Finite Group Databasebeta

← #10710160

Subgroups — #10710160

Order 768.

1026 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2309C211yesmaximal, characteristic
2C22×#834C212yesmaximal
3#67641C214yesmaximal, complemented
4#12582—31nomaximal, Sylow
5C23×C3×D4C2211yescharacteristic
6C2×C3×#117C411yescharacteristic, complemented
7C2×C3×#117C411yescharacteristic, complemented
8C22×#217—22no—
9C3×#390—22no—
10C3×#390C412yescomplemented
11C3×#390C2212yes—
12C3×#379—22no—
13C2×#834—22no—
14C2×#834—24no—
15C2×#834C418yescomplemented
16C2×#834C2214yes—
17C22×#99—32no—
18#2309D311yescharacteristic
19#1393—34no—
20C25×C3D412yes—
21C22×C3×D4—42no—
22C22×C3×D4—42no—
23C22×C3×D4C2×C414yes—
24C22×C3×D4C2×C411yescharacteristic
25C22×C3×D4D412yes—
26C22×C3×D4D412yes—
27C22×C3×D4Q812yes—
28C22×C3×D4C2311yescharacteristic
29C23×C3×C4C2×C411yescharacteristic
30C23×Dic3—42no—
31C3×#117—42no—
32C3×#117—42no—
33C2×C3×C4⋊C4—21no—
34C2×C3×C4⋊C4C2×C412yescomplemented
35C2×C3×C22⋊C4—21no—
36C2×C3×C22⋊C4—21no—
37C2×C3×C22⋊C4—24no—
38C2×C3×C22⋊C4—24no—
39C2×C3×C42—21no—
40C2×C3×C42C2×C412yescomplemented
41C2×#217—28no—
42C2×#217—24no—
43C2×#217—42no—
44C2×#217—44no—
45#834—28no—
46#834—24no—
47#834C2×C418yescomplemented
48C23×D4C2×D311yescharacteristic
49C2×#117Dic311yescharacteristic, complemented
50C2×#117Dic311yescharacteristic, complemented
51C22×C22⋊C4—62no—
52C2×#99—38no—
53C2×#99—34no—
54C2×#99—64no—
55C2×#99—62no—
56#390—22no—
57#390Dic312yescomplemented
58#390C2×D312yes—
59#379—22no—
60C24×C3—22no—
61C24×C3—24no—
62C24×C3—22no—
63C24×C3—84no—
64C24×C3C22⋊C414yes—
65C24×C3C22⋊C412yes—
66C24×C3C22⋊C411yescharacteristic
67C24×C3C4⋊C414yes—
68C24×C3C2×D412yes—
69C2×C3×D4—44no—
70C2×C3×D4—44no—
71C2×C3×D4—41no—
72C2×C3×D4—41no—
73C2×C3×D4—44no—
74C2×C3×D4—42no—
75C2×C3×D4—42no—
76C2×C3×D4—44no—
77C2×C3×D4—41no—
78C2×C3×D4—41no—
79C2×C3×D4C4212yes—
80C2×C3×D4C22⋊C414yes—
81C2×C3×D4C4⋊C414yes—
82C2×C3×D4C4⋊C412yes—
83C2×C3×D4C22×C412yes—
84C2×C3×D4C2×D411yescharacteristic
85C2×C3×D4C2×Q811yescharacteristic
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4C4212yes—
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C4⋊C412yes—
93C22×C3×C4C22×C411yescharacteristic
94C22×Dic3—48no—
95C22×Dic3—44no—
96C22×Dic3—88no—
97C3×C4⋊C4—42no—
98C3×C4⋊C4—42no—
99C3×C22⋊C4—42no—
100C3×C22⋊C4—42no—
101C3×C22⋊C4—84no—
102C3×C22⋊C4—84no—
103C3×C42—42no—
104C3×C42—42no—
105#217—28no—
106#217—48no—
107#217—44no—
108C25#6712yes—
109C22×D4—42no—
110C22×D4—42no—
111C22×D4Dic612yes—
112C22×D4C4×D314yes—
113C22×D4D1212yes—
114C22×D4C2×Dic311yescharacteristic
115C22×D4#6712yes—
116C22×D4C22×D311yescharacteristic
117C23×C4—122no—
118C23×C4C2×Dic311yescharacteristic
119#117—42no—
120#117—42no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4C2×Dic312yescomplemented
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—68no—
128C2×C22⋊C4—64no—
129C2×C22⋊C4—124no—
130C2×C22⋊C4—122no—
131C2×C42—21no—
132C2×C42C2×Dic312yescomplemented
133#99—38no—
134#99—68no—
135#99—64no—
136C23×C3—24no—
137C23×C3—24no—
138C23×C3—22no—
139C23×C3—22no—
140C23×C3—24no—
141C23×C3—24no—
142C23×C3—22no—
143C23×C3—22no—
144C23×C3—42no—
145C23×C3—48no—
146C23×C3—44no—
147C23×C3—44no—
148C23×C3—84no—
149C23×C3—84no—
150C23×C3—84no—
151C23×C3—88no—
152C23×C3#9514yes—
153C23×C3#9512yes—
154C23×C3#9912yes—
155C23×C3#9912yes—
156C23×C3C2×C22⋊C412yes—
157C23×C3C2×C22⋊C411yescharacteristic
158C23×C3C2×C4⋊C412yes—
159C3×D4—42no—
160C3×D4—44no—
161C3×D4—44no—
162C3×D4—42no—
163C3×D4—42no—
164C3×D4—41no—
165C3×D4—41no—
166C2×C3×C4—24no—
167C2×C3×C4—24no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—42no—
173C2×C3×C4—42no—
174C2×C3×C4—44no—
175C2×C3×C4—42no—
176C2×C3×C4—42no—
177C2×C3×C4#9512yes—
178C2×C3×C4C2×C4211yescharacteristic
179C2×C3×C4C2×C4⋊C411yescharacteristic
180C2×Dic3—48no—
181C2×Dic3—816no—
182C2×Dic3—88no—
183C24—22no—
184C24—22no—
185C24—24no—
186C24—84no—
187C24Dic3⋊C414yes—
188C24(C2×D3)⋊C414yes—
189C24#21712yes—
190C24#21711yescharacteristic
191C24C2×#6712yes—
192C2×D4—44no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4C4×Dic312yes—
203C2×D4Dic3⋊C414yes—
204C2×D4C4⋊Dic312yes—
205C2×D4(C2×D3)⋊C414yes—
206C2×D4C2×Dic611yescharacteristic
207C2×D4C2×C4×D312yes—
208C2×D4C2×D1211yescharacteristic
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—124no—
214C22×C4—128no—
215C22×C4—248no—
216C22×C4C4×Dic312yes—
217C22×C4C4⋊Dic312yes—
218C22×C4#21711yescharacteristic
219C22×C4C22×Dic311yescharacteristic
220C4⋊C4—42no—
221C4⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—68no—
225C22⋊C4—84no—
226C22⋊C4—84no—
227C22⋊C4—124no—
228C22⋊C4—128no—
229C42—42no—
230C42—42no—
231C22×C3—22no—
232C22×C3—24no—
233C22×C3—22no—
234C22×C3—24no—
235C22×C3—22no—
236C22×C3—42no—
237C22×C3—42no—
238C22×C3—44no—
239C22×C3—42no—
240C22×C3—42no—
241C22×C3—48no—
242C22×C3—48no—
243C22×C3—44no—
244C22×C3—44no—
245C22×C3—82no—
246C22×C3—82no—
247C22×C3—84no—
248C22×C3—84no—
249C22×C3#34214yes—
250C22×C3C2×#9511yescharacteristic
251C22×C3C2×#9911yescharacteristic
252C22×C3C2×#9911yescharacteristic
253C3×C4—22no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—82no—
257C3×C4—82no—
258Dic3—88no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—24no—
263C23—22no—
264C23—22no—
265C23—24no—
266C23—24no—
267C23—42no—
268C23—48no—
269C23—44no—
270C23—44no—
271C23—84no—
272C23—84no—
273C23—84no—
274C23—88no—
275C23C3⋊#9514yes—
276C23C3⋊#9512yes—
277C23#83412yes—
278C23#83412yes—
279C23C2×Dic3⋊C412yes—
280C23C2×(C2×D3)⋊C412yes—
281C23C2×#21711yescharacteristic
282D4—42no—
283D4—44no—
284D4—42no—
285D4—44no—
286D4—41no—
287D4—42no—
288D4—41no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—128no—
301C2×C4—2416no—
302C2×C4—248no—
303C2×C4C3⋊#9512yes—
304C2×C4C2×C4×Dic311yescharacteristic
305C2×C4C2×C4⋊Dic311yescharacteristic
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—24no—
309C2×C3—42no—
310C2×C3—44no—
311C2×C3—42no—
312C2×C3C2×#34211yescharacteristic
313C2×C3#212712yes—
314C22—22no—
315C22—24no—
316C22—22no—
317C22—24no—
318C22—22no—
319C22—42no—
320C22—42no—
321C22—44no—
322C22—42no—
323C22—42no—
324C22—48no—
325C22—48no—
326C22—44no—
327C22—44no—
328C22—82no—
329C22—82no—
330C22—84no—
331C22—84no—
332C22#452614yes—
333C22C2×C3⋊#9511yescharacteristic
334C22C2×#83411yescharacteristic
335C22C2×#83411yescharacteristic
336C4—22no—
337C4—21no—
338C4—21no—
339C4—82no—
340C4—82no—
341C4—248no—
342C3#1258211yescharacteristic, complemented, Sylow
343C2—21no—
344C2—21no—
345C2—24no—
346C2—42no—
347C2—44no—
348C2—42no—
349C2C2×#452611yescharacteristic
350C2#7186012yes—
subgroupG/H
centerC22C2×#834
commutatorC2×C3×C4C2×C42
FrattiniC22×D4C22×D3
FittingC3×#2309C2
socleC22×C3C2×#99
prime radicalC23×C3×D4C22
cosocleC22×C3×D4C23
hypercenter#2309D3
hypocenterC3#12582
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #2309

Complements

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

NH ≅ G/Ntype
3. #67641C2semidirect
6. C2×C3×#117C4semidirect
7. C2×C3×#117C4semidirect
10. C3×#390C4semidirect
15. C2×#834C4semidirect
34. C2×C3×C4⋊C4C2×C4semidirect
40. C2×C3×C42C2×C4semidirect
47. #834C2×C4semidirect
49. C2×#117Dic3semidirect
50. C2×#117Dic3semidirect
57. #390Dic3semidirect
122. C2×C4⋊C4C2×Dic3semidirect
132. C2×C42C2×Dic3semidirect
342. C3#12582semidirect

Maximal subgroups

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

subgroupindex
C3×#23092
C22×#8342
#676412
#125823