Finite Group Databasebeta

← #128000

Subgroups — #128000

Order 512.

1412 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1412 conjugacy classes of subgroups — too many to draw. Only its 150 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
1#20584C212yesmaximal
2C2×#1780C211yesmaximal, characteristic
3#7305C214yesmaximal, complemented
4C23×#41C2211yescharacteristic
5C2×#410—22no—
6C2×#410C414yescomplemented
7C22×C22⋊C8—22no—
8C22×C22⋊C8C412yescomplemented
9#2280—22no—
10#1782—22no—
11#1782C2214yes—
12#1781—22no—
13#1781C414yescomplemented
14#1780—22no—
15#1780C412yescomplemented
16#1780C2212yes—
17C22×#41—42no—
18C22×#41—42no—
19C22×#41—48no—
20C22×#41C2×C412yes—
21C22×#41C2×C411yescharacteristic
22C22×#41D411yescharacteristic
23C22×#41D412yes—
24C22×#41Q811yescharacteristic
25C22×#41C2311yescharacteristic
26C23×Q8Q811yescharacteristic
27C23×D4C2×C412yes—
28C23×D4D411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D412yes—
31C23×C8—22no—
32C2×#117—24no—
33#410—48no—
34#410—48no—
35C2×#99—24no—
36C2×#99—24no—
37C2×#99—24no—
38C2×#99C2×C418yes—
39C2×C22⋊C8—24no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—24no—
42C2×C22⋊C8—48no—
43C2×C22⋊C8—42no—
44C2×C22⋊C8—42no—
45C2×C22⋊C8C2×C414yes—
46#381—24no—
47#380—44no—
48#379—24no—
49C4×C22⋊C4—44no—
50C25C4211yescharacteristic
51C25C22⋊C412yes—
52C2×#41—28no—
53C2×#41—28no—
54C2×#41—42no—
55C2×#41—44no—
56C2×#41—42no—
57C2×#41—42no—
58C2×#41—41no—
59C2×#41—41no—
60C2×#41—44no—
61C2×#41—41no—
62C2×#41—44no—
63C2×#41—41no—
64C2×#41—42no—
65C2×#41—44no—
66C2×#41—44no—
67C2×#41—44no—
68C2×#41—44no—
69C2×#41—432no—
70C2×#41—432no—
71C22×Q8—42no—
72C22×Q8—44no—
73C22×Q8C4⋊C411yescharacteristic
74C22×Q8C4⋊C412yes—
75C22×Q8C2×Q811yescharacteristic
76C22×D4—44no—
77C22×D4—44no—
78C22×D4—42no—
79C22×D4—44no—
80C22×D4—44no—
81C22×D4C4212yes—
82C22×D4C22⋊C412yes—
83C22×D4C22⋊C412yes—
84C22×D4C4⋊C412yes—
85C22×D4C4⋊C411yescharacteristic
86C22×D4C22×C412yes—
87C22×D4C2×D411yescharacteristic
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—24no—
91C23×C4—24no—
92C23×C4—28no—
93C23×C4—44no—
94C23×C4—44no—
95C23×C4C4211yescharacteristic
96C23×C4C22⋊C412yes—
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C4⋊C411yescharacteristic
101C23×C4C4⋊C412yes—
102C23×C4C4⋊C412yes—
103C23×C4C4⋊C411yescharacteristic
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D412yes—
106C22×C8—24no—
107C22×C8—48no—
108C22×C8—42no—
109C22×C8—42no—
110#117—48no—
111#117—48no—
112C2×C4⋊C4—44no—
113C2×C22⋊C4—28no—
114C2×C22⋊C4—48no—
115C2×C42—44no—
116#99—48no—
117#99—48no—
118#99—48no—
119#99—48no—
120C22⋊C8—48no—
121C22⋊C8—48no—
122C22⋊C8—44no—
123C22⋊C8—44no—
124#95—48no—
125C24—22no—
126C24—24no—
127C24—24no—
128C24—42no—
129C24—42no—
130C24—48no—
131C24—44no—
132C24#9511yescharacteristic
133C24#9512yes—
134C24#9511yescharacteristic
135C24C2×C4211yescharacteristic
136C24C2×C22⋊C412yes—
137#41—48no—
138#41—48no—
139#41—416no—
140#41—416no—
141#41—416no—
142#41—432no—
143#41—416no—
144#41—48no—
145#41—48no—
146C2×Q8—24no—
147C2×Q8—24no—
148C2×Q8—41no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—41no—
152C2×Q8—42no—
153C2×Q8—416no—
154C2×D4—24no—
155C2×D4—24no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—24no—
159C2×D4—24no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—41no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—41no—
173C2×D4—432no—
174C2×D4—416no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—21no—
184C22×C4—24no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—416no—
205C22×C4—416no—
206C22×C4—48no—
207C22×C4—48no—
208C22×C4—48no—
209C22×C4#9511yescharacteristic
210C22×C4#9512yes—
211C22×C4#9511yescharacteristic
212C22×C4#9918yes—
213C22×C4#9912yes—
214C22×C4#9912yes—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C4⋊C411yescharacteristic
217C22×C4C2×C4⋊C412yes—
218C2×C8—48no—
219C2×C8—44no—
220C2×C8—84no—
221C2×C8—84no—
222C4⋊C4—88no—
223C22⋊C4—48no—
224C22⋊C4—48no—
225C22⋊C4—816no—
226C42—88no—
227C23—24no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—22no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—48no—
239C23—44no—
240C23—44no—
241C23—44no—
242C23—416no—
243C23—44no—
244C23—42no—
245C23—42no—
246C23—48no—
247C23—44no—
248C23—84no—
249C23—88no—
250C23—88no—
251C23#34212yes—
252C23C2×#9511yescharacteristic
253C23C2×#9914yes—
254Q8—44no—
255Q8—48no—
256Q8—44no—
257D4—48no—
258D4—48no—
259D4—44no—
260D4—48no—
261D4—48no—
262D4—48no—
263D4—44no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—28no—
268C2×C4—28no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—48no—
279C2×C4—48no—
280C2×C4—48no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—88no—
291C2×C4—816no—
292C2×C4#34214yes—
293C2×C4#34211yescharacteristic
294C2×C4#34214yes—
295C2×C4#34211yescharacteristic
296C2×C4C2×#9912yes—
297C8—84no—
298C22—28no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—24no—
303C22—44no—
304C22—44no—
305C22—44no—
306C22—42no—
307C22—42no—
308C22—44no—
309C22—42no—
310C22—41no—
311C22—41no—
312C22—82no—
313C22—816no—
314C22—82no—
315C22—82no—
316C22—82no—
317C22—88no—
318C22#139114yes—
319C22C2×#34212yes—
320C22C2×#34211yescharacteristic
321C4—22no—
322C4—24no—
323C4—41no—
324C4—42no—
325C4—41no—
326C4—168no—
327C4#130412yes—
328C4#130412yes—
329C2—22no—
330C2—24no—
331C2—44no—
332C2—82no—
333C2—81no—
334C2—81no—
335C2C2×#130411yescharacteristic
336C2C2×#139112yes—
subgroupG/H
centerC2×C4#342
commutatorC24C2×C42
Frattini, cosocleC22×#41C23
socleC22C2×#342
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×#41 ◁ G
LECSG ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7305C2semidirect
6. C2×#410C4semidirect
8. C22×C22⋊C8C4semidirect
13. #1781C4semidirect
15. #1780C4semidirect

Maximal subgroups

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

subgroupindex
#205842
C2×#17802
#73052