Finite Group Databasebeta

← #101102

Subgroups — #101102

Order 512.

794 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 98 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#14822C211yesmaximal, characteristic
2#14342C211yesmaximal, characteristic
3#8713C212yesmaximal
4#8299C211yesmaximal, characteristic
5#7035C212yesmaximal
6C23×C4⋊C4C2211yescharacteristic
7C2×#400C412yescomplemented
8C2×#381—21no—
9C2×#380C2211yescharacteristic
10#1779C2212yes—
11C2×#324—22no—
12#1445—24no—
13#1442—22no—
14#1441C412yescomplemented
15C22⋊#95—21no—
16C22⋊#95—21no—
17C22⋊#95—21no—
18C22⋊#95C2211yescharacteristic
19C22⋊#95C2212yes—
20#1294—24no—
21C24×C4D411yescharacteristic
22C24×C4D411yescharacteristic
23C24×C4C2311yescharacteristic
24C22×C4⋊C4—22no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—44no—
27C22×C4⋊C4C2×C412yes—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4D412yes—
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37#400—44no—
38#381—44no—
39#380—22no—
40#380—22no—
41C2×#95—24no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—21no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95—21no—
50C2×#95—22no—
51C2×#95C2×C412yes—
52C2×#95D414yes—
53#324—44no—
54C25C2×D411yescharacteristic
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—24no—
58C23×C4—21no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—22no—
65C23×C4—24no—
66C23×C4—42no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4—81no—
72C23×C4—81no—
73C23×C4C22⋊C412yes—
74C23×C4C22⋊C412yes—
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D412yes—
77C23×C4C2×D411yescharacteristic
78C23×C4#4112yes—
79C22×C8—42no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—48no—
89C2×C4⋊C4—48no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—81no—
92C2×C22⋊C4—44no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—44no—
96C2×C22⋊C4—44no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C22⋊C8—44no—
103C22⋊C8—44no—
104C22⋊C8—82no—
105#95—42no—
106#95—44no—
107#95—44no—
108#95—48no—
109#95—44no—
110#95—44no—
111#95—42no—
112#95—81no—
113#95—82no—
114#95—82no—
115#95—82no—
116#95—82no—
117C24—22no—
118C24—22no—
119C24—22no—
120C24—22no—
121C24—41no—
122C24—41no—
123C24C2×C22⋊C411yescharacteristic
124C24#12011yescharacteristic
125C24#12311yescharacteristic
126C24#13512yes—
127C24#13712yes—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—42no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—48no—
152C22×C4—48no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—88no—
188C22×C4—88no—
189C22×C4#9914yes—
190C22×C4C4×D412yes—
191C22×C4#12112yes—
192C2×C8—82no—
193C2×C8—82no—
194C2×C8—82no—
195C4⋊C4—48no—
196C4⋊C4—48no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C22⋊C4—84no—
200C22⋊C4—84no—
201C22⋊C4—84no—
202C22⋊C4—82no—
203C22⋊C4—82no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—24no—
213C23—24no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—24no—
221C23—24no—
222C23—42no—
223C23—42no—
224C23—44no—
225C23—44no—
226C23—44no—
227C23—44no—
228C23#38611yescharacteristic
229C23C2×#9912yes—
230C23#41611yescharacteristic
231C23#41911yescharacteristic
232C23#42112yes—
233C23#45212yes—
234C23#45412yes—
235C23#45812yes—
236C23#48512yes—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—48no—
247C2×C4—48no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—82no—
258C2×C4—84no—
259C2×C4—84no—
260C2×C4—84no—
261C2×C4—84no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—84no—
265C2×C4—84no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—84no—
269C2×C4—82no—
270C2×C4—84no—
271C2×C4—82no—
272C2×C4—84no—
273C2×C4—82no—
274C2×C4—84no—
275C2×C4—84no—
276C2×C4—84no—
277C2×C4—84no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—82no—
285C8—162no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—21no—
296C22—22no—
297C22—21no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—48no—
304C22—48no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22#151112yes—
310C22#151612yes—
311C22#161514yes—
312C22#187311yescharacteristic
313C22#188811yescharacteristic
314C22#189011yescharacteristic
315C22#189711yescharacteristic
316C22#189811yescharacteristic
317C22#191211yescharacteristic
318C22#191311yescharacteristic
319C4—42no—
320C4—42no—
321C4—44no—
322C4—82no—
323C4—164no—
324C4—162no—
325C4—161no—
326C4—161no—
327C2—24no—
328C2—22no—
329C2—22no—
330C2—42no—
331C2#902211yescharacteristic
332C2#915411yescharacteristic
333C2#920211yescharacteristic
334C2#941411yescharacteristic
335C2#946611yescharacteristic
336C2#948211yescharacteristic
337C2#950111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×#400C4semidirect
14. #1441C4semidirect

Maximal subgroups

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

subgroupindex
#148222
#143422
#87132
#82992
#70352