Finite Group Databasebeta

← #391607

Subgroups — #391607

Order 512.

686 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23732C211yesmaximal, characteristic
2#11300C212yesmaximal
3#11214C212yesmaximal
4#11212C212yesmaximal
5#11184C212yesmaximal
6#11162C211yesmaximal, characteristic
7#11059C212yesmaximal
8#10957C211yesmaximal, characteristic
9#10903C212yesmaximal
10C2×#428C2212yes—
11#2905C2212yes—
12C2×#419C2212yes—
13C2×#419C2212yes—
14C22×#101C2211yescharacteristic
15#2310C2211yescharacteristic
16C2×#391C2211yescharacteristic
17#2285C2211yescharacteristic
18#2274C2211yescharacteristic
19#2097C414yescomplemented
20#2096C412yescomplemented
21#2095C412yescomplemented
22C2×#344C2212yes—
23C2×#344C2212yes—
24C2×#343C2211yescharacteristic
25#1740C2212yes—
26#1736C2212yes—
27#1733C2212yes—
28#1733C2212yes—
29C2×#339C2212yes—
30C2×#339C2212yes—
31C2×#338C2212yes—
32C2×H(3,ℤ/4)C2212yes—
33#1728C2211yescharacteristic
34C23×Q8C2311yescharacteristic
35C22×M16C2312yes—
36C22×M16C2312yes—
37C22×M16C2312yes—
38C2×#117C2312yes—
39C2×#117C2312yes—
40C2×#117C2312yes—
41C22×C4⋊C4C2311yescharacteristic
42C22×C42C2311yescharacteristic
43#428—42no—
44#428—42no—
45C2×#107C2×C414yes—
46C2×#106C2×C414yes—
47#424—42no—
48#424—42no—
49#423—42no—
50#419—24no—
51#419—42no—
52#419—42no—
53#419—42no—
54C2×#103—42no—
55C2×#103C2×C414yes—
56C2×#103C2×C414yes—
57C2×#101—42no—
58C2×#101—41no—
59#391—42no—
60#391—41no—
61#391C2×C412yes—
62#391C2×C412yes—
63#384C2×C411yescharacteristic
64#384C2×C411yescharacteristic
65#382C2×C412yes—
66#381—41no—
67#380—41no—
68C4×C4⋊C4—41no—
69C4×C4⋊C4—42no—
70C4×C22⋊C4—41no—
71C2×#95—41no—
72#344—24no—
73#344—42no—
74#343—41no—
75#342—42no—
76#340—42no—
77#339—42no—
78#339—42no—
79#339—42no—
80#339—42no—
81#338—42no—
82H(3,ℤ/4)—24no—
83#336C2×C412yes—
84#336C2×C412yes—
85C22×Q8—21no—
86C22×Q8—22no—
87C22×Q8—22no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—22no—
92C22×Q8C22×C412yes—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4C2411yescharacteristic
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8C22×C414yes—
118#117—22no—
119#117—22no—
120#117—22no—
121#117—22no—
122#117—22no—
123#117—22no—
124#117—22no—
125#117—22no—
126#117—22no—
127#117—22no—
128#117—22no—
129#117#4114yes—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4C22×C412yes—
138C2×C4⋊C4C22×C412yes—
139C2×C4⋊C4C22×C412yes—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—21no—
148C2×C42—22no—
149C2×C42—42no—
150C2×C42C22×C411yescharacteristic
151C2×C42C22×C411yescharacteristic
152#107—44no—
153#106—44no—
154C4⋊C8—48no—
155#103—44no—
156#103—44no—
157#103—48no—
158#101—44no—
159#95—42no—
160#95—42no—
161#95—42no—
162#95—42no—
163#95—44no—
164C24H(5,𝔽2)11yescharacteristic
165C2×Q8—22no—
166C2×Q8—24no—
167C2×Q8—22no—
168C2×Q8—24no—
169C2×Q8—42no—
170C2×Q8—41no—
171C2×Q8—42no—
172C2×Q8—41no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—81no—
198C22×C4—82no—
199C22×C4C23×C411yescharacteristic
200C22×C4C2×#4112yes—
201C22×C4H(5,𝔽2)12yes—
202C22×C4H(5,𝔽2)12yes—
203C22×C4H(5,𝔽2)12yes—
204C22×C4H(5,𝔽2)11yescharacteristic
205C22×C42−1+412yes—
206C22×C42−1+412yes—
207M16—42no—
208M16—42no—
209M16—42no—
210M16—42no—
211M16—42no—
212M16—42no—
213M16—42no—
214M16—42no—
215M16—42no—
216M16—42no—
217M16—42no—
218M16—42no—
219C2×C8—24no—
220C2×C8—24no—
221C2×C8—24no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—24no—
232C4⋊C4—22no—
233C4⋊C4—24no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—82no—
239C4⋊C4—82no—
240C22⋊C4—24no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C42—22no—
247C42—22no—
248C42—24no—
249C42—42no—
250C42—41no—
251C42—42no—
252C42—42no—
253C42—41no—
254C42—42no—
255C42—42no—
256C42—162no—
257C23—41no—
258C23—41no—
259C23#51811yescharacteristic
260C23#53912yes—
261C23#56011yescharacteristic
262C23#56011yescharacteristic
263C23#56212yes—
264Q8—44no—
265Q8—44no—
266Q8—82no—
267Q8—82no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—84no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—161no—
294C2×C4C4×#4111yescharacteristic
295C2×C4#51812yes—
296C2×C4#51911yescharacteristic
297C2×C4#52012yes—
298C2×C4#53812yes—
299C2×C4#53912yes—
300C2×C4#54012yes—
301C2×C4#54212yes—
302C2×C4#54212yes—
303C2×C4#56011yescharacteristic
304C2×C4#56012yes—
305C2×C4#56212yes—
306C2×C4#56212yes—
307C2×C4#56311yescharacteristic
308C2×C4#56312yes—
309C2×C4#56312yes—
310C8—44no—
311C8—82no—
312C8—82no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—42no—
321C22#233211yescharacteristic
322C22#237611yescharacteristic
323C22#238011yescharacteristic
324C22#282712yes—
325C22#283712yes—
326C4—42no—
327C4—81no—
328C4—84no—
329C4—82no—
330C4—82no—
331C4—82no—
332C4—82no—
333C4—162no—
334C4#233512yes—
335C4#237812yes—
336C4#282912yes—
337C4#283812yes—
338C2—21no—
339C2—22no—
340C2—22no—
341C2—21no—
342C2#1464611yescharacteristic
343C2#2336011yescharacteristic
344C2#2336211yescharacteristic
subgroupG/H
center, socleC22#2380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #2097C4semidirect
20. #2096C4semidirect
21. #2095C4semidirect

Maximal subgroups

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

subgroupindex
#237322
#113002
#112142
#112122
#111842
#111622
#110592
#109572
#109032