Finite Group Databasebeta

← #329750

Subgroups — #329750

Order 512.

610 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 140 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#29876C211yesmaximal, characteristic
2#29691C212yesmaximal
3#21819C211yesmaximal, characteristic
4#10626C212yesmaximal, complemented
5#10483C212yesmaximal, complemented
6#10482C212yesmaximal, complemented
7#10279C212yesmaximal, complemented
8#9128C212yesmaximal
9#8600C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#500C2211yescharacteristic
12C2×#477C2212yes—
13C2×#477C2212yes—
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2857C2212yes—
17#2086—22no—
18#2075—22no—
19#2057—22no—
20#2043—22no—
21#2036—22no—
22#2036—22no—
23#1990C2212yes—
24#1984—22no—
25#1984C2212yes—
26#1984C2212yes—
27#1834C2211yescharacteristic
28#1818—22no—
29#1803C2212yes—
30#1803C2212yes—
31#1797C2212yes—
32C2×#335C2211yescharacteristic
33C2×#334C2211yescharacteristic
34C2×#333C2212yes—
35C2×#333C2212yes—
36#1296C2212yes—
37#1296C2212yes—
38#1277C2212yes—
39#533—41no—
40#533—41no—
41#533—41no—
42#533—41no—
43C2×C4⋊Q8D411yescharacteristic
44C2×C4⋊Q8D411yescharacteristic
45C2×C4⋊Q8C2312yes—
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8D412yes—
50C2×C4×Q8—21no—
51C2×C4×Q8C2312yes—
52C2×#117—21no—
53C22×C42C2311yescharacteristic
54#500—41no—
55#500—41no—
56#477—42no—
57#477—42no—
58#477—42no—
59#477—42no—
60#431—41no—
61#431—41no—
62#431—41no—
63#431—41no—
64C2×#107—22no—
65C2×#107—22no—
66C2×#107—22no—
67C2×C4⋊C8D412yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—22no—
74C2×#103—22no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8D412yes—
78C2×C4×C8—21no—
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#391—22no—
82#389—22no—
83#389—22no—
84#384C2312yes—
85#380—22no—
86#380—22no—
87#340—22no—
88#340—24no—
89#340—22no—
90#340—22no—
91#336—24no—
92#336—24no—
93#336—22no—
94#335—41no—
95#334—41no—
96#333—42no—
97#333—42no—
98C22×Q8—21no—
99C22×Q8—22no—
100C22×Q8C2×D412yes—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8C2×D412yes—
108C22×C8C2×D412yes—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—41no—
114C4⋊Q8—41no—
115C4⋊Q8—41no—
116C4⋊Q8—41no—
117#125—42no—
118#125—42no—
119C22⋊Q8—42no—
120C22⋊Q8—42no—
121C22⋊Q8—42no—
122C22⋊Q8—42no—
123C22⋊Q8—84no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—42no—
129C4×Q8—42no—
130#117—42no—
131#117—42no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4C2×D412yes—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—42no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—22no—
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42C2411yescharacteristic
155#107—44no—
156#107—44no—
157#107—44no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—42no—
165#103—44no—
166#103—44no—
167#103—82no—
168#103—82no—
169C22⋊C8—42no—
170C22⋊C8—42no—
171C22⋊C8—82no—
172C4×C8—21no—
173C4×C8—21no—
174C4×C8—21no—
175C4×C8—21no—
176C4×C8—42no—
177#95—42no—
178#95—42no—
179C24#12111yescharacteristic
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—82no—
192C2×Q8—82no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4C22×D411yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C2×#4111yescharacteristic
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—24no—
217C2×C8—24no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—44no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—44no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—42no—
241C4⋊C4—82no—
242C4⋊C4—84no—
243C22⋊C4—44no—
244C22⋊C4—84no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—42no—
252C42—42no—
253C42H(5,𝔽2)11yescharacteristic
254C42H(5,𝔽2)11yescharacteristic
255C42H(5,𝔽2)11yescharacteristic
256C42H(5,𝔽2)11yescharacteristic
257C23—21no—
258C23—22no—
259C23—22no—
260C23—21no—
261C23—21no—
262C23C2×#12111yescharacteristic
263Q8—42no—
264Q8—42no—
265Q8—42no—
266Q8—42no—
267Q8—84no—
268Q8—162no—
269Q8—162no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4#47212yes—
298C2×C4#53411yescharacteristic
299C2×C4#53411yescharacteristic
300C2×C4#53411yescharacteristic
301C2×C4#53511yescharacteristic
302C2×C4#53811yescharacteristic
303C2×C4#53811yescharacteristic
304C2×C4#57712yes—
305C2×C4#57812yes—
306C2×C4#57814yes—
307C8—42no—
308C8—42no—
309C8—82no—
310C8—82no—
311C8—82no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—42no—
317C22#261711yescharacteristic
318C22#304412yes—
319C22#306312yes—
320C22C2×#47211yescharacteristic
321C22#306711yescharacteristic
322C4—21no—
323C4—21no—
324C4—22no—
325C4—41no—
326C4—41no—
327C4—82no—
328C4—82no—
329C4—162no—
330C4—162no—
331C4#320112yes—
332C4#320212yes—
333C4#320211yescharacteristic
334C4#320411yescharacteristic
335C4#322312yes—
336C2—81no—
337C2#1957312yes—
338C2#1957412yes—
339C2#2972411yescharacteristic
340C2#2974611yescharacteristic
341C2#2974811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10626C2semidirect
5. #10483C2semidirect
6. #10482C2semidirect
7. #10279C2semidirect

Maximal subgroups

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

subgroupindex
#298762
#296912
#218192
#106262
#104832
#104822
#102792
#91282
#86002