Finite Group Databasebeta

← #207012

Subgroups — #207012

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#20536C211yesmaximal, characteristic
3#13955C211yesmaximal, characteristic
4#9889C214yesmaximal
5#9888C214yesmaximal
6#8026C212yesmaximal
7#8014C212yesmaximal
8C2×#544—21no—
9C2×#431C2212yes—
10#2857C2214yes—
11#2659—22no—
12#2552C414yescomplemented
13#2551C414yescomplemented
14#2550—24no—
15#2316—21no—
16#2285C2212yes—
17C2×#382C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1770—24no—
21#1749C2214yes—
22#1442—22no—
23C4⋊#95—21no—
24C4⋊#95—21no—
25#1434C2212yes—
26#1434C2212yes—
27#1294C2214yes—
28#1294C2214yes—
29#1294C2218yes—
30#544—44no—
31#544—44no—
32C2×C4⋊Q8—24no—
33C2×C4⋊Q8C2×C414yes—
34C2×#125—24no—
35C2×C22⋊Q8—24no—
36C2×#117—22no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C42—21no—
42C22×C42C2312yes—
43C22×C42C2311yescharacteristic
44#431—24no—
45#431Q818yescomplemented
46C2×C4⋊C8—22no—
47C2×C22⋊C8C2314yes—
48C2×#97—24no—
49C2×C4×C8—22no—
50#398—28no—
51#395—28no—
52#391—24no—
53#389—24no—
54#389—24no—
55#389—24no—
56#389C2×C418yes—
57#384C2×C414yes—
58#384C2×C412yes—
59#384C2×C412yes—
60#382—22no—
61#382—24no—
62#380—24no—
63#380—22no—
64#380C2314yes—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—24no—
67C4×C4⋊C4C2×C414yes—
68C4×C4⋊C4C2×C414yes—
69C4×C4⋊C4D414yes—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—21no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95D412yes—
76C2×#95C2311yescharacteristic
77C2×#95C2311yescharacteristic
78C2×#95C2312yes—
79#336—24no—
80C22×Q8—24no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—22no—
84C23×C4C2×D412yes—
85C23×C4C2×D412yes—
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2411yescharacteristic
89C22×C8—24no—
90C4⋊Q8—416no—
91C4⋊Q8—84no—
92#125—416no—
93C22⋊Q8—48no—
94C22⋊Q8—48no—
95#117—44no—
96#117—44no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4C22⋊C414yes—
116C2×C4⋊C4C22⋊C414yes—
117C2×C4⋊C4C22×C414yes—
118C2×C4⋊C4C22×C414yes—
119C2×C22⋊C4—24no—
120C2×C42—24no—
121C2×C42—42no—
122C2×C42—42no—
123C2×C42—42no—
124C2×C42—82no—
125C2×C42C22×C414yes—
126C2×C42C22×C412yes—
127C2×C42C2×D414yes—
128C2×C42C2×D412yes—
129C2×C42C2×Q814yes—
130C2×C42#4114yes—
131C4⋊C8—28no—
132C22⋊C8—24no—
133C22⋊C8C2×Q818yes—
134#97—84no—
135C4×C8—28no—
136#95—24no—
137#95—28no—
138#95—24no—
139#95—22no—
140#95—22no—
141#95—24no—
142#95—44no—
143#95—44no—
144#95—42no—
145#95—82no—
146#95—82no—
147#95C22⋊C418yes—
148C24C22×D411yescharacteristic
149C2×Q8—48no—
150C2×Q8—84no—
151C2×Q8—84no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4C2×C22⋊C414yes—
188C22×C4C2×C22⋊C414yes—
189C22×C4C2×C22⋊C412yes—
190C22×C4C2×C22⋊C412yes—
191C22×C4#12412yes—
192C22×C4#12412yes—
193C22×C4C4⋊Q812yes—
194C22×C4C4⋊Q812yes—
195C22×C4AGL(1,ℤ/8)12yes—
196C22×C4AGL(1,ℤ/8)12yes—
197C22×C4#13712yes—
198C22×C4#13712yes—
199C22×C4C23×C411yescharacteristic
200C22×C4C22×D411yescharacteristic
201C22×C4C22×Q812yes—
202C22×C4C2×#4112yes—
203C2×C8—28no—
204C2×C8—44no—
205C2×C8—44no—
206C4⋊C4—28no—
207C4⋊C4—28no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—48no—
211C4⋊C4—416no—
212C4⋊C4—48no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—82no—
216C4⋊C4—82no—
217C4⋊C4—84no—
218C4⋊C4—84no—
219C4⋊C4—84no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C42—44no—
223C42—44no—
224C42—48no—
225C42—84no—
226C42C4×Q818yes—
227C42C22⋊Q814yes—
228C42C22⋊Q814yes—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23C22×C22⋊C411yescharacteristic
234C23C2×#12411yescharacteristic
235C23C2×C4⋊Q811yescharacteristic
236C23C2×AGL(1,ℤ/8)11yescharacteristic
237C23C2×AGL(1,ℤ/8)11yescharacteristic
238C23C2×#13711yescharacteristic
239C23C2×#13711yescharacteristic
240Q8—168no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4#39114yes—
289C2×C4#39114yes—
290C2×C4#41814yes—
291C2×C4#41914yes—
292C2×C4#48914yes—
293C2×C4#48914yes—
294C2×C4#50114yes—
295C2×C4#50114yes—
296C2×C4C2×C4×Q812yes—
297C2×C4C2×C22⋊Q812yes—
298C8—48no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22C2×#39111yescharacteristic
312C22#288311yescharacteristic
313C22#288411yescharacteristic
314C22C2×#41811yescharacteristic
315C22C2×#41911yescharacteristic
316C22#290111yescharacteristic
317C22#290111yescharacteristic
318C22C2×#48911yescharacteristic
319C22C2×#48911yescharacteristic
320C22C2×#50111yescharacteristic
321C22C2×#50111yescharacteristic
322C4—24no—
323C4—22no—
324C4—22no—
325C4—44no—
326C4—44no—
327C4—82no—
328C4—82no—
329C4—84no—
330C4—84no—
331C4—162no—
332C4#199318yes—
333C2—21no—
334C2—21no—
335C2—21no—
336C2—21no—
337C2#1762611yescharacteristic
338C2#1762711yescharacteristic
339C2#1763111yescharacteristic
340C2#1763211yescharacteristic
341C2#2130711yescharacteristic
342C2C2×#199311yescharacteristic
343C2#2133411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2552C4semidirect
13. #2551C4semidirect
45. #431Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#205362
#139552
#98892
#98882
#80262
#80142