Finite Group Databasebeta

← #329849

Subgroups — #329849

Order 512.

598 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic
2#29922C212yesmaximal
3#21143C211yesmaximal, characteristic
4#10652C212yesmaximal, complemented
5#10602C214yesmaximal, complemented
6#10545C212yesmaximal, complemented
7#9137C212yesmaximal
8#8596C211yesmaximal, characteristic
9C2×#500C2211yescharacteristic
10C2×#424—21no—
11C2×#423C2211yescharacteristic
12C22×C4×C8C2211yescharacteristic
13#2857C2212yes—
14#2285C2211yescharacteristic
15#2096—22no—
16#2072—22no—
17#2035—24no—
18#1990C2212yes—
19#1981C2214yes—
20#1948C2214yes—
21#1942—21no—
22#1942—21no—
23#1939—21no—
24#1939—21no—
25#1939C2212yes—
26#1939C2212yes—
27#1845C2211yescharacteristic
28#1808—21no—
29#1808—21no—
30#1797C2212yes—
31#1729—21no—
32C2×#335C2212yes—
33C2×#332C2214yes—
34#1296C2214yes—
35#1277C2212yes—
36C22×M16—21no—
37C23×C8D411yescharacteristic
38C23×C8D411yescharacteristic
39C2×C4⋊Q8C2312yes—
40C2×C22⋊Q8—22no—
41C2×#117—21no—
42C22×C42C2311yescharacteristic
43#500—42no—
44C2×#107—21no—
45C2×#107—21no—
46C2×#107—21no—
47C2×#107—21no—
48C2×#106—24no—
49#424—44no—
50#423—42no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2314yes—
56C2×#103—24no—
57C2×#103—24no—
58C2×C22⋊C8—22no—
59C2×C4×C8—42no—
60C2×C4×C8D411yescharacteristic
61C2×C4×C8D411yescharacteristic
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#391—22no—
65#389—22no—
66#389—22no—
67#384D411yescharacteristic
68#384D411yescharacteristic
69#384C2312yes—
70#382—21no—
71#380—21no—
72#380—22no—
73#380—22no—
74#380D412yes—
75C4×C4⋊C4—21no—
76C4×C4⋊C4C2312yes—
77#340—22no—
78#340—22no—
79#340—24no—
80#336—22no—
81#336—22no—
82#336—22no—
83#335—42no—
84#332—44no—
85C22×Q8—22no—
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4#4111yescharacteristic
89C2×M16—42no—
90C2×M16—42no—
91C2×M16—42no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—24no—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8SD1614yes—
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C4⋊Q8—24no—
105C4⋊Q8—24no—
106C22⋊Q8—84no—
107#117—42no—
108#117—42no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4C2×D412yes—
120C2×C4⋊C4C2×D412yes—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—42no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42#4111yescharacteristic
129C2×C42C2411yescharacteristic
130#107—42no—
131#107—42no—
132#107—42no—
133#107—42no—
134#106—84no—
135C4⋊C8—24no—
136C4⋊C8—24no—
137C4⋊C8—24no—
138C4⋊C8—42no—
139C4⋊C8—44no—
140C4⋊C8—44no—
141#103—84no—
142#103—84no—
143C22⋊C8—82no—
144C4×C8—22no—
145C4×C8—22no—
146C4×C8—41no—
147C4×C8—41no—
148#95—22no—
149#95—22no—
150#95—42no—
151#95—42no—
152C24#12111yescharacteristic
153C2×Q8—44no—
154C2×Q8—42no—
155C2×Q8—44no—
156C2×Q8—42no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4#12111yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4C2×SD1612yes—
176C22×C4#13712yes—
177C22×C4C22×D411yescharacteristic
178C22×C4C22×D411yescharacteristic
179C22×C4C2×#4111yescharacteristic
180M16—84no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—44no—
191C2×C8—44no—
192C2×C8—42no—
193C2×C8—44no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—44no—
197C2×C8—44no—
198C2×C8C2×SD1612yes—
199C2×C8C2×SD1612yes—
200C2×C8#13712yes—
201C4⋊C4—22no—
202C4⋊C4—24no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—44no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—44no—
214C4⋊C4—82no—
215C4⋊C4—84no—
216C22⋊C4—44no—
217C22⋊C4—84no—
218C42—21no—
219C42—21no—
220C42—22no—
221C42—22no—
222C42—42no—
223C42—42no—
224C42H(5,𝔽2)12yes—
225C422−1+412yes—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23#46112yes—
233C23C2×#12111yescharacteristic
234Q8—84no—
235Q8—84no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4#46112yes—
265C2×C4#47312yes—
266C2×C4#53411yescharacteristic
267C2×C4#53611yescharacteristic
268C2×C4#53712yes—
269C2×C4#53912yes—
270C2×C4C22×SD1611yescharacteristic
271C2×C4C2×#13711yescharacteristic
272C2×C4C2×#13711yescharacteristic
273C2×C4#57511yescharacteristic
274C2×C4#57611yescharacteristic
275C2×C4#57612yes—
276C2×C4#57811yescharacteristic
277C8—22no—
278C8—22no—
279C8—22no—
280C8—41no—
281C8—82no—
282C8—84no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22#261811yescharacteristic
291C22C2×#46111yescharacteristic
292C22#303711yescharacteristic
293C22#305811yescharacteristic
294C22#306111yescharacteristic
295C22C2×#47311yescharacteristic
296C22#306711yescharacteristic
297C4—22no—
298C4—22no—
299C4—22no—
300C4—41no—
301C4—82no—
302C4—82no—
303C4—162no—
304C4—162no—
305C4#318711yescharacteristic
306C4#319511yescharacteristic
307C4#322911yescharacteristic
308C4#323611yescharacteristic
309C4#325912yes—
310C4#326212yes—
311C2—42no—
312C2#1919211yescharacteristic
313C2#1919511yescharacteristic
314C2#1934511yescharacteristic
315C2#1934711yescharacteristic
316C2#2975611yescharacteristic
317C2#2977811yescharacteristic
318C2#2978911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10652C2semidirect
5. #10602C2semidirect
6. #10545C2semidirect

Maximal subgroups

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

subgroupindex
#332092
#299222
#211432
#106522
#106022
#105452
#91372
#85962