Finite Group Databasebeta

← #10575281

Subgroups — #10575281

Order 512.

6366 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62854C212yesmaximal
2#60668C2116yesmaximal, complemented
3#60666C218yesmaximal, complemented
4#60665C218yesmaximal, complemented
5#60664C2112yesmaximal, complemented
6#60660C216yesmaximal, complemented
7C2×#3163C216yesmaximal, complemented
8#60109C212yesmaximal
9#60108C211yesmaximal, characteristic
10C2×#2912C211yesmaximal, characteristic
11#60058C211yesmaximal, characteristic
12C2×#568C2211yescharacteristic
13C22×#131C2211yescharacteristic
14#3504—28no—
15#3504C22116yes—
16#3503—28no—
17#3503C22116yes—
18#3502—24no—
19#3501—212no—
20#3458C22112yes—
21C2×#544C22112yes—
22#3428C2212yes—
23#3428C2211yescharacteristic
24C2×#520C2212yes—
25#3166C22148yescomplemented
26#3165C22148yescomplemented
27#3164C22124yes—
28#3164C22124yescomplemented
29#3164C22148yescomplemented
30#3163C22112yes—
31#3163C22124yes—
32#3163C2214yes—
33#3163C22124yescomplemented
34#3163C22124yescomplemented
35#3163C22112yescomplemented
36#3163C22112yescomplemented
37#3163C22148yescomplemented
38#3162C22124yescomplemented
39C2×#501C22124yescomplemented
40#3160C22124yescomplemented
41C2×#499C22112yescomplemented
42C2×#498C22112yescomplemented
43#2913C2218yes—
44#2913C22116yes—
45#2912C2218yes—
46#2912C2218yes—
47#2912C2214yes—
48#2912C2214yes—
49#2912C22116yes—
50#2911C2216yes—
51#2911C22112yes—
52C2×#428C2216yes—
53C2×#428C22112yes—
54C2×#427C2213yes—
55C2×#427C2216yes—
56C22×#106C2213yes—
57#2875C2218yes—
58C4×#131C2218yes—
59#2873C2216yes—
60C2×#405C2216yes—
61C22×#41C2311yescharacteristic
62#568Q8116yescomplemented
63C2×#131Q8116yescomplemented
64C2×#131C2314yes—
65C2×#131C2318yes—
66C2×#131C2314yes—
67C22×M16C2316yes—
68C22×M16C2313yes—
69C23×C8C2313yes—
70Q82—216no—
71#557—216no—
72#557—216no—
73#557—216no—
74#554—248no—
75#554—248no—
76#554—248no—
77D4×Q8—248no—
78#544—224no—
79#544—224no—
80#544—212no—
81#544—212no—
82#544C23124yes—
83#544C2318yes—
84#544C23148yes—
85#544C23124yes—
86#544C23124yes—
87#543—224no—
88#533—224no—
89#533—224no—
90#533C23148yes—
91C2×C4⋊Q8—212no—
92C2×C4⋊Q8C23124yes—
93C2×#125—212no—
94C2×#125C23124yes—
95C2×C22⋊Q8—224no—
96#520—28no—
97#520D418yes—
98#520C23116yes—
99#520C2318yes—
100#520C2318yes—
101#519—22no—
102#519D412yes—
103#518—26no—
104#518D416yes—
105C4×#41—28no—
106C4×#41C2318yes—
107C2×C4×Q8—22no—
108C2×C4×D4—26no—
109C2×#117C2316yes—
110C2×#117C23112yes—
111C2×#117C2316yes—
112C22×C4⋊C4C2316yes—
113#501C23148yes—
114#501C23196yes—
115#501C23116yes—
116#501C23148yes—
117#501C23148yes—
118#500C23148yes—
119#499C23124yes—
120#499C23148yes—
121#499C2318yes—
122#499C23148yes—
123#498C23124yes—
124#498C23148yes—
125#498C2318yes—
126#428C23124yes—
127#428C2318yes—
128#428C23124yes—
129#428C23148yes—
130#428C23124yes—
131#428C23148yes—
132#428C23124yes—
133#428C23124yes—
134#427C23112yes—
135#427C23112yes—
136#427C23148yes—
137#427C23112yes—
138#427C23112yes—
139#427C2314yes—
140#427C23112yes—
141C2×#107C23124yes—
142C2×#107C23112yes—
143C2×#107C2316yes—
144C2×#106C2316yes—
145C2×#106C23112yes—
146C2×#106C23124yes—
147C2×#106C2316yes—
148C2×#106C23112yes—
149C2×#106C2313yes—
150C2×#106C2316yes—
151C2×#106C2311yescharacteristic
152#405C2314yes—
153#405C23112yes—
154#405C23112yes—
155#405C23112yes—
156#405C23124yes—
157C4×M16C23124yes—
158C2×#97C23112yes—
159C2×C4×C8C23112yes—
160C2×#41C2×D414yes—
161C2×#41C2×Q818yes—
162C2×#41C2×Q818yes—
163C2×#41C2414yes—
164C22×Q8—224no—
165C22×Q8C2×D411yescharacteristic
166C22×D4C2×D413yes—
167C23×C4—26no—
168C23×C4C2413yes—
169#131C2×Q8132yescomplemented
170#131C2×Q8132yescomplemented
171#131C2×Q8164yescomplemented
172C2×M16C2×Q8124yescomplemented
173C2×M16C2×Q8148yescomplemented
174C2×M16C2414yes—
175C2×M16C24112yes—
176C2×M16C24112yes—
177C22×C8C2×Q8124yescomplemented
178C22×C8C2411yescharacteristic
179C22×C8C2413yes—
180C22×C8C2416yes—
181C22×C8C2416yes—
182C22×C8C24112yes—
183C4⋊Q8—224no—
184C4⋊Q8—248no—
185C4⋊Q8—28no—
186C4⋊Q8—224no—
187C4⋊Q8—224no—
188C4⋊Q8—248no—
189C4⋊Q8C24148yes—
190C4⋊Q8C24116yes—
191#125—224no—
192#125—248no—
193#125—28no—
194#125—224no—
195#125—224no—
196#125—248no—
197#125C24148yes—
198#125C24116yes—
199C22⋊Q8—248no—
200C22⋊Q8—248no—
201C22⋊Q8—248no—
202C22⋊Q8—296no—
203C22⋊Q8—216no—
204C22⋊Q8—248no—
205C22⋊Q8—248no—
206C4×Q8—216no—
207C4×Q8—28no—
208C4×Q8—28no—
209C4×Q8—248no—
210C4×Q8—248no—
211C4×Q8C2×D4116yes—
212C4×D4—248no—
213C4×D4—224no—
214C4×D4—224no—
215C4×D4C2×D4148yes—
216#117—212no—
217#117—212no—
218#117—212no—
219#117C2×D4112yes—
220#117C2×D4112yes—
221#117C2418yes—
222#117C24124yes—
223#117C24124yes—
224#117C2414yes—
225#117C24112yes—
226C2×C4⋊C4—26no—
227C2×C4⋊C4—212no—
228C2×C4⋊C4—224no—
229C2×C4⋊C4—224no—
230C2×C4⋊C4—224no—
231C2×C4⋊C4C2×D4112yes—
232C2×C4⋊C4C2416yes—
233C2×C4⋊C4C24124yes—
234C2×C4⋊C4C2412yes—
235C2×C4⋊C4C24112yes—
236C2×C4⋊C4C24112yes—
237C2×C22⋊C4—26no—
238C2×C22⋊C4—212no—
239C2×C22⋊C4C2×D4112yes—
240C2×C42—26no—
241C2×C42—212no—
242C2×C42C24112yes—
243#107C2418yes—
244#107C24148yes—
245#107C24124yes—
246#107C24124yes—
247#107C24116yes—
248#106C2416yes—
249#106C2416yes—
250#106C2414yes—
251#106C2414yes—
252#106C24124yes—
253#106C24148yes—
254#106C2412yes—
255#106C2412yes—
256#106C24112yes—
257#106C24112yes—
258#106C24116yes—
259#97C2418yes—
260#97C24124yes—
261C4×C8C2418yes—
262C4×C8C24124yes—
263C24C22×D413yes—
264#41C4⋊Q8132yes—
265#41C22×Q8116yes—
266#41C22×Q8116yes—
267C2×Q8—216no—
268C2×Q8—248no—
269C2×Q8—1624no—
270C2×Q8C4⋊Q818yes—
271C2×Q8C22×D414yes—
272C2×D4C4⋊Q8124yes—
273C2×D4C22×D4112yes—
274C22×C4—22no—
275C22×C4—212no—
276C22×C4—212no—
277C22×C4—26no—
278C22×C4—224no—
279C22×C4—166no—
280C22×C4C22×D4124yes—
281C22×C4C22×D416yes—
282C22×C4C22×D413yes—
283C22×C4C22×Q8112yes—
284C22×C4C22×Q8112yes—
285C22×C4C2516yes—
286C22×C4C2511yescharacteristic
287M16C22×Q8116yes—
288M16C22×Q8148yes—
289M16C22×Q8148yes—
290C2×C8C22×Q814yes—
291C2×C8C22×Q8112yes—
292C2×C8C22×Q8124yes—
293C2×C8C22×Q8124yes—
294C2×C8C22×Q8148yes—
295C2×C8C2511yescharacteristic
296C2×C8C2518yes—
297C2×C8C2516yes—
298C2×C8C2511yescharacteristic
299C4⋊C4—26no—
300C4⋊C4—224no—
301C4⋊C4—248no—
302C4⋊C4—248no—
303C4⋊C4—248no—
304C4⋊C4—22no—
305C4⋊C4—212no—
306C4⋊C4—212no—
307C4⋊C4—216no—
308C4⋊C4—216no—
309C4⋊C4—216no—
310C4⋊C4C22×D418yes—
311C4⋊C4C22×D4124yes—
312C4⋊C4C25112yes—
313C4⋊C4C2518yes—
314C4⋊C4C2518yes—
315C4⋊C4C2514yes—
316C22⋊C4—28no—
317C22⋊C4—224no—
318C22⋊C4—224no—
319C22⋊C4C22×D418yes—
320C22⋊C4C22×D4124yes—
321C42—28no—
322C42—224no—
323C42—224no—
324C42C22×D4124yes—
325C42C2518yes—
326C23—46no—
327C23C2×C4⋊Q8112yes—
328C23C23×D411yescharacteristic
329C23C23×D416yes—
330Q8—816no—
331Q8—848no—
332Q8C2×C4⋊Q8116yes—
333D4C2×C4⋊Q8148yes—
334C2×C4—24no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—212no—
338C2×C4—424no—
339C2×C4—812no—
340C2×C4—82no—
341C2×C4—86no—
342C2×C4—824no—
343C2×C4—812no—
344C2×C4C2×C4⋊Q8112yes—
345C2×C4C2×C4⋊Q8124yes—
346C2×C4C23×D4116yes—
347C2×C4C23×D416yes—
348C2×C4C23×D411yescharacteristic
349C2×C4C23×Q812yes—
350C2×C4C23×Q8112yes—
351C2×C4C23×Q8112yes—
352C2×C4C23×Q812yes—
353C2×C4C2611yescharacteristic
354C8C23×Q812yes—
355C8C23×Q812yes—
356C8C23×Q8112yes—
357C8C23×Q8116yes—
358C22—212no—
359C22—22no—
360C22—412no—
361C22C22×C4⋊Q812yes—
362C22C22×C4⋊Q8112yes—
363C22C24×D411yescharacteristic
364C4—216no—
365C4—412no—
366C4—48no—
367C4—48no—
368C4—44no—
369C4C22×C4⋊Q812yes—
370C4C22×C4⋊Q8112yes—
371C4C24×Q811yescharacteristic
372C4C24×Q811yescharacteristic
373C2—22no—
374C2—212no—
375C2C23×C4⋊Q811yescharacteristic
376C2#6309212yes—
subgroupG/H
center, socleC22C24×D4
commutator, Frattini, cosocleC2×C4C26
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #3428 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60668C2semidirect
3. #60666C2semidirect
4. #60665C2semidirect
5. #60664C2semidirect
6. #60660C2semidirect
7. C2×#3163C2semidirect
25. #3166C22semidirect
26. #3165C22semidirect
28. #3164C22semidirect
29. #3164C22semidirect
33. #3163C22semidirect
34. #3163C22semidirect
35. #3163C22semidirect
36. #3163C22semidirect
37. #3163C22semidirect
38. #3162C22semidirect
39. C2×#501C22semidirect
40. #3160C22semidirect
41. C2×#499C22semidirect
42. C2×#498C22semidirect
62. #568Q8semidirect
63. C2×#131Q8semidirect
169. #131C2×Q8semidirect
170. #131C2×Q8semidirect
171. #131C2×Q8semidirect
172. C2×M16C2×Q8semidirect
173. C2×M16C2×Q8semidirect
177. C22×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#628542
#606682
#606662
#606652
#606642
#606602
C2×#31632
#601092
#601082
C2×#29122
#600582