Finite Group Databasebeta

← #124279

Subgroups — #124279

Order 512.

510 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 116 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#13911C211yesmaximal, characteristic
2#8558C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic
4#7122C214yesmaximal
5C2×#384C411yescharacteristic
6C2×#382—21no—
7C2×C4×C4⋊C4C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9#1797—21no—
10C4×C22⋊C8—21no—
11C2×#324—21no—
12C2×#324—21no—
13#1442C412yes—
14#1442C2212yes—
15C4⋊#95—21no—
16C4⋊#95—21no—
17C4⋊#95—21no—
18#1280C2212yes—
19#1280C2212yes—
20C22×C4⋊C4—21no—
21C22×C4⋊C4—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4C2×C412yes—
24C22×C4⋊C4D411yescharacteristic
25C22×C4⋊C4D411yescharacteristic
26C22×C42—21no—
27C22×C42C2×C411yescharacteristic
28C22×C42D411yescharacteristic
29C22×C42D411yescharacteristic
30C2×C4⋊C8—41no—
31C2×C22⋊C8—22no—
32C2×C22⋊C8—22no—
33C2×C4×C8—41no—
34#384—41no—
35#384—41no—
36#382—42no—
37#382—42no—
38C4×C4⋊C4—22no—
39C4×C4⋊C4—41no—
40C4×C4⋊C4—41no—
41C4×C4⋊C4C814yescomplemented
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95C2×C412yes—
47C2×#95C2×C411yescharacteristic
48C2×#95C2311yescharacteristic
49#336—41no—
50#336—41no—
51#324—24no—
52#324—42no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4C22⋊C411yescharacteristic
56C23×C4C22⋊C411yescharacteristic
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22⋊C411yescharacteristic
59C23×C4C22×C411yescharacteristic
60C23×C4C2×D411yescharacteristic
61C23×C4C2×D411yescharacteristic
62C22×C8—42no—
63C22×C8—42no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—82no—
81C2×C4⋊C4D812yes—
82C2×C4⋊C4SD1612yes—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42—22no—
86C2×C42—24no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—42no—
92C2×C42—42no—
93C2×C42—42no—
94C2×C42—42no—
95C2×C42C2×C812yes—
96C2×C42M1612yes—
97C4⋊C8—82no—
98C22⋊C8—24no—
99C22⋊C8—42no—
100C22⋊C8—42no—
101C22⋊C8—42no—
102C4×C8—82no—
103#95—22no—
104#95—41no—
105#95—41no—
106#95—44no—
107#95—42no—
108#95—42no—
109#95—82no—
110#95—82no—
111#95C2×C814yes—
112C24C2×C22⋊C411yescharacteristic
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—24no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4C22⋊C812yes—
155C22×C4C22⋊C812yes—
156C22×C4#9911yescharacteristic
157C22×C4#9911yescharacteristic
158C22×C4#10112yes—
159C22×C4#10212yes—
160C22×C4#10212yes—
161C22×C4#10412yes—
162C22×C4C22×C811yescharacteristic
163C22×C4C2×M1611yescharacteristic
164C22×C4C2×D811yescharacteristic
165C22×C4C2×SD1611yescharacteristic
166C22×C4#13712yes—
167C2×C8—44no—
168C2×C8—82no—
169C2×C8—82no—
170C2×C8—82no—
171C2×C8—82no—
172C2×C8—82no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—44no—
176C4⋊C4—48no—
177C4⋊C4—82no—
178C4⋊C4—84no—
179C4⋊C4—84no—
180C4⋊C4—82no—
181C42—22no—
182C42—22no—
183C42—22no—
184C42—22no—
185C42—48no—
186C42—42no—
187C42—42no—
188C42—81no—
189C42—81no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23C2×C22⋊C811yescharacteristic
195C23C2×#9911yescharacteristic
196C23C2×#10111yescharacteristic
197C23C2×#10211yescharacteristic
198C23#41911yescharacteristic
199C23C2×#10411yescharacteristic
200C23#42111yescharacteristic
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—44no—
238C2×C4—41no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—81no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4#32412yes—
255C2×C4#32412yes—
256C2×C4D4⋊C814yes—
257C2×C4#35412yes—
258C2×C4#35512yes—
259C8—84no—
260C8—162no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22C2×#32411yescharacteristic
274C22C2×D4⋊C811yescharacteristic
275C22#147911yescharacteristic
276C22#150811yescharacteristic
277C22#151911yescharacteristic
278C22#152811yescharacteristic
279C22#154611yescharacteristic
280C22#212211yescharacteristic
281C22#212311yescharacteristic
282C22C2×#35411yescharacteristic
283C22C2×#35511yescharacteristic
284C4—24no—
285C4—22no—
286C4—42no—
287C4—41no—
288C4—42no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—84no—
293C4—81no—
294C4—162no—
295C4#132414yes—
296C2—21no—
297C2—21no—
298C2—21no—
299C2—21no—
300C2#757311yescharacteristic
301C2C2×#132411yescharacteristic
302C2#1145411yescharacteristic
303C2#1178711yescharacteristic
304C2#1179411yescharacteristic
305C2#1187811yescharacteristic
306C2#1188211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
41. C4×C4⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
#139112
#85582
C4×#3242
#71222