Finite Group Databasebeta

← #329587

Subgroups — #329587

Order 512.

958 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 154 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#32902C212yesmaximal, complemented
2#29806C211yesmaximal, characteristic, complemented
3#21786C211yesmaximal, characteristic, complemented
4#10621C212yesmaximal, complemented
5#10472C214yesmaximal, complemented
6#10260C212yesmaximal, complemented
7#9134C212yesmaximal, complemented
8#8599C211yesmaximal, characteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#493C2211yescharacteristic, complemented
11C2×#459C2214yes—
12C2×#447—22no—
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2856C2212yes—
16#2054—22no—
17#1999—22no—
18#1957C2212yescomplemented
19#1954—22no—
20#1954C2212yes—
21#1954C2212yes—
22#1879—24no—
23#1874—22no—
24#1834C2211yescharacteristic, complemented
25#1802C2214yes—
26#1797C2212yes—
27#1779—22no—
28C2×#335C2212yescomplemented
29C2×#331C2214yescomplemented
30#1296C2214yescomplemented
31#1277C2212yescomplemented
32C23×D4D411yescharacteristic
33C24×C4—21no—
34C22×D8—24no—
35C22×D8—22no—
36C2×#127C2312yes—
37C2×#121—22no—
38C2×#121—22no—
39C2×#121—22no—
40C2×C4×D4—21no—
41C2×C4×D4—21no—
42C2×C4×D4—42no—
43C2×C4×D4—42no—
44C2×C4×D4D412yes—
45C2×C4×D4C2312yes—
46C22×C4⋊C4D411yescharacteristic
47C22×C22⋊C4—21no—
48C22×C42C2311yescharacteristic
49#493—42no—
50#459—44no—
51#459—44no—
52#447—48no—
53#431—42no—
54#431—42no—
55C2×C4⋊C8D412yes—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2314yes—
58C2×#102—24no—
59C2×#102—24no—
60C2×#102—24no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8D412yes—
66C2×C4×C8—21no—
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69#390—22no—
70#390—22no—
71#390—22no—
72#384C2312yes—
73#340—22no—
74#340—24no—
75#336—22no—
76#335—42no—
77#331—44no—
78C25—21no—
79C22×D4—21no—
80C22×D4—22no—
81C22×D4—22no—
82C22×D4—22no—
83C22×D4—22no—
84C22×D4—42no—
85C22×D4—42no—
86C22×D4—42no—
87C22×D4D814yescomplemented
88C22×D4C2×D412yes—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—84no—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4#4111yescharacteristic
98C2×D8—44no—
99C2×D8—48no—
100C2×D8—48no—
101C2×D8—48no—
102C2×D8—84no—
103C22×C8—24no—
104C22×C8—22no—
105C22×C8C2×D412yes—
106C22×C8C2×D412yes—
107#127—24no—
108#127—24no—
109#121—44no—
110#121—44no—
111#121—84no—
112#121—84no—
113C4×D4—24no—
114C4×D4—24no—
115C4×D4—44no—
116C4×D4—42no—
117C4×D4—42no—
118C4×D4—44no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4C2×D412yes—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—44no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42#4111yescharacteristic
136C2×C42C2411yescharacteristic
137C4⋊C8—24no—
138C4⋊C8—24no—
139C4⋊C8—24no—
140C4⋊C8—42no—
141#102—48no—
142#102—48no—
143#102—84no—
144C22⋊C8—42no—
145C22⋊C8—42no—
146C22⋊C8—44no—
147C22⋊C8—84no—
148C22⋊C8—84no—
149C4×C8—22no—
150C4×C8—22no—
151C4×C8—42no—
152C24—22no—
153C24—24no—
154C24—42no—
155C24—42no—
156C24—42no—
157C24—82no—
158C24—82no—
159C24#12111yescharacteristic
160C2×D4—24no—
161C2×D4—22no—
162C2×D4—24no—
163C2×D4—24no—
164C2×D4—48no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—48no—
168C2×D4—44no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—44no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—44no—
175C2×D4—44no—
176C2×D4—44no—
177C2×D4—82no—
178C2×D4—82no—
179C2×D4—84no—
180C2×D4—84no—
181C2×D4—84no—
182C2×D4C2×D814yes—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—84no—
199C22×C4—82no—
200C22×C4—88no—
201C22×C4—88no—
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4C2×D812yes—
206C22×C4#13512yes—
207C22×C4C22×D411yescharacteristic
208C22×C4C22×D411yescharacteristic
209C22×C4C2×#4111yescharacteristic
210D8—88no—
211D8—816no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—24no—
215C2×C8—24no—
216C2×C8—44no—
217C2×C8—42no—
218C2×C8—44no—
219C2×C8—44no—
220C2×C8—44no—
221C4⋊C4—24no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—82no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C22⋊C4—84no—
230C22⋊C4—84no—
231C22⋊C4—84no—
232C42—21no—
233C42—22no—
234C42—21no—
235C42—22no—
236C42H(5,𝔽2)12yes—
237C42H(5,𝔽2)12yes—
238C23—21no—
239C23—22no—
240C23—21no—
241C23—22no—
242C23—24no—
243C23—48no—
244C23—42no—
245C23—44no—
246C23—42no—
247C23—44no—
248C23—88no—
249C23—84no—
250C23—84no—
251C23—88no—
252C23—82no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23—82no—
257C23—82no—
258C23—82no—
259C23#46612yes—
260C23C2×#12111yescharacteristic
261D4—28no—
262D4—48no—
263D4—44no—
264D4—84no—
265D4—84no—
266D4—84no—
267D4—84no—
268D4—164no—
269D4—168no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—82no—
284C2×C4—88no—
285C2×C4—84no—
286C2×C4—84no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4#46612yes—
293C2×C4#53412yes—
294C2×C4#53411yescharacteristic
295C2×C4#53511yescharacteristic
296C2×C4#53812yes—
297C2×C4C22×D811yescharacteristic
298C2×C4C2×#13511yescharacteristic
299C2×C4#57512yes—
300C2×C4#57614yes—
301C8—42no—
302C8—42no—
303C8—84no—
304C8—82no—
305C22—22no—
306C22—24no—
307C22—41no—
308C22—41no—
309C22—42no—
310C22—44no—
311C22—48no—
312C22—816no—
313C22—82no—
314C22—82no—
315C22—84no—
316C22—162no—
317C22—162no—
318C22—162no—
319C22—162no—
320C22—162no—
321C22—162no—
322C22—162no—
323C22#261711yescharacteristic
324C22#304112yes—
325C22C2×#46611yescharacteristic
326C22#305411yescharacteristic
327C22#306812yes—
328C4—22no—
329C4—22no—
330C4—41no—
331C4—41no—
332C4—82no—
333C4—162no—
334C4#318612yes—
335C4#319112yes—
336C4#319912yes—
337C4#322012yes—
338C2—42no—
339C2—84no—
340C2—322no—
341C2#1938612yes—
342C2#1942512yes—
343C2#2972111yescharacteristic
344C2#2973011yescharacteristic
345C2#2975111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
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
1. #32902C2semidirect
2. #29806C2semidirect
3. #21786C2semidirect
4. #10621C2semidirect
5. #10472C2semidirect
6. #10260C2semidirect
7. #9134C2semidirect
8. #8599C2semidirect
10. C2×#493C22semidirect
18. #1957C22semidirect
24. #1834C22semidirect
28. C2×#335C22semidirect
29. C2×#331C22semidirect
30. #1296C22semidirect
31. #1277C22semidirect
87. C22×D4D8semidirect

Maximal subgroups

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

subgroupindex
#329022
#298062
#217862
#106212
#104722
#102602
#91342
#85992