Finite Group Databasebeta

← #128451

Subgroups — #128451

Order 512.

418 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 120 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#22063C211yesmaximal, characteristic
2C4×#352C211yesmaximal, characteristic
3#11472C211yesmaximal, characteristic
4C8×#100C211yesmaximal, characteristic
5C8×#99C211yesmaximal, characteristic
6#7204C211yesmaximal, characteristic
7#7203C211yesmaximal, characteristic
8C2×C8×D4C2211yescharacteristic
9C22×C4⋊C8C411yescharacteristic, complemented
10C4×#123C2211yescharacteristic
11C2×#352C412yes—
12#1939—21no—
13#1923—21no—
14#1916—21no—
15#1869C2211yescharacteristic
16#1850—21no—
17#1850C411yescharacteristic, complemented
18#1817C411yescharacteristic, complemented
19C4×#100C2211yescharacteristic
20C4×#99C2211yescharacteristic
21C8×C22⋊C4—21no—
22C8×C22⋊C4C411yescharacteristic, complemented
23#1460C2211yescharacteristic
24#1459C2211yescharacteristic
25C8×M16—21no—
26#1390C412yes—
27#1326C412yescomplemented
28#1317C412yescomplemented
29#1283—21no—
30#1282—21no—
31C23×C8C2×C411yescharacteristic
32C23×C8C2×C411yescharacteristic
33C23×C8D411yescharacteristic
34C2×#123C2×C411yescharacteristic
35C2×C4×D4C2311yescharacteristic
36C22×C42C2×C411yescharacteristic
37C8×D4—41no—
38C8×D4—41no—
39#431—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—41no—
44C2×C4⋊C8—41no—
45C2×C4⋊C8D411yescharacteristic
46C2×#100C2×C412yes—
47C2×#99C2×C412yes—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8C2×C411yescharacteristic
51C2×C22⋊C8C2×C411yescharacteristic
52C2×C22⋊C8D411yescharacteristic
53C4×M16—21no—
54C2×C4×C8—21no—
55C2×C4×C8—41no—
56C2×C4×C8D411yescharacteristic
57#388—21no—
58#387—21no—
59#386C2×C411yescharacteristic
60#385C2×C411yescharacteristic
61#381C2×C411yescharacteristic
62C4×C4⋊C4—21no—
63C4×C22⋊C4—21no—
64C4×C22⋊C4—21no—
65C4×C22⋊C4C2×C411yescharacteristic
66#352C818yescomplemented
67#343—21no—
68#342—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336—41no—
75#324—21no—
76#323—21no—
77#323C2×C412yes—
78#323C2×C412yes—
79#322—41no—
80C82—41no—
81C22×D4C22×C411yescharacteristic
82C23×C4C4211yescharacteristic
83C23×C4C4211yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C2×M16—22no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8—81no—
96C22×C8—81no—
97C22×C8—81no—
98C22×C8C22⋊C411yescharacteristic
99C22×C8C22⋊C411yescharacteristic
100C22×C8C22⋊C411yescharacteristic
101C22×C8C22⋊C411yescharacteristic
102#123—22no—
103#123C2×C814yes—
104C4×D4—41no—
105C4×D4—41no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4#4111yescharacteristic
109C2×C22⋊C4—21no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4C4211yescharacteristic
113C2×C22⋊C4C4211yescharacteristic
114C2×C22⋊C4C22×C411yescharacteristic
115C2×C22⋊C4#4111yescharacteristic
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42C2×D411yescharacteristic
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—42no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—81no—
129#100—22no—
130#100C2×C814yes—
131#99—22no—
132#99C2×C814yes—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—81no—
140#97—41no—
141C4×C8—21no—
142C4×C8—21no—
143C4×C8—41no—
144C4×C8—41no—
145C4×C8—81no—
146C4×C8—82no—
147#95—21no—
148#95—21no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—41no—
153C24C2×C4211yescharacteristic
154C24C4×D411yescharacteristic
155C2×D4—41no—
156C2×D4—41no—
157C2×D4C22×C812yes—
158C2×D4#13112yes—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—81no—
177C22×C4C4×C812yescomplemented
178C22×C4C2×C22⋊C411yescharacteristic
179C22×C4#11711yescharacteristic
180C22×C4C4×D411yescharacteristic
181C22×C4C4×D411yescharacteristic
182C22×C4C4×D411yescharacteristic
183M16—24no—
184M16—42no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—42no—
195C2×C8—41no—
196C2×C8—42no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—81no—
203C2×C8—81no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8—81no—
207C2×C8—82no—
208C2×C8#9911yescharacteristic
209C2×C8#9911yescharacteristic
210C4⋊C4—24no—
211C4⋊C4—41no—
212C4⋊C4—81no—
213C22⋊C4—21no—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—42no—
219C22⋊C4—41no—
220C22⋊C4—81no—
221C22⋊C4C4×C812yescomplemented
222C42—21no—
223C42—21no—
224C42—41no—
225C42—41no—
226C42—42no—
227C42—41no—
228C42—41no—
229C42—81no—
230C42—81no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23C4×C22⋊C411yescharacteristic
240C23C2×C4×C811yescharacteristic
241C23#40511yescharacteristic
242C23C8×D412yes—
243D4—42no—
244D4—42no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—44no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4C2×#9911yescharacteristic
280C2×C4#41011yescharacteristic
281C2×C4C8×D412yes—
282C8—21no—
283C8—21no—
284C8—44no—
285C8—41no—
286C8—41no—
287C8—41no—
288C8—81no—
289C8—82no—
290C8—82no—
291C8—81no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—41no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—81no—
306C22C8×C22⋊C411yescharacteristic
307C22C4×#9911yescharacteristic
308C22#175711yescharacteristic
309C4—21no—
310C4—22no—
311C4—42no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—42no—
316C4—81no—
317C4—84no—
318C4—81no—
319C4#212111yescharacteristic
320C4#212311yescharacteristic
321C2—21no—
322C2—21no—
323C2—42no—
324C2—41no—
325C2—41no—
326C2C8×#9911yescharacteristic
327C2#1262111yescharacteristic
328C2#1262411yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC23C2×C4×C8
Frattini, cosocleC2×C4×D4C23
socleC22C4×#99
prime radicalC22×D4C22×C4
LCSG ▷ C23 ▷ C22 ▷ 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. C22×C4⋊C8C4semidirect
17. #1850C4semidirect
18. #1817C4semidirect
22. C8×C22⋊C4C4semidirect
27. #1326C4semidirect
28. #1317C4semidirect
66. #352C8semidirect
177. C22×C4C4×C8semidirect
221. C22⋊C4C4×C8semidirect

Maximal subgroups

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

subgroupindex
#220632
C4×#3522
#114722
C8×#1002
C8×#992
#72042
#72032