Finite Group Databasebeta

← #98921

Subgroups — #98921

Order 512.

490 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C8×C22⋊Q8C211yesmaximal, characteristic
2#10861C211yesmaximal, characteristic
3#8595C211yesmaximal, characteristic
4#8572C212yesmaximal
5#7039C212yesmaximal
6C2×C8×Q8C411yescharacteristic, complemented
7C22×#106—21no—
8C22×C4×C8C2211yescharacteristic
9C4×C22⋊Q8C2211yescharacteristic
10#1984C411yescharacteristic, complemented
11#1939C411yescharacteristic, complemented
12#1916—21no—
13#1850—21no—
14#1797C2212yes—
15#1797C2211yescharacteristic
16C8×C4⋊C4—21no—
17C8×C4⋊C4C411yescharacteristic, complemented
18C8×C22⋊C4—21no—
19C2×#340—21no—
20C2×#334—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
31C8×Q8—42no—
32C2×#106—24no—
33C2×#106—42no—
34C2×#106—42no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C22⋊C8—21no—
41C2×C22⋊C8—22no—
42C2×C4×C8—21no—
43C2×C4×C8—21no—
44C2×C4×C8—42no—
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49#391Q811yescharacteristic
50#389Q811yescharacteristic
51#387—21no—
52#385—21no—
53#380D411yescharacteristic
54C4×C4⋊C4—21no—
55C4×C4⋊C4C2×C411yescharacteristic
56C4×C22⋊C4—21no—
57C2×#95—22no—
58C2×#95—22no—
59#340—42no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—22no—
66#334—42no—
67#328—44no—
68C22×Q8C4⋊C411yescharacteristic
69C23×C4—42no—
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4#4111yescharacteristic
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—42no—
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86C22×C8SD1612yes—
87C22×C8Q1612yes—
88C22⋊Q8—22no—
89C22⋊Q8SD1614yes—
90C4×Q8—41no—
91C4×Q8—41no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4C4⋊C411yescharacteristic
102C2×C4⋊C4C4⋊C411yescharacteristic
103C2×C4⋊C4C4⋊C411yescharacteristic
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—41no—
109C2×C42—41no—
110C2×C42C22×C411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×Q811yescharacteristic
113C2×C42#4111yescharacteristic
114#106—48no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—42no—
118C4⋊C8—42no—
119C4⋊C8—42no—
120C4⋊C8—42no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C4×C8—22no—
126C4×C8—22no—
127C4×C8—41no—
128C4×C8—41no—
129C4×C8—42no—
130C4×C8—42no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—42no—
137#95—42no—
138#95—82no—
139C24#12111yescharacteristic
140C2×Q8—21no—
141C2×Q8—41no—
142C2×Q8—41no—
143C2×Q8#10612yescomplemented
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—44no—
158C22×C4—82no—
159C22×C4—82no—
160C22×C4—82no—
161C22×C4—82no—
162C22×C4—82no—
163C22×C4—82no—
164C22×C4C2×C22⋊C411yescharacteristic
165C22×C4C2×C4⋊C411yescharacteristic
166C22×C4#11711yescharacteristic
167C22×C4#12111yescharacteristic
168C22×C4C22⋊Q811yescharacteristic
169C22×C4C22⋊Q811yescharacteristic
170C22×C4C2×SD1611yescharacteristic
171C22×C4C2×SD1612yes—
172C22×C4C2×Q1611yescharacteristic
173C22×C4#13512yes—
174C22×C4AGL(1,ℤ/8)11yescharacteristic
175C22×C4#13711yescharacteristic
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—24no—
179C2×C8—24no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—82no—
195C2×C8—82no—
196C2×C8—82no—
197C2×C8#9912yes—
198C2×C8#10312yes—
199C2×C8#10312yes—
200C2×C8#10412yes—
201C4⋊C4—21no—
202C4⋊C4—24no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C4⋊C4#10612yescomplemented
212C22⋊C4—24no—
213C22⋊C4—42no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—41no—
219C42—41no—
220C42—41no—
221C42—41no—
222C42—41no—
223C42—41no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23#38011yescharacteristic
229C23#46011yescharacteristic
230C23#46211yescharacteristic
231C23#46512yes—
232C23#47112yes—
233Q8—42no—
234Q8—42no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—44no—
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—41no—
269C2×C4—41no—
270C2×C4—84no—
271C2×C4—84no—
272C2×C4—84no—
273C2×C4—162no—
274C2×C4—162no—
275C2×C4—162no—
276C2×C4—162no—
277C2×C4C2×#9911yescharacteristic
278C2×C4#41011yescharacteristic
279C2×C4C2×#10311yescharacteristic
280C2×C4#41711yescharacteristic
281C2×C4C2×#10411yescharacteristic
282C2×C4#42111yescharacteristic
283C2×C4C2×#10611yescharacteristic
284C2×C4#42711yescharacteristic
285C2×C4#46512yes—
286C2×C4#47511yescharacteristic
287C2×C4#47711yescharacteristic
288C8—22no—
289C8—41no—
290C8—41no—
291C8—81no—
292C8—81no—
293C8—162no—
294C8#32814yescomplemented
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—22no—
303C22#166612yes—
304C22#167212yes—
305C22#179911yescharacteristic
306C22#180311yescharacteristic
307C22#180611yescharacteristic
308C22#180711yescharacteristic
309C22#181211yescharacteristic
310C22#181411yescharacteristic
311C22#181511yescharacteristic
312C4—21no—
313C4—21no—
314C4—22no—
315C4—21no—
316C4—21no—
317C4—42no—
318C4—41no—
319C4—41no—
320C4—42no—
321C4—81no—
322C4—81no—
323C4—164no—
324C4C2×#32811yescharacteristic
325C4#150911yescharacteristic
326C4#151311yescharacteristic
327C4#151811yescharacteristic
328C2—22no—
329C2—22no—
330C2#857211yescharacteristic
331C2#860811yescharacteristic
332C2#862611yescharacteristic
333C2#864311yescharacteristic
334C2#866611yescharacteristic
335C2#867511yescharacteristic
336C2#869411yescharacteristic
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×C8×Q8C4semidirect
10. #1984C4semidirect
11. #1939C4semidirect
17. C8×C4⋊C4C4semidirect
143. C2×Q8#106semidirect
211. C4⋊C4#106semidirect
294. C8#328semidirect

Maximal subgroups

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

subgroupindex
C8×C22⋊Q82
#108612
#85952
#85722
#70392