Finite Group Databasebeta

← #146355

Subgroups — #146355

Order 512.

1168 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 250 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×C4×#99C212yesmaximal
2#13785C211yesmaximal, characteristic
3C2×#1393C214yesmaximal
4C22×C4×D4C2211yescharacteristic
5C22×#99C2214yes—
6C2×#390C2212yes—
7C2×C4×C4⋊C4C411yescharacteristic, complemented
8C2×C4×C22⋊C4—22no—
9C2×#376C411yescharacteristic, complemented
10C4×#99—24no—
11C4×#99C418yescomplemented
12C2×#342—22no—
13#1437—22no—
14C4⋊#95C412yescomplemented
15C22⋊#95—22no—
16#1393C4116yescomplemented
17C23×D4C2311yescharacteristic
18C24×C4D412yes—
19C2×C4×D4—42no—
20C2×C4×D4—42no—
21C2×C4×D4C2×C414yes—
22C2×C4×D4D412yes—
23C2×C4×D4Q812yes—
24C22×C4⋊C4C2×C412yes—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C22⋊C4—24no—
27C22×C22⋊C4—24no—
28C22×C22⋊C4D412yes—
29C22×C42—21no—
30C22×C42—42no—
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C412yes—
33C2×#99—24no—
34C2×#99—28no—
35C2×#99C2×C4116yes—
36C2×#99C2×C418yes—
37#390—24no—
38#390C2×C418yes—
39C4×C4⋊C4—42no—
40C4×C22⋊C4—28no—
41C4×C22⋊C4—44no—
42#376—42no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—24no—
47C2×#95—42no—
48#342—28no—
49C25C2×D412yes—
50C22×D4—41no—
51C22×D4—41no—
52C22×D4—42no—
53C22×D4C22×C412yes—
54C22×D4C22×C414yes—
55C22×D4C2×D411yescharacteristic
56C22×D4C2×Q811yescharacteristic
57C22×D4#4114yes—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—24no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—44no—
64C23×C4—84no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C412yes—
67C23×C4C22⋊C414yes—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C4⋊C414yes—
70C23×C4C22×C411yescharacteristic
71C4×D4—44no—
72C4×D4—44no—
73C4×D4—44no—
74C4×D4—44no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4C4212yes—
81C2×C4⋊C4C4214yescomplemented
82C2×C4⋊C4C4⋊C412yes—
83C2×C22⋊C4—28no—
84C2×C22⋊C4—216no—
85C2×C22⋊C4—216no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—48no—
89C2×C22⋊C4—44no—
90C2×C22⋊C4—84no—
91C2×C22⋊C4—84no—
92C2×C22⋊C4C22⋊C414yes—
93C2×C22⋊C4C4⋊C414yes—
94C2×C42—22no—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42—42no—
98C2×C42—42no—
99C2×C42—48no—
100C2×C42—42no—
101C2×C42—82no—
102C2×C42—88no—
103C2×C42C4212yes—
104C2×C42C4214yescomplemented
105C2×C42C4⋊C412yes—
106#99—28no—
107#99—216no—
108#99C42116yescomplemented
109#95—42no—
110#95—48no—
111#95—42no—
112#95—42no—
113#95—84no—
114C24—22no—
115C24—22no—
116C24—24no—
117C24—42no—
118C24—44no—
119C24—42no—
120C24C2×C22⋊C411yescharacteristic
121C24C2×C22⋊C412yes—
122C24C2×C4⋊C412yes—
123C24C4×D418yes—
124C2×D4—42no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—42no—
128C2×D4—44no—
129C2×D4—44no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—44no—
133C2×D4—44no—
134C2×D4C2×C4214yes—
135C2×D4#11714yes—
136C2×D4C4×D414yes—
137C2×D4C4×Q814yes—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—42no—
152C22×C4—48no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—44no—
156C22×C4—48no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—416no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—84no—
169C22×C4—84no—
170C22×C4—84no—
171C22×C4—88no—
172C22×C4—88no—
173C22×C4—84no—
174C22×C4—84no—
175C22×C4#9512yes—
176C22×C4#9514yes—
177C22×C4#9512yes—
178C22×C4#9912yes—
179C22×C4#9912yes—
180C22×C4C2×C4211yescharacteristic
181C22×C4C2×C4212yes—
182C22×C4C2×C4⋊C411yescharacteristic
183C22×C4#11712yes—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C22⋊C4—216no—
189C22⋊C4—44no—
190C22⋊C4—44no—
191C22⋊C4—48no—
192C22⋊C4—416no—
193C22⋊C4—816no—
194C42—24no—
195C42—24no—
196C42—44no—
197C42—44no—
198C42—816no—
199C42—84no—
200C23—22no—
201C23—21no—
202C23—21no—
203C23—22no—
204C23—22no—
205C23—28no—
206C23—22no—
207C23—22no—
208C23—28no—
209C23—48no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—48no—
215C23—44no—
216C23—42no—
217C23—42no—
218C23—82no—
219C23—82no—
220C23—84no—
221C23—84no—
222C23C2×#9511yescharacteristic
223C23C4×C22⋊C412yes—
224C23C4×C22⋊C414yes—
225C23C4×C4⋊C414yes—
226C23C2×#9911yescharacteristic
227C23C2×#9911yescharacteristic
228C23#41012yes—
229D4—44no—
230D4—44no—
231D4—44no—
232D4—44no—
233C2×C4—24no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—48no—
247C2×C4—48no—
248C2×C4—42no—
249C2×C4—44no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—416no—
255C2×C4—44no—
256C2×C4—48no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—84no—
261C2×C4—84no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—84no—
268C2×C4—88no—
269C2×C4—88no—
270C2×C4—816no—
271C2×C4—816no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4#34214yes—
275C2×C4C4311yescharacteristic
276C2×C4#37611yescharacteristic
277C2×C4C4×C4⋊C412yes—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—24no—
287C22—24no—
288C22—44no—
289C22—44no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—48no—
295C22—48no—
296C22—88no—
297C22—82no—
298C22—82no—
299C22C4×#9511yescharacteristic
300C22C2×#34211yescharacteristic
301C22#174011yescharacteristic
302C22C4×#9912yes—
303C22C4×#9912yes—
304C22#212712yes—
305C22#212811yescharacteristic
306C22#212911yescharacteristic
307C4—24no—
308C4—21no—
309C4—21no—
310C4—22no—
311C4—44no—
312C4—42no—
313C4—84no—
314C4—816no—
315C4—84no—
316C2—22no—
317C2—21no—
318C2—21no—
319C2—44no—
320C2—44no—
321C2C4×#34211yescharacteristic
322C2#1258211yescharacteristic
323C2#1258311yescharacteristic
324C2#1262712yes—
325C2#1262812yes—
subgroupG/H
center, socleC23C2×#99
commutatorC2×C4C43
Frattini, prime radical, cosocleC23×D4C23
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×C4×C4⋊C4C4semidirect
9. C2×#376C4semidirect
11. C4×#99C4semidirect
14. C4⋊#95C4semidirect
16. #1393C4semidirect
81. C2×C4⋊C4C42semidirect
104. C2×C42C42semidirect
108. #99C42semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#992
#137852
C2×#13932