Finite Group Databasebeta

← #128992

Subgroups — #128992

Order 512.

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

normal subgroups only

This group has 387 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#22209C211yesmaximal, characteristic
2#11728C211yesmaximal, characteristic
3#11728C211yesmaximal, characteristic
4#8160C211yesmaximal, characteristic
5C8×#99C211yesmaximal, characteristic
6#7287C211yesmaximal, characteristic
7#7283C211yesmaximal, characteristic
8C2×C8×Q8C411yescharacteristic, complemented
9C2×C8×D4C2211yescharacteristic
10#2398C2211yescharacteristic
11#1985C411yescharacteristic, complemented
12#1955C2211yescharacteristic
13#1923—21no—
14#1923—21no—
15#1850—21no—
16#1850—21no—
17#1843C411yescharacteristic, complemented
18C4×#99C2211yescharacteristic
19C4×#99C2211yescharacteristic
20#1752—21no—
21C8×C22⋊C4—21no—
22#1746C411yescharacteristic, complemented
23#1459C2211yescharacteristic
24#1459C2211yescharacteristic
25#1351C412yescomplemented
26#1351C412yescomplemented
27#1344C412yescomplemented
28#1344C412yescomplemented
29#1282—21no—
30#1282—21no—
31C23×C8D411yescharacteristic
32C23×C8D411yescharacteristic
33C2×C4×Q8C2×C411yescharacteristic
34C2×C4×D4C2311yescharacteristic
35C8×Q8—41no—
36C8×Q8—41no—
37C8×D4—41no—
38C8×D4—41no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8C2×C411yescharacteristic
43C2×C4⋊C8C2×C411yescharacteristic
44C2×#99—21no—
45C2×#99—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8D411yescharacteristic
51C2×C22⋊C8D411yescharacteristic
52C2×#97—41no—
53C2×#97C2×C411yescharacteristic
54C2×C4×C8—21no—
55C2×C4×C8—41no—
56C2×C4×C8C2×C411yescharacteristic
57#400—21no—
58#397C2×C411yescharacteristic
59#397C2×C411yescharacteristic
60#394—21no—
61#393C2×C411yescharacteristic
62#393C2×C411yescharacteristic
63C4×C22⋊C4—21no—
64C4×C22⋊C4—21no—
65C4×C22⋊C4—21no—
66C4×C22⋊C4—21no—
67#376C2×C411yescharacteristic
68#342C2×C412yes—
69#342C2×C412yes—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336—41no—
75#336—41no—
76#324—21no—
77#324—21no—
78#323C2×C412yes—
79#323C2×C412yes—
80C22×Q8C4211yescharacteristic
81C22×D4#4111yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8—81no—
93C22×C8—81no—
94C22×C8C22⋊C411yescharacteristic
95C22×C8C22⋊C411yescharacteristic
96C22×C8C22⋊C411yescharacteristic
97C22×C8C22⋊C411yescharacteristic
98C4×Q8—41no—
99C4×Q8—41no—
100C4×D4—41no—
101C4×D4—41no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4C4211yescharacteristic
104C2×C4⋊C4#4111yescharacteristic
105C2×C22⋊C4—22no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4C22×C411yescharacteristic
112C2×C22⋊C4C22×C411yescharacteristic
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—41no—
116C2×C42—41no—
117C2×C42C22×C411yescharacteristic
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—81no—
123C4⋊C8—81no—
124#99—24no—
125#99—42no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—41no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—81no—
133C22⋊C8—81no—
134#97—41no—
135#97—82no—
136C4×C8—21no—
137C4×C8—21no—
138C4×C8—42no—
139C4×C8—82no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95—41no—
146#95—41no—
147#95C4211yescharacteristic
148#95C4211yescharacteristic
149C24C4×D411yescharacteristic
150C24C4×D411yescharacteristic
151C2×Q8—42no—
152C2×Q8—41no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—41no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4C2×C4211yescharacteristic
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4#11711yescharacteristic
173C22×C4C4×D411yescharacteristic
174C22×C4C4×D411yescharacteristic
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—41no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—81no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8#9911yescharacteristic
199C2×C8#9911yescharacteristic
200C4⋊C4—41no—
201C4⋊C4—41no—
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—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C42—21no—
214C42—21no—
215C42—41no—
216C42—41no—
217C42—41no—
218C42—41no—
219C42—81no—
220C42—81no—
221C42—81no—
222C42—81no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23C4×C22⋊C411yescharacteristic
232C23#44312yes—
233C23#44412yes—
234Q8—42no—
235Q8—42no—
236D4—44no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—44no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4C2×#9911yescharacteristic
273C2×C4#41011yescharacteristic
274C8—21no—
275C8—21no—
276C8—41no—
277C8—41no—
278C8—41no—
279C8—81no—
280C8—82no—
281C8—82no—
282C8—81no—
283C22—21no—
284C22—21no—
285C22—22no—
286C22—22no—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—81no—
295C22C4×#9911yescharacteristic
296C22#176411yescharacteristic
297C22#176511yescharacteristic
298C4—21no—
299C4—41no—
300C4—42no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—81no—
305C4—81no—
306C4—84no—
307C4—81no—
308C4—162no—
309C4#212711yescharacteristic
310C4#212911yescharacteristic
311C2—21no—
312C2—21no—
313C2—42no—
314C2—42no—
315C2#824011yescharacteristic
316C2#1262311yescharacteristic
317C2#1262711yescharacteristic
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
8. C2×C8×Q8C4semidirect
11. #1985C4semidirect
17. #1843C4semidirect
22. #1746C4semidirect
25. #1351C4semidirect
26. #1351C4semidirect
27. #1344C4semidirect
28. #1344C4semidirect

Maximal subgroups

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

subgroupindex
#222092
#117282
#81602
C8×#992
#72872
#72832