Finite Group Databasebeta

← #127807

Subgroups — #127807

Order 512.

846 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 138 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×#1955C211yesmaximal, characteristic
2#20722C212yesmaximal
3#7283C214yesmaximal, complemented
4C22×C4×D4C2211yescharacteristic
5C22×C4⋊C8C411yescharacteristic, complemented
6C22×C22⋊C8—22no—
7C22×#97C411yescharacteristic, complemented
8#2280—22no—
9#1955—22no—
10#1955C412yescomplemented
11#1955C2212yes—
12#1819C414yescomplemented
13#1799—22no—
14#1799C414yescomplemented
15#1781—22no—
16C4×#99—22no—
17C4×#99C2214yes—
18C23×D4D411yescharacteristic
19C24×C4D412yes—
20C23×C8—21no—
21C23×C8—21no—
22C2×C4×D4—41no—
23C2×C4×D4—41no—
24C2×C4×D4—41no—
25C2×C4×D4—41no—
26C2×C4×D4C2×C412yes—
27C2×C4×D4C2×C411yescharacteristic
28C2×C4×D4D412yes—
29C2×C4×D4D411yescharacteristic
30C2×C4×D4Q811yescharacteristic
31C2×C4×D4C2311yescharacteristic
32C2×#117—42no—
33C22×C4⋊C4Q811yescharacteristic
34C22×C22⋊C4C2×C412yes—
35C22×C42C2×C411yescharacteristic
36C2×C4⋊C8—21no—
37C2×C4⋊C8—41no—
38C2×C4⋊C8—41no—
39C2×C4⋊C8C2×C412yes—
40C2×#99—24no—
41C2×#99—24no—
42C2×C22⋊C8—24no—
43C2×C22⋊C8—24no—
44C2×C22⋊C8—48no—
45C2×#97—21no—
46C2×#97—41no—
47C2×#97—41no—
48C2×#97C2×C412yes—
49#381—42no—
50#380—24no—
51#380—42no—
52#379—24no—
53C4×C22⋊C4—24no—
54C4×C22⋊C4—42no—
55#342—24no—
56#342C2×C418yes—
57C25C22⋊C412yes—
58C22×D4—41no—
59C22×D4—41no—
60C22×D4—44no—
61C22×D4C22⋊C412yes—
62C22×D4C4⋊C411yescharacteristic
63C22×D4C2×D411yescharacteristic
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—44no—
69C23×C4—44no—
70C23×C4C4211yescharacteristic
71C23×C4C22⋊C412yes—
72C23×C4C22⋊C412yes—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C4⋊C412yes—
75C23×C4C4⋊C411yescharacteristic
76C23×C4C4⋊C412yes—
77C23×C4C2×D412yes—
78C22×C8—24no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—42no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8—44no—
85C22×C8—41no—
86C22×C8—41no—
87C4×D4—42no—
88C4×D4—42no—
89C4×D4—41no—
90C4×D4—41no—
91C4×D4—41no—
92C4×D4—41no—
93C4×D4—42no—
94C4×D4—42no—
95C4×D4—41no—
96C4×D4—41no—
97C4×D4—41no—
98C4×D4—41no—
99#117—84no—
100#117—84no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—44no—
104C2×C4⋊C4C4⋊C412yes—
105C2×C4⋊C4C4⋊C411yescharacteristic
106C2×C4⋊C4C2×Q811yescharacteristic
107C2×C22⋊C4—28no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—44no—
111C2×C22⋊C4—44no—
112C2×C22⋊C4C4212yes—
113C2×C22⋊C4C22⋊C412yes—
114C2×C22⋊C4C4⋊C412yes—
115C2×C22⋊C4C22×C412yes—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—44no—
121C2×C42C4211yescharacteristic
122C2×C42C22⋊C411yescharacteristic
123C2×C42C4⋊C411yescharacteristic
124C2×C42C22×C411yescharacteristic
125C4⋊C8—42no—
126C4⋊C8—42no—
127#99—48no—
128#99—48no—
129C22⋊C8—48no—
130C22⋊C8—48no—
131#97—42no—
132#97—42no—
133#95—44no—
134#95—44no—
135C24—24no—
136C24—22no—
137C24—22no—
138C24—42no—
139C24—44no—
140C24—44no—
141C24#9511yescharacteristic
142C24#9512yes—
143C24C2×C22⋊C412yes—
144C2×D4—24no—
145C2×D4—24no—
146C2×D4—42no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—48no—
152C2×D4—48no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—48no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4—84no—
184C22×C4—84no—
185C22×C4#9511yescharacteristic
186C22×C4#9511yescharacteristic
187C22×C4#9511yescharacteristic
188C22×C4#9512yes—
189C22×C4#9911yescharacteristic
190C22×C4#9911yescharacteristic
191C22×C4#9911yescharacteristic
192C22×C4#9911yescharacteristic
193C22×C4C2×C4211yescharacteristic
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4C2×C4⋊C411yescharacteristic
196C22×C4C2×C4⋊C412yes—
197C2×C8—24no—
198C2×C8—44no—
199C2×C8—42no—
200C2×C8—44no—
201C2×C8—42no—
202C2×C8—44no—
203C2×C8—42no—
204C2×C8—82no—
205C2×C8—82no—
206C4⋊C4—42no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—88no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—48no—
215C22⋊C4—48no—
216C22⋊C4—88no—
217C22⋊C4—88no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—84no—
227C42—84no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—44no—
237C23—44no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—416no—
243C23—44no—
244C23—44no—
245C23—82no—
246C23—82no—
247C23—88no—
248C23#34211yescharacteristic
249C23#34211yescharacteristic
250C23C2×#9511yescharacteristic
251C23#42114yes—
252D4—44no—
253D4—48no—
254D4—44no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—88no—
291C2×C4—88no—
292C2×C4—88no—
293C2×C4#34211yescharacteristic
294C2×C4#34211yescharacteristic
295C2×C4#35312yes—
296C2×C4#35412yes—
297C2×C4C2×#9911yescharacteristic
298C2×C4C2×#9911yescharacteristic
299C8—44no—
300C8—82no—
301C22—28no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—44no—
310C22—44no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—82no—
317C22—88no—
318C22—88no—
319C22—82no—
320C22#139112yes—
321C22#139312yes—
322C22#173312yes—
323C22C2×#34211yescharacteristic
324C4—21no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—42no—
329C4—42no—
330C4—41no—
331C4—41no—
332C4—168no—
333C4#139312yes—
334C4C2×#35311yescharacteristic
335C4C2×#35411yescharacteristic
336C2—22no—
337C2—22no—
338C2—22no—
339C2—44no—
340C2—82no—
341C2#804911yescharacteristic
342C2C2×#139111yescharacteristic
343C2C2×#139311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7283C2semidirect
5. C22×C4⋊C8C4semidirect
7. C22×#97C4semidirect
10. #1955C4semidirect
12. #1819C4semidirect
14. #1799C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19552
#207222
#72832