Finite Group Databasebeta

← #6372214

Subgroups — #6372214

Order 512.

3498 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3503C211yesmaximal, characteristic
2C2×#2875C211yesmaximal, characteristic
3#34184C211yesmaximal, characteristic
4#21326C2112yesmaximal, complemented
5#21324C213yesmaximal, complemented
6#21323C213yesmaximal, complemented
7#21321C213yesmaximal, complemented
8C2×#1993C213yesmaximal, complemented
9#20348C214yesmaximal
10C22×#131C2212yes—
11#3503—28no—
12#3503C4116yescomplemented
13C2×Q82—21no—
14C2×#557—21no—
15C2×#554—23no—
16C2×D4×Q8—23no—
17C2×#544C2216yes—
18C2×#533C2213yes—
19C22×C4⋊Q8C2213yes—
20C2×#520C2212yes—
21C2×#520C2212yes—
22C2×C4×#41C2211yescharacteristic
23#2875—28no—
24C2×#405C2216yes—
25C2×C4×M16C2213yes—
26C22×#97C2213yes—
27#2351—21no—
28#2351—21no—
29#2349—23no—
30#2349—23no—
31#2317C2216yes—
32#2285C2213yes—
33C2×#382C2213yes—
34#1993C2213yes—
35#1993C2216yes—
36#1993C22112yes—
37#1993C2211yescharacteristic
38#1993C2213yes—
39#1993C2213yes—
40#1992C22112yes—
41#1991C22112yes—
42#1990C22112yes—
43#1736C22112yes—
44#1736C22112yes—
45#1735C22112yes—
46C2×#340C22112yes—
47C22×#41C2311yescharacteristic
48C23×Q8—23no—
49C2×#131—28no—
50C2×#131—28no—
51C2×#131Q8116yescomplemented
52C22×M16C2316yes—
53C23×C8C2316yes—
54Q82—216no—
55#557—216no—
56#554—248no—
57D4×Q8—248no—
58#544—224no—
59#544C2×C4148yescomplemented
60#533—212no—
61#533C2×C4124yescomplemented
62C2×C4⋊Q8—26no—
63C2×C4⋊Q8—23no—
64C2×C4⋊Q8—23no—
65C2×C4⋊Q8—212no—
66C2×C4⋊Q8C2×C4124yescomplemented
67C2×C4⋊Q8C2313yes—
68C2×C4⋊Q8C2316yes—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#125—26no—
71C2×#125C2316yes—
72C2×C22⋊Q8—23no—
73C2×C22⋊Q8—23no—
74C2×C22⋊Q8—212no—
75#520—28no—
76#520—216no—
77#520C2×C4116yes—
78C4×#41C2×C418yes—
79C4×#41D418yes—
80C2×C4×Q8—22no—
81C2×C4×Q8—22no—
82C2×C4×Q8—23no—
83C2×C4×Q8—23no—
84C2×C4×Q8D411yescharacteristic
85C2×C4×Q8D411yescharacteristic
86C2×C4×D4—26no—
87C2×C4×D4—26no—
88C2×C4×D4D413yes—
89C2×C4×D4D413yes—
90C2×#117C2316yes—
91C2×#117C2316yes—
92C2×#117C2313yes—
93C22×C4⋊C4—23no—
94C22×C4⋊C4C2316yes—
95C22×C4⋊C4C2316yes—
96C22×C42C2313yes—
97#405—224no—
98C4×M16—212no—
99C2×#97—212no—
100C2×#97C2311yescharacteristic
101C2×#97C2313yes—
102C2×#97C2316yes—
103C2×C4×C8C2316yes—
104#389—26no—
105#389—26no—
106#385—212no—
107#384C2316yes—
108#382—23no—
109#382—23no—
110#382C2316yes—
111#382C2313yes—
112#382C2311yescharacteristic
113#380—23no—
114#380—23no—
115C4×C4⋊C4—23no—
116C4×C4⋊C4—23no—
117C2×#95—23no—
118C2×#95—23no—
119#340C23112yes—
120#340C23124yes—
121#340C23112yes—
122#340C2314yes—
123#340C23112yes—
124C2×#41C22×C414yes—
125C2×#41C2×D414yes—
126C2×#41C2×Q818yes—
127C2×#41#4118yes—
128C22×Q8—224no—
129C22×Q8—21no—
130C22×Q8—23no—
131C22×Q8—26no—
132C22×Q8—812no—
133C22×Q8C2×D411yescharacteristic
134C22×D4C2×D413yes—
135C23×C4—26no—
136C23×C4—26no—
137C23×C4C2×D413yes—
138C23×C4C2×D413yes—
139C23×C4C2413yes—
140#131—264no—
141C2×M16—212no—
142C2×M16—212no—
143C2×M16C2×Q8124yescomplemented
144C22×C8—212no—
145C22×C8—212no—
146C22×C8C2×Q8124yescomplemented
147C22×C8C2412yes—
148C22×C8C2416yes—
149C4⋊Q8—248no—
150C4⋊Q8—224no—
151C4⋊Q8—224no—
152C4⋊Q8—26no—
153C4⋊Q8—212no—
154C4⋊Q8—22no—
155C4⋊Q8C22×C4112yes—
156C4⋊Q8C22×C4124yes—
157C4⋊Q8C22×C414yes—
158#125—248no—
159#125—212no—
160#125C22×C4124yes—
161C22⋊Q8—224no—
162C22⋊Q8—224no—
163C22⋊Q8—296no—
164C4×Q8—216no—
165C4×Q8—224no—
166C4×Q8—224no—
167C4×Q8—48no—
168C4×Q8C22⋊C418yes—
169C4×Q8C22⋊C418yes—
170C4×D4—248no—
171C4×D4—424no—
172C4×D4C22⋊C4124yes—
173C4×D4C22⋊C4124yes—
174#117—212no—
175#117—224no—
176#117C22×C4124yes—
177#117C22×C4112yes—
178#117C2×D4112yes—
179C2×C4⋊C4—224no—
180C2×C4⋊C4—23no—
181C2×C4⋊C4—26no—
182C2×C4⋊C4—26no—
183C2×C4⋊C4—212no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—26no—
186C2×C4⋊C4—26no—
187C2×C4⋊C4—224no—
188C2×C4⋊C4—26no—
189C2×C4⋊C4—26no—
190C2×C4⋊C4—26no—
191C2×C4⋊C4C22×C4124yes—
192C2×C4⋊C4C2×D413yes—
193C2×C4⋊C4C2×D413yes—
194C2×C4⋊C4C2416yes—
195C2×C4⋊C4C2412yes—
196C2×C4⋊C4C2416yes—
197C2×C4⋊C4C2412yes—
198C2×C22⋊C4—26no—
199C2×C22⋊C4—26no—
200C2×C22⋊C4C2×D416yes—
201C2×C42—26no—
202C2×C42—26no—
203C2×C42C22×C4112yes—
204C2×C42C2×D413yes—
205C2×C42C2×D413yes—
206C2×C42C2×D4112yes—
207C2×C42C2413yes—
208C2×C42C2411yescharacteristic
209#97—22no—
210#97—26no—
211#97—212no—
212C4×C8—212no—
213#95—23no—
214#95—23no—
215#95—26no—
216#95—26no—
217#95—21no—
218#95—21no—
219C24C22×D413yes—
220#41C4×Q8132yes—
221#41C22⋊Q8132yes—
222C2×Q8—24no—
223C2×Q8—212no—
224C2×Q8—224no—
225C2×Q8—44no—
226C2×Q8—412no—
227C2×Q8—812no—
228C2×Q8—848no—
229C2×Q8C2×C22⋊C414yes—
230C2×Q8#12414yes—
231C2×Q8C4⋊Q814yes—
232C2×D4C2×C22⋊C4112yes—
233C2×D4#124112yes—
234C2×D4C4⋊Q8112yes—
235C22×C4—22no—
236C22×C4—26no—
237C22×C4—224no—
238C22×C4—26no—
239C22×C4—22no—
240C22×C4—46no—
241C22×C4—46no—
242C22×C4—412no—
243C22×C4—812no—
244C22×C4—86no—
245C22×C4—86no—
246C22×C4C2×C22⋊C4112yes—
247C22×C4C2×C22⋊C4112yes—
248C22×C4C23×C416yes—
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D413yes—
252C22×C4C22×D413yes—
253C22×C4C22×D413yes—
254C22×C4C22×D416yes—
255C22×C4C22×Q8112yes—
256C22×C4C2×#41112yes—
257C22×C4C2511yescharacteristic
258M16—248no—
259C2×C8—248no—
260C2×C8—26no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—26no—
264C2×C8C22×Q814yes—
265C2×C8C22×Q8112yes—
266C4⋊C4—212no—
267C4⋊C4—224no—
268C4⋊C4—224no—
269C4⋊C4—24no—
270C4⋊C4—224no—
271C4⋊C4—224no—
272C4⋊C4—26no—
273C4⋊C4—22no—
274C4⋊C4—212no—
275C4⋊C4—24no—
276C4⋊C4—412no—
277C4⋊C4—86no—
278C4⋊C4—86no—
279C4⋊C4C2×C22⋊C4112yes—
280C4⋊C4C2×C22⋊C4112yes—
281C4⋊C4C23×C4112yes—
282C4⋊C4C23×C414yes—
283C22⋊C4—224no—
284C22⋊C4—412no—
285C22⋊C4C2×C22⋊C4124yes—
286C42—224no—
287C42—412no—
288C42C2×C22⋊C4112yes—
289C42C2×C22⋊C4112yes—
290C42C23×C412yes—
291C42C23×C416yes—
292C42C22×D416yes—
293C42C22×D412yes—
294C23—26no—
295C23—26no—
296C23C22×C22⋊C416yes—
297C23C2×#12416yes—
298C23C2×C4⋊Q816yes—
299C23C23×D411yescharacteristic
300Q8—48no—
301Q8—424no—
302Q8—824no—
303Q8#391116yes—
304D4#391148yes—
305C2×C4—26no—
306C2×C4—22no—
307C2×C4—26no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—212no—
311C2×C4—412no—
312C2×C4—412no—
313C2×C4—412no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—46no—
317C2×C4—42no—
318C2×C4—46no—
319C2×C4—42no—
320C2×C4—46no—
321C2×C4—824no—
322C2×C4—1612no—
323C2×C4C22×C22⋊C412yes—
324C2×C4C22×C22⋊C412yes—
325C2×C4C22×C22⋊C416yes—
326C2×C4C22×C22⋊C416yes—
327C2×C4C22×C22⋊C416yes—
328C2×C4C2×C4×Q8124yes—
329C2×C4C2×C22⋊Q8124yes—
330C2×C4C2×#12416yes—
331C2×C4C2×C4⋊Q816yes—
332C2×C4C24×C411yescharacteristic
333C2×C4C23×D411yescharacteristic
334C2×C4C23×Q812yes—
335C2×C4C22×#4112yes—
336C8—24no—
337C8—212no—
338C22—26no—
339C22—26no—
340C22—212no—
341C22—212no—
342C22C2×#391112yes—
343C22C23×C22⋊C411yescharacteristic
344C22C22×#12411yescharacteristic
345C22C22×C4⋊Q811yescharacteristic
346C22#358512yes—
347C22#358612yes—
348C4—22no—
349C4—26no—
350C4—22no—
351C4—26no—
352C4—412no—
353C4—44no—
354C4—82no—
355C4—86no—
356C4C2×#391112yes—
357C4C22×C4×Q812yes—
358C4C22×C22⋊Q812yes—
359C2—212no—
360C2C22×#39111yescharacteristic
361C2#6008711yescharacteristic
362C2#6008811yescharacteristic
363C2#6062512yes—
364C2#6066812yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21326C2semidirect
5. #21324C2semidirect
6. #21323C2semidirect
7. #21321C2semidirect
8. C2×#1993C2semidirect
12. #3503C4semidirect
51. C2×#131Q8semidirect
59. #544C2×C4semidirect
61. #533C2×C4semidirect
66. C2×C4⋊Q8C2×C4semidirect
143. C2×M16C2×Q8semidirect
146. C22×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#35032
C2×#28752
#341842
#213262
#213242
#213232
#213212
C2×#19932
#203482