Finite Group Databasebeta

← #474687

Subgroups — #474687

Order 512.

782 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23554C212yesmaximal
2#21300C211yesmaximal, characteristic
3C2×#1942C211yesmaximal, characteristic
4#11331C212yesmaximal
5#11290C214yesmaximal
6#11270C212yesmaximal
7#11204C212yesmaximal
8#10822C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#428C2214yes—
11C22×#107C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#389C2212yes—
16#2059—24no—
17#2058—22no—
18#2057—22no—
19#1984—22no—
20#1984—22no—
21#1942—22no—
22#1942—22no—
23C2×#343C2214yes—
24C2×#343C2214yes—
25C2×#341C2212yes—
26C2×#341C2212yes—
27#1736C2214yes—
28C2×#340C2212yes—
29#1729C2212yes—
30#1729C2212yes—
31#1728C2212yes—
32C22×M16C2312yes—
33C22×M16C2314yes—
34C23×C8C2311yescharacteristic
35C23×C8C2311yescharacteristic
36#533—42no—
37#533—42no—
38C2×C4⋊Q8D411yescharacteristic
39C2×C4⋊Q8D411yescharacteristic
40C2×#125D412yes—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×#117C2312yes—
46C22×C4⋊C4—22no—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C42C2311yescharacteristic
50#428—44no—
51#428—44no—
52C2×#107—22no—
53C2×#107—24no—
54C2×#107—22no—
55C2×#107—24no—
56C2×#107—42no—
57C2×#106—24no—
58#423—42no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4×C8—42no—
62C2×C4×C8D412yes—
63C2×C4×C8D412yes—
64#389—22no—
65#389—22no—
66#389—22no—
67#389—22no—
68C2×#95—22no—
69#343—28no—
70#343—44no—
71#341—42no—
72#341Q818yescomplemented
73#340—24no—
74#340—24no—
75#340—24no—
76#336—24no—
77#336—24no—
78#336—22no—
79#336—22no—
80C22×Q8—21no—
81C22×Q8—21no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4#4111yescharacteristic
85C23×C4#4111yescharacteristic
86C23×C4C2411yescharacteristic
87C2×M16—24no—
88C2×M16—24no—
89C2×M16—24no—
90C2×M16—24no—
91C2×M16—24no—
92C2×M16—24no—
93C2×M16—24no—
94C2×M16—24no—
95C2×M16C2×Q818yes—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—24no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8C2×D412yes—
104C22×C8C2×D412yes—
105C22×C8C2×Q814yes—
106C4⋊Q8—42no—
107C4⋊Q8—42no—
108#125—44no—
109C22⋊Q8—44no—
110C22⋊Q8—44no—
111C4×Q8—42no—
112C4×Q8—42no—
113C4×Q8—42no—
114C4×Q8—42no—
115#117—24no—
116#117—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—84no—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4#4114yes—
136C2×C22⋊C4—22no—
137C2×C42—22no—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144#107—28no—
145#107—48no—
146#107—44no—
147#106—48no—
148C4⋊C8—42no—
149C4⋊C8—42no—
150C4×C8—42no—
151C4×C8—42no—
152#95—42no—
153#95—42no—
154C24C2×#4111yescharacteristic
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—42no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4—82no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4#12111yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4C22×D411yescharacteristic
183C22×C4C22×D411yescharacteristic
184C22×C4C22×Q812yes—
185C22×C4C2×#4112yes—
186C22×C4H(5,𝔽2)12yes—
187C22×C4H(5,𝔽2)12yes—
188C22×C42−1+412yes—
189C22×C42−1+412yes—
190M16—28no—
191M16—28no—
192M16—44no—
193M16—44no—
194M16—48no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—28no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—28no—
202C2×C8—44no—
203C2×C8—44no—
204C2×C8—44no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8C4⋊Q814yes—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—84no—
223C4⋊C4—84no—
224C22⋊C4—44no—
225C42—21no—
226C42—21no—
227C42—22no—
228C42—22no—
229C42—21no—
230C42—21no—
231C42—42no—
232C42—42no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23C2×#12111yescharacteristic
238C23#53812yes—
239C23#55212yes—
240C23#55712yes—
241Q8—84no—
242Q8—84no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—84no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—84no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—164no—
270C2×C4#47212yes—
271C2×C4#47312yes—
272C2×C4#52511yescharacteristic
273C2×C4C2×C4⋊Q811yescharacteristic
274C2×C4#53411yescharacteristic
275C2×C4#53412yes—
276C2×C4#53511yescharacteristic
277C2×C4#53611yescharacteristic
278C2×C4#53611yescharacteristic
279C2×C4#54214yes—
280C2×C4#54414yes—
281C2×C4#54812yes—
282C2×C4D4×Q812yes—
283C2×C4#55414yes—
284C2×C4#56314yes—
285C8—24no—
286C8—42no—
287C8—48no—
288C8—82no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22#253611yescharacteristic
296C22#254611yescharacteristic
297C22#261711yescharacteristic
298C22#277414yes—
299C22C2×#47211yescharacteristic
300C22C2×#47311yescharacteristic
301C22#306711yescharacteristic
302C22#306711yescharacteristic
303C4—22no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—82no—
308C4—82no—
309C4—84no—
310C4—162no—
311C4#258311yescharacteristic
312C4#263211yescharacteristic
313C4#266011yescharacteristic
314C4#272914yes—
315C4#281711yescharacteristic
316C2—24no—
317C2#1699311yescharacteristic
318C2#2821711yescharacteristic
319C2#2821911yescharacteristic
320C2#2869911yescharacteristic
321C2#2870211yescharacteristic
322C2#2974811yescharacteristic
323C2#2974911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
72. #341Q8semidirect

Maximal subgroups

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

subgroupindex
#235542
#213002
C2×#19422
#113312
#112902
#112702
#112042
#108222