Finite Group Databasebeta

← #194899

Subgroups — #194899

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#21804C212yesmaximal, complemented
4C2×#1460C211yesmaximal, characteristic
5#9016C218yesmaximal, complemented
6#8014C212yesmaximal, complemented
7C22×C4⋊Q8C2211yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C2×#431C2212yes—
10#2563—24no—
11#2563C418yescomplemented
12C2×#393C2214yes—
13#2357—24no—
14#2354—24no—
15#2312C2214yes—
16#2309—21no—
17C2×#390—21no—
18#2285C2212yes—
19C2×#382—21no—
20#2280—22no—
21C2×#380—22no—
22#2274—21no—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1981—24no—
25#1956—24no—
26#1787C2218yes—
27#1460—24no—
28C2×#323C2214yes—
29#1294C2218yes—
30C23×Q8C2312yes—
31C23×D4D411yescharacteristic
32C24×C4D412yes—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—24no—
37C2×C4⋊Q8C2×C414yes—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—24no—
41C2×C4×D4C2×C414yes—
42C2×C4×D4D414yes—
43C2×#117—22no—
44C2×#117—24no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4D411yescharacteristic
49C22×C22⋊C4—24no—
50C22×C22⋊C4C2312yes—
51C22×C42—21no—
52C22×C42C2311yescharacteristic
53#431—24no—
54#431—24no—
55C2×C4⋊C8—22no—
56C2×#103—216no—
57C2×C22⋊C8C2314yes—
58C2×C4×C8—22no—
59#393—28no—
60#393C2×C4116yes—
61#390—28no—
62#390—28no—
63#388—216no—
64#384D414yes—
65#382—28no—
66#382—22no—
67#381—24no—
68#380—24no—
69#380—216no—
70#380—22no—
71#380C2314yes—
72#379—24no—
73C4×C4⋊C4—28no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4C2×C414yes—
77C4×C4⋊C4D414yes—
78C4×C22⋊C4—24no—
79C4×C22⋊C4—24no—
80C2×#95—24no—
81C2×#95C2312yes—
82#323—28no—
83C25C2×D412yes—
84C22×Q8—22no—
85C22×Q8—24no—
86C22×Q8—24no—
87C22×Q8—84no—
88C22×Q8C22×C414yes—
89C22×D4—22no—
90C22×D4—22no—
91C22×D4—24no—
92C22×D4—24no—
93C22×D4C22⋊C414yes—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—24no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—48no—
100C23×C4C22⋊C418yes—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2411yescharacteristic
105C22×C8—24no—
106C4⋊Q8—28no—
107C4⋊Q8—28no—
108C4⋊Q8—44no—
109C4⋊Q8—44no—
110C4×D4—28no—
111C4×D4—28no—
112C4×D4—28no—
113C4×D4—28no—
114#117—44no—
115#117—44no—
116#117—88no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C22⋊C414yes—
131C2×C4⋊C4C2×D418yes—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—216no—
138C2×C22⋊C4—88no—
139C2×C22⋊C4C22×C414yes—
140C2×C22⋊C4C2×D414yes—
141C2×C42—24no—
142C2×C42—24no—
143C2×C42—28no—
144C2×C42—44no—
145C2×C42C22×C412yes—
146C2×C42C2×D412yes—
147C2×C42C2×D412yes—
148C2×C42C2×D412yes—
149C2×C42#4114yes—
150C4⋊C8—44no—
151#103—816no—
152C22⋊C8—24no—
153C22⋊C8—24no—
154C4×C8—44no—
155#95—28no—
156#95—216no—
157#95—24no—
158#95—22no—
159#95—24no—
160#95C22×C414yes—
161C24—22no—
162C24—24no—
163C24—22no—
164C24—24no—
165C24—44no—
166C24—44no—
167C24C2×C22⋊C414yes—
168C24C22×D411yescharacteristic
169C2×Q8—28no—
170C2×Q8—416no—
171C2×Q8—44no—
172C2×Q8—84no—
173C2×Q8—84no—
174C2×Q8—88no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—28no—
178C2×D4—44no—
179C2×D4—44no—
180C2×D4—48no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—28no—
185C22×C4—28no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—28no—
189C22×C4—42no—
190C22×C4—416no—
191C22×C4—48no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—48no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—88no—
200C22×C4—84no—
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C22⋊C414yes—
203C22×C4C2×C22⋊C412yes—
204C22×C4#12011yescharacteristic
205C22×C4#12012yes—
206C22×C4#12011yescharacteristic
207C22×C4#12312yes—
208C22×C4#12312yes—
209C22×C4#13514yes—
210C22×C4#13714yes—
211C22×C4C23×C411yescharacteristic
212C22×C4C22×D412yes—
213C22×C4C22×D411yescharacteristic
214C22×C4C2×#4112yes—
215C2×C8—48no—
216C2×C8—44no—
217C4⋊C4—28no—
218C4⋊C4—48no—
219C4⋊C4—48no—
220C4⋊C4—44no—
221C4⋊C4—48no—
222C4⋊C4—44no—
223C4⋊C4—48no—
224C4⋊C4—48no—
225C22⋊C4—28no—
226C22⋊C4—28no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C22⋊C4—88no—
230C22⋊C4—816no—
231C22⋊C4—88no—
232C42—48no—
233C42—48no—
234C42—88no—
235C42C4×D414yes—
236C42C4×D414yes—
237C42#12114yes—
238C42#12114yes—
239C23—22no—
240C23—22no—
241C23—24no—
242C23—24no—
243C23—24no—
244C23—44no—
245C23—44no—
246C23—416no—
247C23—44no—
248C23—44no—
249C23—44no—
250C23—44no—
251C23—82no—
252C23—82no—
253C23—88no—
254C23C22×C22⋊C411yescharacteristic
255C23C2×#12011yescharacteristic
256C23C2×#12311yescharacteristic
257C23C2×#13512yes—
258C23C2×#13712yes—
259Q8—416no—
260Q8—88no—
261D4—416no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—416no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—48no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—48no—
278C2×C4—48no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—88no—
282C2×C4—82no—
283C2×C4—88no—
284C2×C4—84no—
285C2×C4—84no—
286C2×C4#38612yes—
287C2×C4#38614yes—
288C2×C4#38612yes—
289C2×C4#41614yes—
290C2×C4#41914yes—
291C2×C4#45214yes—
292C2×C4#45214yes—
293C2×C4#45514yes—
294C2×C4#48518yes—
295C2×C4C2×C4×D412yes—
296C2×C4C2×#12112yes—
297C8—84no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—24no—
303C22—24no—
304C22—48no—
305C22—44no—
306C22—44no—
307C22—44no—
308C22—84no—
309C22—88no—
310C22—88no—
311C22C2×#38611yescharacteristic
312C22C2×#41611yescharacteristic
313C22C2×#41911yescharacteristic
314C22C2×#45212yes—
315C22C2×#45512yes—
316C22#302712yes—
317C22C2×#48512yes—
318C4—22no—
319C4—22no—
320C4—22no—
321C4—44no—
322C4—84no—
323C4—84no—
324C4—88no—
325C4#187318yes—
326C4#188514yes—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—84no—
331C2#1840614yes—
332C2C2×#187311yescharacteristic
333C2C2×#188511yescharacteristic
334C2#2096111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21804C2semidirect
5. #9016C2semidirect
6. #8014C2semidirect
11. #2563C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
#341862
#218042
C2×#14602
#90162
#80142