Finite Group Databasebeta

← #128494

Subgroups — #128494

Order 512.

411 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 102 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#21633C211yesmaximal, characteristic
2C4×#354C211yesmaximal, characteristic
3#11477C211yesmaximal, characteristic
4#8160C211yesmaximal, characteristic
5C8×#99C211yesmaximal, characteristic
6#7205C211yesmaximal, characteristic
7#7205C211yesmaximal, characteristic
8C2×#446C411yescharacteristic, complemented
9C2×C8×D4C2211yescharacteristic
10C4×#124C2211yescharacteristic
11C2×#354C412yes—
12#1984C411yescharacteristic, complemented
13#1954C2211yescharacteristic
14#1924—21no—
15#1924—21no—
16#1831C411yescharacteristic, complemented
17C4×#99C2211yescharacteristic
18C4×#99C2211yescharacteristic
19#1752—21no—
20#1752—21no—
21#1752—21no—
22C8×C22⋊C4—21no—
23C4×#97C411yescharacteristic, complemented
24#1459C2211yescharacteristic
25#1459C2211yescharacteristic
26#1393C412yes—
27#1324C412yescomplemented
28#1319C412yescomplemented
29#1282—21no—
30#1282—21no—
31C23×C8D411yescharacteristic
32C23×C8D411yescharacteristic
33C2×#124C2×C411yescharacteristic
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×D4C2311yescharacteristic
36#446—41no—
37#446—41no—
38C8×D4—41no—
39C8×D4—41no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8C2×C411yescharacteristic
42C2×C4⋊C8C2×C411yescharacteristic
43C2×#99C2×C412yes—
44C2×#99C2×C412yes—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8D411yescharacteristic
50C2×C22⋊C8D411yescharacteristic
51C2×#97—21no—
52C2×#97—21no—
53C2×#97—21no—
54C2×#97—41no—
55C2×C4×C8—41no—
56C2×C4×C8C2×C411yescharacteristic
57C2×C4×C8C2×C411yescharacteristic
58#391C2×C411yescharacteristic
59#390C2×C411yescharacteristic
60#388—21no—
61#388—21no—
62#382C2×C411yescharacteristic
63C4×C22⋊C4—21no—
64C4×C22⋊C4—21no—
65C4×C22⋊C4—21no—
66C4×C22⋊C4—21no—
67C43C2×C411yescharacteristic
68#354—24no—
69#342—21no—
70#342—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336—41no—
76#336—41no—
77#324C2×C412yes—
78#324C2×C412yes—
79#323—21no—
80#323—21no—
81C22×Q8C4211yescharacteristic
82C22×D4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—42no—
92C22×C8—42no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8C22⋊C411yescharacteristic
96C22×C8C22⋊C411yescharacteristic
97C22×C8C22⋊C411yescharacteristic
98C22×C8C22⋊C411yescharacteristic
99#124—22no—
100#124—22no—
101C4×Q8—41no—
102C4×Q8—41no—
103C4×D4—41no—
104C4×D4—41no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4C4211yescharacteristic
107C2×C4⋊C4C22×C411yescharacteristic
108C2×C22⋊C4—22no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4#4111yescharacteristic
115C2×C22⋊C4#4111yescharacteristic
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42C4211yescharacteristic
122C2×C42C4211yescharacteristic
123C2×C42C22×C411yescharacteristic
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—42no—
127#99—22no—
128#99—22no—
129#99—22no—
130#99M1614yes—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139#97—42no—
140#97—42no—
141#97—41no—
142#97—41no—
143#97—82no—
144C4×C8—21no—
145C4×C8—21no—
146C4×C8—41no—
147C4×C8—82no—
148#95—21no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—41no—
153#95—41no—
154C24C4×D411yescharacteristic
155C24C4×D411yescharacteristic
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4C2×M1612yes—
163C2×D4#13112yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4C2×C4211yescharacteristic
179C22×C4C2×C22⋊C411yescharacteristic
180C22×C4#11711yescharacteristic
181C22×C4C4×D411yescharacteristic
182C22×C4C4×D411yescharacteristic
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C2×C8—81no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8#9911yescharacteristic
207C2×C8#9911yescharacteristic
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C22⋊C4—24no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—81no—
220C22⋊C4—81no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—41no—
226C42—41no—
227C42—41no—
228C42—41no—
229C42—41no—
230C42—81no—
231C42—81no—
232C42—81no—
233C42—81no—
234C42—81no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23C4×C22⋊C411yescharacteristic
244C23#43512yes—
245C23#43612yes—
246Q8—81no—
247Q8—81no—
248D4—42no—
249D4—42no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—44no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4C4×M1611yescharacteristic
285C2×C4#40511yescharacteristic
286C2×C4C2×#9911yescharacteristic
287C2×C4#41011yescharacteristic
288C8—21no—
289C8—42no—
290C8—41no—
291C8—41no—
292C8—41no—
293C8—82no—
294C8—82no—
295C8—81no—
296C8#35412yescomplemented
297C22—21no—
298C22—21no—
299C22—22no—
300C22—22no—
301C22—42no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—81no—
309C22#175311yescharacteristic
310C22C4×#9911yescharacteristic
311C22#175711yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—81no—
319C4—81no—
320C4—84no—
321C4—81no—
322C4—81no—
323C4—162no—
324C4C2×#35411yescharacteristic
325C4#212611yescharacteristic
326C2—21no—
327C2—21no—
328C2—42no—
329C2—42no—
330C2#816111yescharacteristic
331C2C4×#35411yescharacteristic
332C2#1263411yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C4×#99
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×#446C4semidirect
12. #1984C4semidirect
16. #1831C4semidirect
23. C4×#97C4semidirect
27. #1324C4semidirect
28. #1319C4semidirect
296. C8#354semidirect

Maximal subgroups

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

subgroupindex
#216332
C4×#3542
#114772
#81602
C8×#992
#72052