Finite Group Databasebeta

← #5022218

Subgroups — #5022218

Order 512.

2608 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16152C216yesmaximal, complemented
2#15948C213yesmaximal
3#15668C213yesmaximal, complemented
4#15636C213yesmaximal, complemented
5#15468C216yesmaximal, complemented
6#15467C213yesmaximal, complemented
7#14706C213yesmaximal, complemented
8#14649C213yesmaximal, complemented
9#13827C211yesmaximal, characteristic, complemented
10C22×#123C2213yes—
11C22×#120C2211yescharacteristic
12C23×C42C2211yescharacteristic, complemented
13C2×#393C2213yes—
14C2×#393C2216yes—
15C2×#393C2213yescomplemented
16C2×#391C2213yescomplemented
17C2×#386C2216yes—
18C2×#381C2213yescomplemented
19C2×#379C2213yes—
20C2×#379C2213yes—
21#1446C2216yescomplemented
22#1445C2213yescomplemented
23#1443C2216yescomplemented
24#1443C2216yescomplemented
25#1442C2216yescomplemented
26#1440C2213yes—
27#1440C2213yes—
28#1440C2216yes—
29#1440C2216yes—
30#1440C2216yes—
31#1439C2216yes—
32#1439C2216yes—
33#1438C2216yes—
34#1438C2213yes—
35#1437C2213yes—
36#1437C2216yes—
37#1437C2213yes—
38#1437C2213yes—
39#1437C2216yes—
40#1437C2211yescharacteristic, complemented
41#1437C2212yescomplemented
42#1436C2216yescomplemented
43#1436C2213yescomplemented
44#1436C2213yescomplemented
45#1434C2216yescomplemented
46C22⋊#95C2216yescomplemented
47C26C2311yescharacteristic
48C23×Q8C2313yes—
49C23×D4C2313yes—
50C24×C4C2313yes—
51C2×#123—212no—
52C2×#123—46no—
53C2×#123—46no—
54C2×#120—46no—
55C22×C4⋊C4C2313yes—
56C22×C4⋊C4C2313yes—
57C22×C22⋊C4C2313yes—
58C22×C22⋊C4C2313yes—
59C22×C22⋊C4C2316yes—
60C22×C22⋊C4C2316yes—
61C22×C22⋊C4C2316yes—
62C22×C22⋊C4C2316yes—
63C22×C22⋊C4C2316yes—
64C22×C22⋊C4C2313yes—
65C22×C22⋊C4C2313yes—
66C22×C22⋊C4C2313yes—
67C22×C22⋊C4C2313yes—
68C22×C42—46no—
69C22×C42C2311yescharacteristic
70C22×C42C2313yes—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73#393—212no—
74#393—412no—
75#393—46no—
76#391—46no—
77#386—412no—
78#381—46no—
79#379—212no—
80#379—46no—
81C2×#95C2316yes—
82C2×#95C2316yes—
83C2×#95C2316yes—
84C2×#95C2313yes—
85C2×#95C2316yes—
86C2×#95C2316yes—
87C2×#95C2313yes—
88C2×#95C2316yes—
89C2×#95C2316yes—
90C2×#95C2316yes—
91C2×#95C2316yes—
92C2×#95C2316yes—
93C2×#95C2316yes—
94C2×#95C2313yes—
95C2×#95C2313yes—
96C2×#95C2316yes—
97C25—26no—
98C25—43no—
99C25—46no—
100C25—43no—
101C25C2413yes—
102C25C2411yescharacteristic
103C22×Q8—26no—
104C22×Q8—46no—
105C22×Q8—43no—
106C22×Q8—46no—
107C22×Q8—43no—
108C22×D4—26no—
109C22×D4—26no—
110C22×D4—212no—
111C22×D4—43no—
112C22×D4—46no—
113C23×C4—26no—
114C23×C4—26no—
115C23×C4—26no—
116C23×C4—212no—
117C23×C4—46no—
118C23×C4C2413yes—
119C23×C4C2416yes—
120C23×C4C2416yes—
121C23×C4C2413yes—
122C23×C4C2413yes—
123C23×C4C2413yes—
124C23×C4C2413yes—
125#123—448no—
126#120—416no—
127C2×C4⋊C4—26no—
128C2×C4⋊C4—26no—
129C2×C4⋊C4—26no—
130C2×C4⋊C4—43no—
131C2×C4⋊C4—43no—
132C2×C4⋊C4—43no—
133C2×C22⋊C4—26no—
134C2×C22⋊C4—26no—
135C2×C22⋊C4—26no—
136C2×C22⋊C4—26no—
137C2×C22⋊C4—26no—
138C2×C22⋊C4—212no—
139C2×C22⋊C4—212no—
140C2×C22⋊C4—212no—
141C2×C22⋊C4—212no—
142C2×C22⋊C4—212no—
143C2×C22⋊C4—43no—
144C2×C22⋊C4—46no—
145C2×C22⋊C4—43no—
146C2×C22⋊C4—43no—
147C2×C22⋊C4—43no—
148C2×C22⋊C4—43no—
149C2×C22⋊C4—46no—
150C2×C22⋊C4—46no—
151C2×C22⋊C4—46no—
152C2×C22⋊C4—46no—
153C2×C22⋊C4—46no—
154C2×C22⋊C4—46no—
155C2×C22⋊C4—46no—
156C2×C22⋊C4—43no—
157C2×C22⋊C4—46no—
158C2×C22⋊C4—46no—
159C2×C22⋊C4—46no—
160C2×C22⋊C4—46no—
161C2×C22⋊C4—46no—
162C2×C22⋊C4#41112yes—
163C2×C42—43no—
164C2×C42—46no—
165C2×C42—43no—
166C2×C42—416no—
167C2×C42—43no—
168C2×C42—46no—
169#95—26no—
170#95—212no—
171#95—212no—
172#95—212no—
173#95—46no—
174#95—46no—
175#95—46no—
176#95—46no—
177#95—43no—
178#95—43no—
179#95—46no—
180#95—46no—
181#95—46no—
182#95—43no—
183#95—46no—
184#95—46no—
185C24—23no—
186C24—23no—
187C24—26no—
188C24—28no—
189C24—26no—
190C24—26no—
191C24—23no—
192C24—23no—
193C24—26no—
194C24—26no—
195C24—26no—
196C24—26no—
197C24—44no—
198C24—412no—
199C24—86no—
200C24—86no—
201C24—812no—
202C24—812no—
203C24—86no—
204C24—812no—
205C24—82no—
206C24—84no—
207C24C2×#4116yes—
208C24H(5,𝔽2)16yes—
209C24C2511yescharacteristic
210C2×Q8—412no—
211C2×Q8—86no—
212C2×Q8—812no—
213C2×Q8—812no—
214C2×Q8—86no—
215C2×D4—412no—
216C2×D4—424no—
217C2×D4—412no—
218C2×D4—86no—
219C2×D4—812no—
220C22×C4—224no—
221C22×C4—23no—
222C22×C4—26no—
223C22×C4—23no—
224C22×C4—26no—
225C22×C4—26no—
226C22×C4—23no—
227C22×C4—23no—
228C22×C4—23no—
229C22×C4—23no—
230C22×C4—26no—
231C22×C4—26no—
232C22×C4—26no—
233C22×C4—26no—
234C22×C4—26no—
235C22×C4—26no—
236C22×C4—23no—
237C22×C4—23no—
238C22×C4—26no—
239C22×C4—26no—
240C22×C4—26no—
241C22×C4—26no—
242C22×C4—23no—
243C22×C4—26no—
244C22×C4—26no—
245C22×C4—23no—
246C22×C4—26no—
247C22×C4—23no—
248C22×C4—23no—
249C22×C4—26no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4—26no—
255C22×C4—26no—
256C22×C4—26no—
257C22×C4—424no—
258C22×C4—424no—
259C22×C4—412no—
260C22×C4—412no—
261C22×C4C2×#4116yes—
262C22×C4C2×#4116yes—
263C22×C4H(5,𝔽2)16yes—
264C4⋊C4—412no—
265C4⋊C4—412no—
266C22⋊C4—412no—
267C22⋊C4—412no—
268C22⋊C4—412no—
269C22⋊C4—412no—
270C22⋊C4—812no—
271C22⋊C4—812no—
272C22⋊C4—812no—
273C22⋊C4—86no—
274C22⋊C4—812no—
275C22⋊C4—812no—
276C22⋊C4—86no—
277C42—44no—
278C42—412no—
279C42—82no—
280C42—84no—
281C23—26no—
282C23—212no—
283C23—26no—
284C23—212no—
285C23—26no—
286C23—26no—
287C23—26no—
288C23—212no—
289C23—212no—
290C23—41no—
291C23—43no—
292C23—46no—
293C23—43no—
294C23—41no—
295C23—42no—
296C23—46no—
297C23—46no—
298C23—46no—
299C23—46no—
300C23—46no—
301C23—46no—
302C23—43no—
303C23—43no—
304C23—46no—
305C23—43no—
306C23—46no—
307C23—46no—
308C23—46no—
309C23—46no—
310C23—43no—
311C23—46no—
312C23—824no—
313C23—824no—
314C23—824no—
315C23—848no—
316C23H(3,𝔽4)14yes—
317C23C22×#4113yes—
318C23C2×H(5,𝔽2)13yes—
319C23C2×H(5,𝔽2)13yes—
320C23#58516yes—
321Q8—824no—
322D4—824no—
323C2×C4—26no—
324C2×C4—26no—
325C2×C4—26no—
326C2×C4—212no—
327C2×C4—212no—
328C2×C4—212no—
329C2×C4—448no—
330C2×C4—46no—
331C2×C4—43no—
332C2×C4—43no—
333C2×C4—43no—
334C2×C4—43no—
335C2×C4—43no—
336C2×C4—46no—
337C2×C4—46no—
338C2×C4—43no—
339C2×C4—43no—
340C2×C4—43no—
341C2×C4—43no—
342C2×C4—46no—
343C2×C4—46no—
344C2×C4—43no—
345C2×C4—46no—
346C2×C4—46no—
347C2×C4—46no—
348C2×C4—46no—
349C2×C4—46no—
350C2×C4—46no—
351C2×C4—46no—
352C2×C4—43no—
353C2×C4—46no—
354C2×C4—46no—
355C2×C4—46no—
356C2×C4—43no—
357C2×C4—46no—
358C2×C4—46no—
359C2×C4—46no—
360C2×C4—46no—
361C2×C4—46no—
362C2×C4—46no—
363C22—212no—
364C22—46no—
365C22—46no—
366C22—412no—
367C22—46no—
368C22—46no—
369C22—412no—
370C22—46no—
371C22—412no—
372C22—832no—
373C22—86no—
374C22—86no—
375C22—86no—
376C22—86no—
377C22#344411yescharacteristic
378C22#349013yes—
379C22#351013yes—
380C22#351316yes—
381C22#351416yes—
382C22C2×H(3,𝔽4)11yescharacteristic
383C22#352913yes—
384C22#353313yes—
385C22#353416yes—
386C22#354911yescharacteristic
387C22#356812yes—
388C4—46no—
389C4—46no—
390C4—46no—
391C4—83no—
392C4—83no—
393C4—86no—
394C4—86no—
395C2—44no—
396C2—86no—
397C2#3540813yes—
398C2#4556813yes—
399C2#4606716yes—
400C2#4628713yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #16152C2semidirect
3. #15668C2semidirect
4. #15636C2semidirect
5. #15468C2semidirect
6. #15467C2semidirect
7. #14706C2semidirect
8. #14649C2semidirect
9. #13827C2semidirect
12. C23×C42C22semidirect
15. C2×#393C22semidirect
16. C2×#391C22semidirect
18. C2×#381C22semidirect
21. #1446C22semidirect
22. #1445C22semidirect
23. #1443C22semidirect
24. #1443C22semidirect
25. #1442C22semidirect
40. #1437C22semidirect
41. #1437C22semidirect
42. #1436C22semidirect
43. #1436C22semidirect
44. #1436C22semidirect
45. #1434C22semidirect
46. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
#161522
#159482
#156682
#156362
#154682
#154672
#147062
#146492
#138272