Finite Group Databasebeta

← #98941

Subgroups — #98941

Order 512.

454 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 110 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#21597C211yesmaximal, characteristic
2#10820C211yesmaximal, characteristic
3#8596C211yesmaximal, characteristic
4#8572C212yesmaximal
5#7039C212yesmaximal
6C2×#446C411yescharacteristic, complemented
7C22×C4×C8C2211yescharacteristic
8C4×C22⋊Q8C2211yescharacteristic
9#1984C411yescharacteristic, complemented
10#1939C411yescharacteristic, complemented
11#1916—21no—
12#1850—21no—
13#1797C2212yes—
14#1797C2211yescharacteristic
15#1771—21no—
16#1770C411yescharacteristic, complemented
17#1752—21no—
18C2×#340—21no—
19C2×#340—21no—
20C2×#335—21no—
21C2×#328—22no—
22C4⋊#95C2212yes—
23#1294—22no—
24#1277—21no—
25C23×C8D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C22⋊Q8D411yescharacteristic
28C2×C4×Q8C2×C411yescharacteristic
29C22×C4⋊C4—22no—
30C22×C42C2311yescharacteristic
31#446—42no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C22⋊C8—21no—
38C2×C22⋊C8—22no—
39C2×#97—21no—
40C2×#97—21no—
41C2×C4×C8—42no—
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#391Q811yescharacteristic
47#389Q811yescharacteristic
48#387—21no—
49#385—21no—
50#380D411yescharacteristic
51C4×C4⋊C4—21no—
52C4×C4⋊C4C2×C411yescharacteristic
53C4×C22⋊C4—21no—
54C2×#95—22no—
55C2×#95—22no—
56#340—24no—
57#340—42no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—22no—
64#335—42no—
65#328—44no—
66C22×Q8C4⋊C411yescharacteristic
67C23×C4—42no—
68C23×C4C2×D411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4#4111yescharacteristic
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8—42no—
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22⋊Q8—22no—
85C22⋊Q8—22no—
86C4×Q8—41no—
87C4×Q8—41no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4C4⋊C411yescharacteristic
98C2×C4⋊C4C4⋊C411yescharacteristic
99C2×C4⋊C4C4⋊C411yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42C22×C411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×Q811yescharacteristic
109C2×C42#4111yescharacteristic
110C4⋊C8—41no—
111C4⋊C8—41no—
112C4⋊C8—42no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115C4⋊C8—42no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119#97—42no—
120#97—42no—
121C4×C8—22no—
122C4×C8—22no—
123C4×C8—41no—
124C4×C8—41no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—42no—
131#95—42no—
132#95—82no—
133C24#12111yescharacteristic
134C2×Q8—22no—
135C2×Q8—41no—
136C2×Q8—41no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—44no—
152C22×C4—82no—
153C22×C4—82no—
154C22×C4—82no—
155C22×C4—82no—
156C22×C4—82no—
157C22×C4—82no—
158C22×C4C2×C22⋊C411yescharacteristic
159C22×C4C2×C4⋊C411yescharacteristic
160C22×C4#11711yescharacteristic
161C22×C4#12111yescharacteristic
162C22×C4C22⋊Q811yescharacteristic
163C22×C4C22⋊Q811yescharacteristic
164C22×C4#13512yes—
165C22×C4AGL(1,ℤ/8)12yes—
166C22×C4#13712yes—
167C22×C4#13712yes—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—44no—
179C2×C8—44no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—82no—
187C2×C8—82no—
188C2×C8—82no—
189C2×C8#9912yes—
190C2×C8#10412yes—
191C4⋊C4—22no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—82no—
201C4⋊C4—82no—
202C22⋊C4—42no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—41no—
210C42—41no—
211C42—41no—
212C42—41no—
213C42—41no—
214C42—41no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23#38011yescharacteristic
220C23#46412yes—
221C23#46812yes—
222C23#47211yescharacteristic
223C23#47311yescharacteristic
224Q8—82no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—84no—
258C2×C4—84no—
259C2×C4—84no—
260C2×C4—162no—
261C2×C4—162no—
262C2×C4—162no—
263C2×C4—162no—
264C2×C4C2×#9911yescharacteristic
265C2×C4#41011yescharacteristic
266C2×C4#41611yescharacteristic
267C2×C4#41911yescharacteristic
268C2×C4C2×#10411yescharacteristic
269C2×C4#42111yescharacteristic
270C2×C4#42812yes—
271C2×C4#46911yescharacteristic
272C2×C4#47011yescharacteristic
273C2×C4#47912yes—
274C8—22no—
275C8—22no—
276C8—41no—
277C8—41no—
278C8—81no—
279C8—81no—
280C8—162no—
281C22—22no—
282C22—22no—
283C22—21no—
284C22—21no—
285C22—22no—
286C22—22no—
287C22—21no—
288C22—21no—
289C22#169812yes—
290C22#170012yes—
291C22#179911yescharacteristic
292C22#180511yescharacteristic
293C22#180611yescharacteristic
294C22#180911yescharacteristic
295C22#181211yescharacteristic
296C22#181411yescharacteristic
297C22#181611yescharacteristic
298C4—22no—
299C4—21no—
300C4—22no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4—164no—
309C4#150011yescharacteristic
310C4#150811yescharacteristic
311C4#151511yescharacteristic
312C4#151611yescharacteristic
313C2—22no—
314C2—22no—
315C2#857411yescharacteristic
316C2#861011yescharacteristic
317C2#862711yescharacteristic
318C2#865511yescharacteristic
319C2#866911yescharacteristic
320C2#867911yescharacteristic
321C2#869711yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#446C4semidirect
9. #1984C4semidirect
10. #1939C4semidirect
16. #1770C4semidirect

Maximal subgroups

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

subgroupindex
#215972
#108202
#85962
#85722
#70392