Finite Group Databasebeta

← #128985

Subgroups — #128985

Order 512.

371 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22064C211yesmaximal, characteristic
2#11735C211yesmaximal, characteristic
3#11727C211yesmaximal, characteristic
4#8164C211yesmaximal, characteristic
5C8×#99C211yesmaximal, characteristic
6#7285C211yesmaximal, characteristic
7#7282C211yesmaximal, characteristic
8C2×C8×D4C2211yescharacteristic
9C2×#424C411yescharacteristic, complemented
10#2395C2211yescharacteristic
11#1939—21no—
12#1923—21no—
13#1916—21no—
14#1870C2211yescharacteristic
15#1851C411yescharacteristic, complemented
16#1850—21no—
17#1818C411yescharacteristic, complemented
18C4×#100C2211yescharacteristic
19C4×#99C2211yescharacteristic
20C8×C22⋊C4—21no—
21C8×C22⋊C4C411yescharacteristic, complemented
22#1460C2211yescharacteristic
23#1459C2211yescharacteristic
24#1451—21no—
25#1353C412yescomplemented
26#1352C412yes—
27#1346C412yescomplemented
28#1344C412yes—
29#1283—21no—
30#1282—21no—
31C22×M16C2×C411yescharacteristic
32C23×C8C2×C411yescharacteristic
33C23×C8D411yescharacteristic
34C2×C4×D4C2311yescharacteristic
35C2×#117C2×C411yescharacteristic
36C8×D4—41no—
37C8×D4—41no—
38#431—21no—
39#424—41no—
40#424—41no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×#100—21no—
46C2×#99—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8C2×C411yescharacteristic
50C2×C22⋊C8C2×C411yescharacteristic
51C2×C22⋊C8D411yescharacteristic
52C4×M16—21no—
53C2×C4×C8—21no—
54C2×C4×C8—41no—
55C2×C4×C8D411yescharacteristic
56#400C2×C411yescharacteristic
57#399C2×C411yescharacteristic
58#398C2×C411yescharacteristic
59#397—21no—
60#397C2×C411yescharacteristic
61#394—21no—
62C4×C4⋊C4—21no—
63C4×C22⋊C4—21no—
64C4×C22⋊C4—21no—
65C4×C22⋊C4C2×C411yescharacteristic
66#343C2×C412yes—
67#342C2×C412yes—
68#336—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—41no—
74#324C2×C412yes—
75#323—21no—
76#323—21no—
77#323C2×C412yes—
78#322—41no—
79#322—41no—
80C22×D4#4111yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C2×M16—22no—
84C2×M16—41no—
85C2×M16—81no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—41no—
93C22×C8—81no—
94C22×C8—81no—
95C22×C8C22⋊C411yescharacteristic
96C22×C8C22⋊C411yescharacteristic
97C22×C8C22⋊C411yescharacteristic
98C22×C8C22⋊C411yescharacteristic
99C4×D4—41no—
100C4×D4—41no—
101#117—41no—
102#117—41no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4C4211yescharacteristic
106C2×C4⋊C4C4211yescharacteristic
107C2×C4⋊C4C22×C411yescharacteristic
108C2×C22⋊C4—22no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4C22×C411yescharacteristic
114C2×C22⋊C4#4111yescharacteristic
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—41no—
118C2×C42C2×D411yescharacteristic
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—42no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—81no—
125#100—42no—
126#99—24no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—81no—
134#97—41no—
135C4×C8—21no—
136C4×C8—21no—
137C4×C8—41no—
138C4×C8—41no—
139C4×C8—81no—
140C4×C8—82no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95—41no—
146#95C4211yescharacteristic
147#95C4211yescharacteristic
148C24#11711yescharacteristic
149C24C4×D411yescharacteristic
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—41no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4C2×C4211yescharacteristic
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C4×D411yescharacteristic
169C22×C4C4×D411yescharacteristic
170C22×C4C4×D411yescharacteristic
171M16—42no—
172M16—42no—
173M16—82no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—41no—
178C2×C8—42no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—41no—
185C2×C8—42no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—81no—
192C2×C8—81no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8#9911yescharacteristic
197C2×C8#9911yescharacteristic
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C22⋊C4—24no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C42—21no—
212C42—21no—
213C42—41no—
214C42—41no—
215C42—41no—
216C42—41no—
217C42—81no—
218C42—81no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23C4×C22⋊C411yescharacteristic
229C23#44412yes—
230D4—44no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—44no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—81no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4C2×#9911yescharacteristic
263C2×C4#41011yescharacteristic
264C2×C4#44312yes—
265C8—21no—
266C8—21no—
267C8—41no—
268C8—41no—
269C8—41no—
270C8—81no—
271C8—82no—
272C8—82no—
273C8—82no—
274C8—81no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22C4×#9911yescharacteristic
288C22#176411yescharacteristic
289C22#176511yescharacteristic
290C4—22no—
291C4—21no—
292C4—42no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—81no—
297C4—84no—
298C4—81no—
299C4—81no—
300C4#212111yescharacteristic
301C4#212311yescharacteristic
302C2—21no—
303C2—21no—
304C2—42no—
305C2—81no—
306C2#824011yescharacteristic
307C2#1262111yescharacteristic
308C2#1263311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C4×#99
prime radicalC22×D4#41
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C2×#424C4semidirect
15. #1851C4semidirect
17. #1818C4semidirect
21. C8×C22⋊C4C4semidirect
25. #1353C4semidirect
27. #1346C4semidirect

Maximal subgroups

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

subgroupindex
#220642
#117352
#117272
#81642
C8×#992
#72852
#72822