Finite Group Databasebeta

← #124530

Subgroups — #124530

Order 512.

506 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 94 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#13848C211yesmaximal, characteristic
2#8699C211yesmaximal, characteristic
3#8699C211yesmaximal, characteristic
4#7123C212yesmaximal
5#7122C212yesmaximal
6C2×C4×C4⋊C4—21no—
7C2×C4×C4⋊C4C411yescharacteristic
8C2×C4×C4⋊C4C411yescharacteristic
9C2×#376C2211yescharacteristic
10#1817—21no—
11#1817—21no—
12#1444C411yescharacteristic
13#1444C2211yescharacteristic
14#1442C411yescharacteristic
15#1442C2211yescharacteristic
16#1441—21no—
17#1441—21no—
18C4×#95—21no—
19#1280—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C42—21no—
29C22×C42—21no—
30C22×C42C2×C411yescharacteristic
31C22×C42D411yescharacteristic
32C22×C42D411yescharacteristic
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C4×C4⋊C4—22no—
36C4×C4⋊C4—44no—
37C4×C4⋊C4—42no—
38C4×C4⋊C4C814yescomplemented
39#376—42no—
40C2×#95—21no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95C2×C411yescharacteristic
48C2×#95C2×C411yescharacteristic
49C2×#95C2×C411yescharacteristic
50C2×#95D411yescharacteristic
51C2×#95D411yescharacteristic
52C2×#95C2311yescharacteristic
53#336—42no—
54#336—42no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4C22⋊C411yescharacteristic
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22×C411yescharacteristic
64C23×C4C2×D411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C22×C8—42no—
67C22×C8—42no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
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—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C42—22no—
82C2×C42—22no—
83C2×C42—24no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42—41no—
89C2×C42—41no—
90C2×C42—42no—
91C2×C42—42no—
92C2×C42—42no—
93C2×C42—42no—
94C2×C42—42no—
95C2×C42C2×C812yes—
96C2×C42M1612yes—
97C22⋊C8—42no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100C22⋊C8—42no—
101#95—21no—
102#95—21no—
103#95—41no—
104#95—41no—
105#95—41no—
106#95—41no—
107#95—42no—
108#95—42no—
109#95—42no—
110#95—42no—
111#95—42no—
112#95—42no—
113#95—42no—
114#95C2×C812yes—
115#95C2×C812yes—
116C24C2×C22⋊C411yescharacteristic
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4C22⋊C812yes—
176C22×C4C22⋊C812yes—
177C22×C4#9912yes—
178C22×C4#10112yes—
179C22×C4C22×C811yescharacteristic
180C22×C4C2×M1611yescharacteristic
181C22×C4AGL(1,ℤ/8)12yes—
182C22×C4#13712yes—
183C2×C8—82no—
184C2×C8—82no—
185C2×C8—82no—
186C2×C8—82no—
187C2×C8—82no—
188C2×C8—82no—
189C4⋊C4—48no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—48no—
193C4⋊C4—48no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C42—24no—
197C42—24no—
198C42—44no—
199C42—44no—
200C42—48no—
201C42—84no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23C2×C22⋊C811yescharacteristic
207C23C2×#9911yescharacteristic
208C23C2×#10111yescharacteristic
209C23#41811yescharacteristic
210C23#41911yescharacteristic
211C23#42111yescharacteristic
212C23#42111yescharacteristic
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4#32414yes—
280C8—162no—
281C8—162no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22C2×#32411yescharacteristic
295C22#147811yescharacteristic
296C22#147911yescharacteristic
297C22#151011yescharacteristic
298C22#151111yescharacteristic
299C22#153411yescharacteristic
300C22#155611yescharacteristic
301C22#212011yescharacteristic
302C22#212011yescharacteristic
303C22#212611yescharacteristic
304C22#213211yescharacteristic
305C4—24no—
306C4—44no—
307C4—42no—
308C4—42no—
309C4—42no—
310C4—41no—
311C4—41no—
312C4—82no—
313C4—82no—
314C4—81no—
315C4—82no—
316C4—81no—
317C2—21no—
318C2—21no—
319C2—21no—
320C2—21no—
321C2#759311yescharacteristic
322C2#1144711yescharacteristic
323C2#1145211yescharacteristic
324C2#1179811yescharacteristic
325C2#1180411yescharacteristic
326C2#1181011yescharacteristic
327C2#1181611yescharacteristic
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×#376 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
38. C4×C4⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
#138482
#86992
#71232
#71222