Finite Group Databasebeta

← #353144

Subgroups — #353144

Order 512.

2466 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#405C212yesmaximal
2C22×#405C211yesmaximal, characteristic
3C2×#1738C211yesmaximal, characteristic
4#20347C211yesmaximal, characteristic
5C2×#1733C212yesmaximal
6#10707C214yesmaximal, complemented
7#10706C214yesmaximal, complemented
8C23×M16C2212yes—
9C23×M16C2211yescharacteristic
10C24×C8C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#117C2212yes—
13C2×#405—22no—
14C2×#405—28no—
15C2×#405—28no—
16C2×#405—22no—
17C2×#405C4116yescomplemented
18C2×#405C418yescomplemented
19C2×#405C2214yes—
20C2×#405C2214yes—
21C22×#97—22no—
22C22×#97C2212yes—
23C22×C4×C8—22no—
24C22×C4×C8C2212yes—
25#1738C418yescomplemented
26C2×#341C2214yes—
27#1736C418yescomplemented
28#1735C2214yes—
29#1733C4116yescomplemented
30C2×#339C2218yes—
31C24×C4C2311yescharacteristic
32C22×M16—24no—
33C22×M16—22no—
34C22×M16—24no—
35C22×M16—24no—
36C22×M16—24no—
37C22×M16—24no—
38C22×M16C2×C418yes—
39C22×M16C2×C418yes—
40C22×M16C2×C418yes—
41C22×M16C2×C414yes—
42C22×M16C2×C414yes—
43C22×M16C2×C414yes—
44C22×M16C2314yes—
45C23×C8—22no—
46C23×C8C2×C418yes—
47C23×C8C2×C414yes—
48C23×C8D412yes—
49C23×C8D418yes—
50C23×C8D412yes—
51C23×C8Q814yes—
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—22no—
58C2×#117—24no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117C2×C418yes—
62C2×#117C2×C418yes—
63C2×#117C2×C414yes—
64C2×#117C2×C414yes—
65C2×#117C2×C418yes—
66C2×#117C2314yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C414yes—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C42—22no—
72C22×C42C2312yes—
73#405—216no—
74#405—216no—
75#405—232no—
76#405—28no—
77#405—28no—
78#405C2×C4116yescomplemented
79#405C2×C4116yescomplemented
80C2×#97—216no—
81C2×#97—24no—
82C2×#97—24no—
83C2×#97—28no—
84C2×#97C2×C418yescomplemented
85C2×C4×C8—216no—
86C2×C4×C8—24no—
87C2×C4×C8—28no—
88C2×C4×C8—24no—
89C2×C4×C8C2×C418yescomplemented
90#341C2×C4116yescomplemented
91#340C2×C4116yes—
92#339C2×C4132yescomplemented
93C25#4111yescharacteristic
94C23×C4—24no—
95C23×C4—22no—
96C23×C4C22×C414yes—
97C23×C4C22×C412yes—
98C23×C4C22×C412yes—
99C23×C4C22×C412yes—
100C23×C4C22×C414yes—
101C23×C4C2×D414yes—
102C23×C4C2×D412yes—
103C23×C4C2×Q812yes—
104C23×C4#4111yescharacteristic
105C23×C4#4112yes—
106C23×C4#4114yes—
107C23×C4C2411yescharacteristic
108C2×M16—28no—
109C2×M16—28no—
110C2×M16—28no—
111C2×M16—216no—
112C2×M16—28no—
113C2×M16—28no—
114C2×M16—216no—
115C2×M16—28no—
116C2×M16—28no—
117C2×M16—216no—
118C2×M16—216no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16—24no—
122C2×M16C42132yes—
123C2×M16C42116yes—
124C2×M16C22×C418yes—
125C2×M16C22×C418yes—
126C2×M16C22×C418yes—
127C22×C8—28no—
128C22×C8—28no—
129C22×C8—28no—
130C22×C8—24no—
131C22×C8—24no—
132C22×C8—24no—
133C22×C8C42116yes—
134C22×C8C22⋊C418yes—
135C22×C8C22⋊C418yes—
136C22×C8C22⋊C418yes—
137C22×C8C22⋊C4116yes—
138C22×C8C22⋊C418yes—
139C22×C8C4⋊C418yes—
140C22×C8C4⋊C4132yes—
141C22×C8C4⋊C418yes—
142C22×C8C22×C414yes—
143C22×C8C22×C414yes—
144C22×C8C22×C414yes—
145C22×C8C22×C412yes—
146C22×C8C22×C412yes—
147C22×C8C22×C414yes—
148C22×C8C22×C414yes—
149C22×C8C2×D412yes—
150C22×C8C2×D414yes—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D414yes—
153C22×C8C2×D411yescharacteristic
154C22×C8C2×Q812yes—
155C22×C8C2×Q812yes—
156#117—28no—
157#117—28no—
158#117—28no—
159#117—28no—
160#117—216no—
161#117—28no—
162#117—28no—
163#117—24no—
164#117—24no—
165#117C42116yes—
166#117C42116yes—
167#117C22×C418yes—
168#117C22×C418yes—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4C42116yes—
174C2×C4⋊C4C22×C418yes—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—24no—
181C2×C42—28no—
182C2×C42—28no—
183C2×C42—24no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—22no—
187C2×C42C22×C418yes—
188C2×C42C22×C414yes—
189#97—216no—
190#97—216no—
191#97—216no—
192C4×C8—216no—
193C4×C8—216no—
194C4×C8—216no—
195C24—44no—
196C24#11714yes—
197C24#11712yes—
198C24C4×D412yes—
199C24C4×D414yes—
200C24C4×Q812yes—
201C24C2×#4111yescharacteristic
202C22×C4—24no—
203C22×C4—28no—
204C22×C4—28no—
205C22×C4—216no—
206C22×C4—48no—
207C22×C4—416no—
208C22×C4C2×C4214yes—
209C22×C4C2×C4218yes—
210C22×C4C2×C4214yes—
211C22×C4C2×C4214yes—
212C22×C4C2×C4214yes—
213C22×C4C2×C4214yes—
214C22×C4C2×C22⋊C414yes—
215C22×C4C2×C22⋊C414yes—
216C22×C4C2×C22⋊C414yes—
217C22×C4C2×C4⋊C414yes—
218C22×C4C2×C4⋊C418yes—
219C22×C4#11714yes—
220C22×C4#11718yes—
221C22×C4#11712yes—
222C22×C4#11714yes—
223C22×C4C4×D414yes—
224C22×C4C4×D418yes—
225C22×C4C4×D414yes—
226C22×C4C4×D414yes—
227C22×C4C4×D414yes—
228C22×C4C4×D418yes—
229C22×C4C4×D414yes—
230C22×C4C4×D412yes—
231C22×C4C4×D414yes—
232C22×C4C4×Q814yes—
233C22×C4C4×Q818yes—
234C22×C4C4×Q812yes—
235C22×C4C23×C411yescharacteristic
236C22×C4C23×C412yes—
237C22×C4C23×C411yescharacteristic
238C22×C4C23×C412yes—
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D412yes—
242C22×C4C22×Q811yescharacteristic
243C22×C4C2×#4111yescharacteristic
244C22×C4C2×#4112yes—
245M16—28no—
246M16—28no—
247M16—28no—
248M16—232no—
249M16—232no—
250M16C2×C42116yes—
251C2×C8—28no—
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—28no—
255C2×C8—28no—
256C2×C8—28no—
257C2×C8—28no—
258C2×C8—232no—
259C2×C8#95132yes—
260C2×C8#95132yes—
261C2×C8C2×C4218yes—
262C2×C8C2×C4214yes—
263C2×C8C2×C4214yes—
264C2×C8C2×C22⋊C414yes—
265C2×C8C2×C22⋊C414yes—
266C2×C8C2×C22⋊C414yes—
267C2×C8C2×C22⋊C414yes—
268C2×C8C2×C22⋊C414yes—
269C2×C8C2×C22⋊C414yes—
270C2×C8C2×C4⋊C412yes—
271C2×C8C2×C4⋊C414yes—
272C2×C8C2×C4⋊C418yes—
273C2×C8C2×C4⋊C418yes—
274C2×C8C2×C4⋊C412yes—
275C4⋊C4—28no—
276C4⋊C4—216no—
277C4⋊C4—216no—
278C4⋊C4C2×C42116yes—
279C22⋊C4—28no—
280C22⋊C4—28no—
281C22⋊C4—216no—
282C22⋊C4—216no—
283C42—28no—
284C42—24no—
285C42—28no—
286C42—28no—
287C42—216no—
288C42C2×C4218yes—
289C23—28no—
290C23—24no—
291C23—48no—
292C23—44no—
293C23—48no—
294C23—44no—
295C23#37614yes—
296C23C4×C22⋊C414yes—
297C23C4×C22⋊C414yes—
298C23C4×C22⋊C414yes—
299C23C4×C4⋊C414yes—
300C23C4×C4⋊C418yes—
301C23C2×#11712yes—
302C23C2×#11711yescharacteristic
303C23C2×C4×D412yes—
304C23C2×C4×D411yescharacteristic
305C23C2×C4×Q811yescharacteristic
306C2×C4—216no—
307C2×C4—28no—
308C2×C4—416no—
309C2×C4—48no—
310C2×C4—432no—
311C2×C4—416no—
312C2×C4C4318yes—
313C2×C4C2×#9518yes—
314C2×C4#37614yes—
315C2×C4C4×C22⋊C414yes—
316C2×C4C4×C22⋊C414yes—
317C2×C4C4×C22⋊C414yes—
318C2×C4C4×C22⋊C418yes—
319C2×C4C4×C22⋊C418yes—
320C2×C4C4×C22⋊C418yes—
321C2×C4C4×C4⋊C418yes—
322C2×C4C4×C4⋊C414yes—
323C2×C4C4×C4⋊C418yes—
324C2×C4C4×C4⋊C418yes—
325C2×C4C4×C4⋊C418yes—
326C2×C4C22×C4211yescharacteristic
327C2×C4C22×C4211yescharacteristic
328C2×C4C22×C4212yes—
329C2×C4C22×C4211yescharacteristic
330C2×C4C22×C4212yes—
331C2×C4C22×C22⋊C411yescharacteristic
332C2×C4C22×C22⋊C412yes—
333C2×C4C22×C4⋊C412yes—
334C2×C4C22×C4⋊C411yescharacteristic
335C2×C4C2×#11711yescharacteristic
336C2×C4C2×#11712yes—
337C2×C4C2×C4×D411yescharacteristic
338C2×C4C2×C4×D412yes—
339C2×C4C2×C4×D412yes—
340C2×C4C2×C4×D412yes—
341C2×C4C2×C4×D412yes—
342C2×C4C2×C4×Q812yes—
343C2×C4C2×C4×Q811yescharacteristic
344C8—216no—
345C8—28no—
346C8C2×#9514yes—
347C8C2×#9518yes—
348C8C2×#9514yes—
349C22—24no—
350C22—28no—
351C22—24no—
352C22—28no—
353C22—22no—
354C22—22no—
355C22—28no—
356C22—44no—
357C22—44no—
358C22—48no—
359C22C4×#9518yes—
360C22C2×#37611yescharacteristic
361C22C2×C4×C22⋊C411yescharacteristic
362C22C2×C4×C22⋊C412yes—
363C22C2×C4×C4⋊C412yes—
364C22C2×C4×C4⋊C411yescharacteristic
365C22#295514yes—
366C4—48no—
367C4—416no—
368C4C4×#9518yes—
369C4C2×C4311yescharacteristic
370C4C22×#9511yescharacteristic
371C4C2×C4×C22⋊C412yes—
372C4C2×C4×C22⋊C411yescharacteristic
373C4C2×C4×C4⋊C411yescharacteristic
374C4C2×C4×C4⋊C412yes—
375C2—21no—
376C2—28no—
377C2—21no—
378C2—22no—
379C2C2×C4×#9511yescharacteristic
380C2#2042014yes—
381C2#2044412yes—
subgroupG/H
centerC22×C8C2×D4
commutatorC4C2×C43
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C8 ◁ C24×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #10707C2semidirect
7. #10706C2semidirect
17. C2×#405C4semidirect
18. C2×#405C4semidirect
25. #1738C4semidirect
27. #1736C4semidirect
29. #1733C4semidirect
78. #405C2×C4semidirect
79. #405C2×C4semidirect
84. C2×#97C2×C4semidirect
89. C2×C4×C8C2×C4semidirect
90. #341C2×C4semidirect
92. #339C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4052
C2×#17382
#203472
C2×#17332
#107072
#107062