yescharacteristic
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—82no—
206C22×C4—124no—
207C22×C4C4×Dic312yes—
208C22×C4Dic3⋊C414yes—
209C22×C4C4⋊Dic312yes—
210C22×C4(C2×D3)⋊C414yes—
211C22×C4#21712yes—
212C22×C4#21711yescharacteristic
213C22×C4C22×Dic311yescharacteristic
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—128no—
218C22⋊C4—42no—
219C22⋊C4—68no—
220C22⋊C4—84no—
221C22⋊C4—124no—
222C42—42no—
223C42—128no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—22no—
227C22×C3—22no—
228C22×C3—22no—
229C22×C3—42no—
230C22×C3—42no—
231C22×C3—42no—
232C22×C3—42no—
233C22×C3—42no—
234C22×C3—41no—
235C22×C3—42no—
236C22×C3—41no—
237C22×C3—82no—
238C22×C3C2×#9511yescharacteristic
239C22×C3C2×#9911yescharacteristic
240C22×C3C2×#9911yescharacteristic
241C3×C4—24no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—82no—
245C3×C4—82no—
246C3×C4#34214yes—
247Dic3—88no—
248C23—21no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—41no—
253C23—41no—
254C23—42no—
255C23—42no—
256C23C3⋊#9512yes—
257C23C2×Dic3⋊C412yes—
258C23C2×(C2×D3)⋊C412yes—
259C23C2×#21711yescharacteristic
260Q8—42no—
261Q8—42no—
262D4—44no—
263D4—44no—
264D4—41no—
265D4—42no—
266D4—41no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—84no—
283C2×C4—128no—
284C2×C4—244no—
285C2×C4—244no—
286C2×C4C3⋊#9514yes—
287C2×C4C3⋊#9512yes—
288C2×C4#83412yes—
289C2×C4#83412yes—
290C2×C4C2×C4×Dic311yescharacteristic
291C2×C4C2×C4⋊Dic311yescharacteristic
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—42no—
295C2×C3—42no—
296C2×C3—42no—
297C2×C3C2×#34211yescharacteristic
298C2×C3#212811yescharacteristic
299C2×C3#212911yescharacteristic
300C22—21no—
301C22—21no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—41no—
310C22—42no—
311C22—42no—
312C22—41no—
313C22—82no—
314C22C2×C3⋊#9511yescharacteristic
315C22C2×#83411yescharacteristic
316C22C2×#83411yescharacteristic
317C4—24no—
318C4—21no—
319C4—21no—
320C4—82no—
321C4—82no—
322C4—248no—
323C4#452614yes—
324C3#1258311yescharacteristic, complemented, Sylow
325C2—21no—
326C2—21no—
327C2—42no—
328C2—42no—
329C2—42no—
330C2C2×#452611yescharacteristic
331C2#7186111yescharacteristic
332C2#7187111yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC2×C3×C4C2×C42
FrattiniC22×D4C22×D3
FittingC3×#2311C2
socleC22×C3C2×#99
prime radicalC22×C3×#41C22
cosocleC22×C3×D4C23
hypercenter#2311D3
hypocenterC3#12583
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
3. #67641C2semidirect
6. C2×C3×#117C4semidirect
7. C22×C3×C4⋊C4C4semidirect
9. C3×#391C4semidirect
14. #5240C4semidirect
47. #834C2×C4semidirect
50. C2×#117Dic3semidirect
51. C22×C4⋊C4Dic3semidirect
56. #391Dic3semidirect
324. C3#12583semidirect

Maximal subgroups

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

subgroupindex
C3×#23112
C2×#52402
#676412
#125833