Finite Group Databasebeta

← #10710175

Subgroups — #10710175

Order 768.

794 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2311C211yesmaximal, characteristic
2C2×#5237C212yesmaximal
3#67644C214yesmaximal, complemented
4#12587—31nomaximal, Sylow
5C22×C3×#41C2211yescharacteristic
6C2×C3×#117C411yescharacteristic
7C22×C3×C4⋊C4C411yescharacteristic
8C22×#208—22no—
9C3×#391C2212yes—
10C3×#390C412yes—
11C3×#386—22no—
12C3×#380—22no—
13#5237—24no—
14#5237C418yescomplemented
15C2×#836C2214yes—
16C2×#833—22no—
17C2×#413—32no—
18#2311D311yescharacteristic
19#1394—34no—
20C2×C3×#41—42no—
21C2×C3×#41—42no—
22C2×C3×#41C2×C414yes—
23C2×C3×#41D412yes—
24C2×C3×#41Q812yes—
25C22×C3×Q8C2311yescharacteristic
26C22×C3×D4C2×C411yescharacteristic
27C22×C3×D4D412yes—
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×#208—28no—
40C2×#208—24no—
41C2×#208—42no—
42C2×#208—44no—
43C22×C3⋊C8—42no—
44C3×#95—24no—
45#836—24no—
46#836C2×C418yescomplemented
47#833—28no—
48C22×#41C2×D311yescharacteristic
49C22×M16—62no—
50C2×#117Dic311yescharacteristic
51C22×C4⋊C4Dic311yescharacteristic
52#413—38no—
53#413—64no—
54C2×#101—34no—
55C2×#100—62no—
56#391C2×D312yes—
57#390Dic312yes—
58#386—22no—
59#380—22no—
60C24×C3C22⋊C412yes—
61C24×C3C22⋊C411yescharacteristic
62C3×#41—44no—
63C3×#41—44no—
64C3×#41—44no—
65C3×#41—44no—
66C2×C3×Q8—41no—
67C2×C3×Q8—41no—
68C2×C3×Q8C22×C412yes—
69C2×C3×Q8C2×D411yescharacteristic
70C2×C3×Q8C2×Q811yescharacteristic
71C2×C3×D4—41no—
72C2×C3×D4—41no—
73C2×C3×D4—42no—
74C2×C3×D4—42no—
75C2×C3×D4C4212yes—
76C2×C3×D4C22⋊C414yes—
77C2×C3×D4C4⋊C414yes—
78C2×C3×D4C4⋊C412yes—
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⋊C411yescharacteristic
91C22×C3×C4C4⋊C414yes—
92C22×C3×C4C4⋊C412yes—
93C22×C3×C4C22×C411yescharacteristic
94C22×C3×C4C2×D412yes—
95C3×C4⋊C4—42no—
96C3×C4⋊C4—42no—
97C3×C4⋊C4—42no—
98C3×C22⋊C4—42no—
99C3×C22⋊C4—84no—
100C3×C42—42no—
101#208—28no—
102#208—44no—
103#208—48no—
104C2×C3⋊C8—44no—
105C2×C3⋊C8—48no—
106C2×#41—42no—
107C2×#41—42no—
108C2×#41Dic612yes—
109C2×#41C4×D314yes—
110C2×#41D1212yes—
111C22×Q8C22×D311yescharacteristic
112C22×D4C2×Dic311yescharacteristic
113C22×D4#6712yes—
114C23×C4—21no—
115C23×C4C2×Dic311yescharacteristic
116C23×C4#6712yes—
117C2×M16—68no—
118C2×M16—64no—
119C2×M16—122no—
120C2×M16—124no—
121C22×C8—122no—
122#117—42no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4C2×Dic312yes—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—24no—
129C2×C42C2×Dic312yes—
130#101—38no—
131#101—64no—
132#100—68no—
133#95—24no—
134C23×C3—21no—
135C23×C3—21no—
136C23×C3—22no—
137C23×C3—22no—
138C23×C3—41no—
139C23×C3—41no—
140C23×C3—42no—
141C23×C3—42no—
142C23×C3#9512yes—
143C23×C3#9514yes—
144C23×C3C2×C22⋊C411yescharacteristic
145C3×Q8—42no—
146C3×Q8—41no—
147C3×Q8—41no—
148C3×D4—42no—
149C3×D4—44no—
150C3×D4—44no—
151C3×D4—42no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—24no—
157C2×C3×C4—24no—
158C2×C3×C4—24no—
159C2×C3×C4—24no—
160C2×C3×C4—24no—
161C2×C3×C4—24no—
162C2×C3×C4—42no—
163C2×C3×C4—44no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—84no—
168C2×C3×C4#9512yes—
169C2×C3×C4#9912yes—
170C2×C3×C4#9912yes—
171C2×C3×C4C2×C4211yescharacteristic
172C2×C3×C4C2×C22⋊C412yes—
173C2×C3×C4C2×C4⋊C412yes—
174C2×C3×C4C2×C4⋊C411yescharacteristic
175C3⋊C8—48no—
176C24#21712yes—
177C24#21711yescharacteristic
178#41—44no—
179#41—44no—
180#41—44no—
181#41—44no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8C2×Dic611yescharacteristic
185C2×Q8C2×C4×D312yes—
186C2×Q8C2×D1211yescharacteristic
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4C4×Dic312yes—
192C2×D4Dic3⋊C414yes—
193C2×D4C4⋊Dic312yes—
194C2×D4(C2×D3)⋊C414yes—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—82no—
204C22×C4C4×Dic312yes—
205C22×C4Dic3⋊C414yes—
206C22×C4C4⋊Dic312yes—
207C22×C4(C2×D3)⋊C414yes—
208C22×C4#21711yescharacteristic
209C22×C4C22×Dic311yescharacteristic
210C22×C4C2×#6712yes—
211M16—68no—
212M16—124no—
213M16—128no—
214C2×C8—124no—
215C2×C8—128no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—84no—
221C42—42no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—22no—
225C22×C3—24no—
226C22×C3—42no—
227C22×C3—44no—
228C22×C3—42no—
229C22×C3—41no—
230C22×C3—42no—
231C22×C3—41no—
232C22×C3—42no—
233C22×C3—82no—
234C22×C3C2×#9511yescharacteristic
235C22×C3C2×#9911yescharacteristic
236C22×C3C2×#9911yescharacteristic
237C3×C4—22no—
238C3×C4—21no—
239C3×C4—22no—
240C3×C4—21no—
241C3×C4—82no—
242C3×C4—82no—
243C3×C4#34214yes—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23C3⋊#9512yes—
253C23C3⋊#9514yes—
254C23C2×#21711yescharacteristic
255Q8—41no—
256Q8—42no—
257Q8—41no—
258D4—42no—
259D4—44no—
260D4—42no—
261D4—44no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—84no—
278C2×C4C3⋊#9512yes—
279C2×C4#83412yes—
280C2×C4#83412yes—
281C2×C4C2×C4×Dic311yescharacteristic
282C2×C4C2×Dic3⋊C412yes—
283C2×C4C2×C4⋊Dic311yescharacteristic
284C2×C4C2×(C2×D3)⋊C412yes—
285C8—128no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—42no—
289C2×C3—44no—
290C2×C3C2×#34211yescharacteristic
291C2×C3#213411yescharacteristic
292C2×C3#213511yescharacteristic
293C22—21no—
294C22—21no—
295C22—22no—
296C22—24no—
297C22—42no—
298C22—44no—
299C22—42no—
300C22—41no—
301C22—42no—
302C22—41no—
303C22—42no—
304C22—82no—
305C22C2×C3⋊#9511yescharacteristic
306C22C2×#83411yescharacteristic
307C22C2×#83411yescharacteristic
308C4—22no—
309C4—21no—
310C4—22no—
311C4—21no—
312C4—82no—
313C4—82no—
314C4#452614yes—
315C3#1258711yescharacteristic, complemented, Sylow
316C2—21no—
317C2—21no—
318C2—42no—
319C2—44no—
320C2C2×#452611yescharacteristic
321C2#7186211yescharacteristic
322C2#7187311yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC2×C3×C4C2×C42
FrattiniC22×Q8C22×D3
FittingC3×#2311C2
socleC22×C3C2×#99
prime radicalC22×C3×#41C22
cosocleC22×C3×Q8C23
hypercenter#2311D3
hypocenterC3#12587
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. #67644C2semidirect
14. #5237C4semidirect
46. #836C2×C4semidirect
315. C3#12587semidirect

Maximal subgroups

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

subgroupindex
C3×#23112
C2×#52372
#676442
#125873