Finite Group Databasebeta

← #6432107

Subgroups — #6432107

Order 512.

3570 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62844C211yesmaximal, characteristic
2#34199C213yesmaximal, complemented
3#23598C218yesmaximal, complemented
4#23596C218yesmaximal, complemented
5#23565C211yesmaximal, characteristic, complemented
6#23515C213yesmaximal, complemented
7C2×#2030C211yesmaximal, characteristic, complemented
8C2×#2022C213yesmaximal, complemented
9#20392C213yesmaximal, complemented
10#3505—212no—
11#3491—26no—
12#3490—26no—
13#3459C2218yes—
14C2×#543C2211yescharacteristic
15C2×#534C2213yes—
16C22×#123C2213yes—
17C2×#518C2213yes—
18C22×#117C2211yescharacteristic
19C22×#99C2216yes—
20#2359—212no—
21#2344—26no—
22#2342—26no—
23#2322C22112yescomplemented
24#2315C2213yes—
25#2308C2213yes—
26C4×#123C22112yescomplemented
27C2×#381C2213yes—
28#2273C2213yes—
29#2062C2218yescomplemented
30#2044C22124yescomplemented
31#2030C2214yescomplemented
32#2030C2214yescomplemented
33#2022C22112yescomplemented
34#2022C22112yescomplemented
35C4×#99C22124yescomplemented
36C2×#342C2213yes—
37C2×#342C2213yes—
38C23×D4C2313yes—
39C24×C4C2311yescharacteristic
40#553—224no—
41#552—224no—
42#551—248no—
43#548—28no—
44#546—224no—
45#543—26no—
46#543C2314yes—
47#539—224no—
48#538—224no—
49#534—212no—
50#534—26no—
51#534C23112yes—
52#529—224no—
53#525—224no—
54C2×#123—26no—
55C2×#123—26no—
56C2×#123—212no—
57C2×#123—212no—
58C2×#123C2316yes—
59C2×#123C2316yes—
60C2×C22⋊Q8—26no—
61C2×#121—26no—
62C2×#120—26no—
63#518—212no—
64#518—212no—
65C2×C4×D4D4112yes—
66C2×C4×D4C23112yes—
67C2×#117—26no—
68C2×#117—26no—
69C2×#117D416yes—
70C22×C4⋊C4—23no—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—23no—
73C22×C22⋊C4D416yes—
74C22×C22⋊C4C2311yescharacteristic
75C22×C22⋊C4C2313yes—
76C22×C22⋊C4C2313yes—
77C22×C22⋊C4C2313yes—
78C22×C22⋊C4C2311yescharacteristic
79C22×C42C2312yes—
80C2×#99—224no—
81C2×#99C23124yes—
82#389—212no—
83#388—212no—
84#388—212no—
85#387—212no—
86#387—212no—
87#386—26no—
88#386C23112yes—
89#385—26no—
90#385—212no—
91#385C23112yes—
92#381—23no—
93#381—26no—
94#381C2316yes—
95#381C2316yes—
96#380—26no—
97#379—23no—
98C4×C4⋊C4—212no—
99C4×C22⋊C4—212no—
100C4×C22⋊C4—26no—
101C4×C22⋊C4—26no—
102C4×C22⋊C4C23112yes—
103C2×#95—26no—
104#342C23112yes—
105#342C23112yes—
106C25C2×D413yes—
107C25C2411yescharacteristic
108C2×#41—28no—
109C22×Q8—22no—
110C22×D4—26no—
111C22×D4—26no—
112C22×D4—26no—
113C22×D4—26no—
114C22×D4—212no—
115C22×D4—26no—
116C22×D4C2×D416yes—
117C22×D4C2416yes—
118C23×C4—23no—
119C23×C4—23no—
120C23×C4—26no—
121C23×C4—23no—
122C23×C4—23no—
123C23×C4C2×D413yes—
124C23×C4C2×D4112yes—
125C23×C4C2×D416yes—
126C23×C4#4114yescomplemented
127C23×C4C2411yescharacteristic
128C23×C4C2411yescharacteristic
129C23×C4C2412yes—
130C23×C4C2412yes—
131#126—248no—
132#125—224no—
133#124—224no—
134#123—248no—
135#123—224no—
136#123—248no—
137#123—224no—
138#123—212no—
139#123—212no—
140#123—212no—
141#123—212no—
142C22⋊Q8—224no—
143C22⋊Q8—224no—
144C22⋊Q8—224no—
145#121—224no—
146#121—224no—
147#121—224no—
148#120—224no—
149#120—224no—
150C4×Q8—28no—
151C4×D4—224no—
152C4×D4—248no—
153C4×D4—224no—
154C4×D4—224no—
155#117—224no—
156#117—224no—
157#117—412no—
158#117—44no—
159C2×C4⋊C4—224no—
160C2×C4⋊C4—26no—
161C2×C4⋊C4—26no—
162C2×C4⋊C4—26no—
163C2×C4⋊C4—26no—
164C2×C4⋊C4—212no—
165C2×C4⋊C4—212no—
166C2×C4⋊C4C2×D4112yes—
167C2×C22⋊C4—224no—
168C2×C22⋊C4—212no—
169C2×C22⋊C4—26no—
170C2×C22⋊C4—26no—
171C2×C22⋊C4—26no—
172C2×C22⋊C4—26no—
173C2×C22⋊C4—23no—
174C2×C22⋊C4—26no—
175C2×C22⋊C4—224no—
176C2×C22⋊C4—212no—
177C2×C22⋊C4—23no—
178C2×C22⋊C4—23no—
179C2×C22⋊C4—23no—
180C2×C22⋊C4—212no—
181C2×C22⋊C4C2×D416yes—
182C2×C22⋊C4C2×D4112yes—
183C2×C22⋊C4C2×D416yes—
184C2×C22⋊C4C2×D416yes—
185C2×C22⋊C4#41112yescomplemented
186C2×C22⋊C4C2416yes—
187C2×C22⋊C4C2416yes—
188C2×C22⋊C4C2416yes—
189C2×C42—212no—
190C2×C42—212no—
191C2×C42C2×D4112yes—
192#99—248no—
193#95—212no—
194#95—212no—
195#95—212no—
196#95—212no—
197#95—212no—
198C24—26no—
199C24—26no—
200C24—26no—
201C24—82no—
202C24—86no—
203C24C22×D416yes—
204C24C22×D413yes—
205C24C22×D413yes—
206C24C2×#4112yes—
207C24C2×#4116yes—
208C24C2511yescharacteristic
209#41—168no—
210C2×Q8—28no—
211C2×Q8—162no—
212C2×D4—26no—
213C2×D4—26no—
214C2×D4—212no—
215C2×D4—212no—
216C2×D4—212no—
217C2×D4—212no—
218C2×D4—26no—
219C2×D4—26no—
220C2×D4—224no—
221C2×D4—412no—
222C2×D4—412no—
223C2×D4—412no—
224C2×D4—412no—
225C2×D4—166no—
226C22×C4—26no—
227C22×C4—212no—
228C22×C4—26no—
229C22×C4—212no—
230C22×C4—26no—
231C22×C4—23no—
232C22×C4—26no—
233C22×C4—23no—
234C22×C4—23no—
235C22×C4—23no—
236C22×C4—26no—
237C22×C4—26no—
238C22×C4—212no—
239C22×C4—224no—
240C22×C4—26no—
241C22×C4—26no—
242C22×C4—43no—
243C22×C4—41no—
244C22×C4—46no—
245C22×C4—46no—
246C22×C4—46no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—86no—
250C22×C4—812no—
251C22×C4—86no—
252C22×C4—86no—
253C22×C4C22×D416yes—
254C22×C4C22×D4112yes—
255C22×C4C22×D4112yes—
256C22×C4C2×#4112yes—
257C22×C4C2×#4112yes—
258C4⋊C4—224no—
259C4⋊C4—224no—
260C4⋊C4—224no—
261C4⋊C4—224no—
262C4⋊C4—224no—
263C4⋊C4—48no—
264C4⋊C4—812no—
265C22⋊C4—224no—
266C22⋊C4—224no—
267C22⋊C4—248no—
268C22⋊C4—212no—
269C22⋊C4—212no—
270C22⋊C4—212no—
271C22⋊C4—26no—
272C22⋊C4—224no—
273C22⋊C4—212no—
274C22⋊C4—43no—
275C22⋊C4—41no—
276C22⋊C4—46no—
277C22⋊C4—812no—
278C22⋊C4#120112yes—
279C22⋊C4#12014yes—
280C22⋊C4C2×#41112yes—
281C42—224no—
282C42—48no—
283C42—812no—
284C23—26no—
285C23—26no—
286C23—26no—
287C23—46no—
288C23—42no—
289C23—46no—
290C23—412no—
291C23—412no—
292C23—812no—
293C23—82no—
294C23—86no—
295C23—812no—
296C23—166no—
297C23C2×#12013yes—
298C23C2×#12011yescharacteristic
299C23#52516yes—
300C23#52516yes—
301C23#53216yes—
302C23#53216yes—
303C23C23×D413yes—
304C23C22×#4111yescharacteristic
305C23C22×#4113yes—
306Q8—88no—
307D4—412no—
308D4—412no—
309D4—412no—
310D4—412no—
311D4—824no—
312C2×C4—26no—
313C2×C4—212no—
314C2×C4—212no—
315C2×C4—26no—
316C2×C4—26no—
317C2×C4—26no—
318C2×C4—26no—
319C2×C4—412no—
320C2×C4—412no—
321C2×C4—412no—
322C2×C4—44no—
323C2×C4—412no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—43no—
327C2×C4—43no—
328C2×C4—412no—
329C2×C4—412no—
330C2×C4—412no—
331C2×C4—42no—
332C2×C4—46no—
333C2×C4—824no—
334C2×C4—86no—
335C2×C4—86no—
336C2×C4—812no—
337C2×C4—166no—
338C2×C4C2×#12018yes—
339C22—26no—
340C22—22no—
341C22—26no—
342C22—46no—
343C22—46no—
344C22—46no—
345C22—46no—
346C22—41no—
347C22—43no—
348C22—412no—
349C22—46no—
350C22—412no—
351C22—86no—
352C22—1612no—
353C22#238816yes—
354C22#238812yes—
355C22C22×#12011yescharacteristic
356C22C2×#52513yes—
357C22C2×#53213yes—
358C4—28no—
359C4—412no—
360C4—42no—
361C4—42no—
362C4—46no—
363C4—412no—
364C4—86no—
365C4—82no—
366C2—23no—
367C2—23no—
368C2—26no—
369C2—22no—
370C2—86no—
371C2C2×#238811yescharacteristic
372C2#6040112yes—
subgroupG/H
center, socleC22C22×#120
commutator, Frattini, cosocleC24C25
prime radical#62844C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C22⋊C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34199C2semidirect
3. #23598C2semidirect
4. #23596C2semidirect
5. #23565C2semidirect
6. #23515C2semidirect
7. C2×#2030C2semidirect
8. C2×#2022C2semidirect
9. #20392C2semidirect
23. #2322C22semidirect
26. C4×#123C22semidirect
29. #2062C22semidirect
30. #2044C22semidirect
31. #2030C22semidirect
32. #2030C22semidirect
33. #2022C22semidirect
34. #2022C22semidirect
35. C4×#99C22semidirect
126. C23×C4#41semidirect
185. C2×C22⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
#628442
#341992
#235982
#235962
#235652
#235152
C2×#20302
C2×#20222
#203922