Finite Group Databasebeta

← #147015

Subgroups — #147015

Order 512.

1072 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 202 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×#1799C212yesmaximal
2#13785C211yesmaximal, characteristic
3C2×#1393C214yesmaximal
4C22×C4×D4C2211yescharacteristic
5C22×#99C2214yes—
6C2×#390C2212yes—
7C2×#380—22no—
8C2×C4×C4⋊C4C411yescharacteristic, complemented
9C2×#376C411yescharacteristic, complemented
10#1799—24no—
11#1799C418yescomplemented
12C2×#342—22no—
13#1437—22no—
14C4⋊#95C412yescomplemented
15C22⋊#95—22no—
16#1393—28no—
17C23×D4C2311yescharacteristic
18C24×C4D412yes—
19C2×C4×D4—42no—
20C2×C4×D4—42no—
21C2×C4×D4C2×C414yes—
22C2×C4×D4D412yes—
23C2×C4×D4Q812yes—
24C22×C4⋊C4—42no—
25C22×C4⋊C4C2×C412yes—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C22⋊C4—24no—
28C22×C22⋊C4—24no—
29C22×C22⋊C4D412yes—
30C22×C42—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C412yes—
33C2×#99—28no—
34C2×#99—28no—
35C2×#99—24no—
36C2×#99C2×C418yes—
37#390—24no—
38#390D414yes—
39#390Q814yes—
40#380—28no—
41#380—44no—
42C4×C4⋊C4—42no—
43#376—42no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—24no—
48C2×#95—42no—
49#342—28no—
50C25C2×D412yes—
51C22×D4—41no—
52C22×D4—41no—
53C22×D4—42no—
54C22×D4C22×C412yes—
55C22×D4C2×D412yes—
56C22×D4C2×D411yescharacteristic
57C22×D4C2×Q812yes—
58C22×D4C2×Q811yescharacteristic
59C22×D4#4114yes—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—24no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—44no—
66C23×C4—84no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C412yes—
69C23×C4C22⋊C414yes—
70C23×C4C22⋊C411yescharacteristic
71C23×C4C4⋊C414yes—
72C23×C4C22×C411yescharacteristic
73C4×D4—44no—
74C4×D4—44no—
75C4×D4—44no—
76C4×D4—44no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—48no—
83C2×C4⋊C4—88no—
84C2×C4⋊C4C4212yes—
85C2×C4⋊C4C4⋊C412yes—
86C2×C4⋊C4C4⋊C414yescomplemented
87C2×C22⋊C4—28no—
88C2×C22⋊C4—28no—
89C2×C22⋊C4—28no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—48no—
93C2×C22⋊C4—48no—
94C2×C22⋊C4—44no—
95C2×C22⋊C4—84no—
96C2×C22⋊C4—84no—
97C2×C22⋊C4C22⋊C414yes—
98C2×C22⋊C4C4⋊C414yes—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—42no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42—82no—
106C2×C42C4212yes—
107C2×C42C4⋊C412yes—
108C2×C42C4⋊C414yescomplemented
109#99—216no—
110#99—44no—
111#99—44no—
112#95—42no—
113#95—48no—
114#95—42no—
115#95—42no—
116#95—84no—
117C24—22no—
118C24—22no—
119C24—24no—
120C24—42no—
121C24—44no—
122C24—42no—
123C24C2×C22⋊C411yescharacteristic
124C24C2×C22⋊C412yes—
125C24C2×C4⋊C412yes—
126C24#12114yes—
127C24C22⋊Q814yes—
128C2×D4—42no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4—41no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—41no—
138C2×D4—44no—
139C2×D4—44no—
140C2×D4C4×D414yes—
141C2×D4C4×Q814yes—
142C2×D4#12412yes—
143C2×D4#12512yes—
144C2×D4#12711yescharacteristic
145C2×D4C4⋊Q812yes—
146C2×D4C4⋊Q811yescharacteristic
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—42no—
161C22×C4—48no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—48no—
176C22×C4—84no—
177C22×C4—84no—
178C22×C4—84no—
179C22×C4—88no—
180C22×C4—88no—
181C22×C4—88no—
182C22×C4—84no—
183C22×C4—84no—
184C22×C4#9512yes—
185C22×C4#9514yes—
186C22×C4#9512yes—
187C22×C4#9912yes—
188C22×C4#9912yes—
189C22×C4C2×C4211yescharacteristic
190C22×C4C2×C4⋊C412yes—
191C22×C4C2×C4⋊C411yescharacteristic
192C22×C4#11712yes—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—816no—
198C22⋊C4—44no—
199C22⋊C4—44no—
200C22⋊C4—48no—
201C22⋊C4—48no—
202C22⋊C4—88no—
203C22⋊C4—88no—
204C22⋊C4—88no—
205C42—24no—
206C42—24no—
207C42—44no—
208C42—44no—
209C42—84no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23—24no—
214C23—24no—
215C23—22no—
216C23—22no—
217C23—24no—
218C23—24no—
219C23—22no—
220C23—22no—
221C23—44no—
222C23—44no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—44no—
228C23—44no—
229C23—44no—
230C23—42no—
231C23—42no—
232C23—82no—
233C23—84no—
234C23—82no—
235C23—84no—
236C23C2×#9511yescharacteristic
237C23#38012yes—
238C23#38914yes—
239C23#39012yes—
240C23#39112yes—
241C23C2×#9911yescharacteristic
242C23C2×#9911yescharacteristic
243C23#41012yes—
244D4—44no—
245D4—44no—
246D4—42no—
247D4—42no—
248D4—41no—
249D4—42no—
250D4—41no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—48no—
265C2×C4—48no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—48no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—84no—
278C2×C4—84no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—84no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—88no—
288C2×C4—88no—
289C2×C4—88no—
290C2×C4—168no—
291C2×C4—168no—
292C2×C4#34214yes—
293C2×C4C4×C4⋊C412yes—
294C2×C4#38211yescharacteristic
295C2×C4#38411yescharacteristic
296C22—22no—
297C22—22no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—24no—
306C22—22no—
307C22—44no—
308C22—42no—
309C22—42no—
310C22—44no—
311C22—44no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—44no—
317C22—44no—
318C22—84no—
319C22—84no—
320C22—82no—
321C22—82no—
322C22C4⋊#9511yescharacteristic
323C22C2×#34211yescharacteristic
324C22#174011yescharacteristic
325C22#179912yes—
326C22#179912yes—
327C22#212712yes—
328C22#212811yescharacteristic
329C22#212911yescharacteristic
330C4—24no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—44no—
335C4—42no—
336C4—84no—
337C4—84no—
338C4—168no—
339C2—22no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—44no—
344C2—42no—
345C2#1084611yescharacteristic
346C2#1258211yescharacteristic
347C2#1258311yescharacteristic
348C2#1279912yes—
349C2#1280012yes—
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, prime radical, cosocleC23×D4C23
LCSG ▷ C22×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
8. C2×C4×C4⋊C4C4semidirect
9. C2×#376C4semidirect
11. #1799C4semidirect
14. C4⋊#95C4semidirect
86. C2×C4⋊C4C4⋊C4semidirect
108. C2×C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17992
#137852
C2×#13932