Finite Group Databasebeta

← #142039

Subgroups — #142039

Order 512.

464 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 92 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#15086C211yesmaximal, characteristic
2#10802C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic
4#7251C212yesmaximal
5#7251C212yesmaximal
6C2×#382C411yescharacteristic
7C2×C4×C4⋊C4C2211yescharacteristic
8C4×C22⋊C8—21no—
9C2×#343—21no—
10C2×#343C2212yes—
11#1728—21no—
12C2×#324C2212yes—
13#1445—21no—
14#1444C411yescharacteristic
15#1442—21no—
16#1442C411yescharacteristic
17#1441C2211yescharacteristic
18#1441C2211yescharacteristic
19#1436—21no—
20#1434—21no—
21#1434C411yescharacteristic
22#1280—21no—
23C22×M16—22no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28C22×C4⋊C4C2311yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42D411yescharacteristic
31C22×C42D411yescharacteristic
32C2×C22⋊C8—22no—
33C2×C4×C8—41no—
34#382—41no—
35#382—41no—
36C4×C4⋊C4—41no—
37C4×C4⋊C4—41no—
38C2×#95—21no—
39C2×#95—21no—
40C2×#95—21no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95C2×C411yescharacteristic
50C2×#95C2×C411yescharacteristic
51C2×#95C2×C411yescharacteristic
52C2×#95D411yescharacteristic
53#343—24no—
54#343—24no—
55#336—41no—
56#336—41no—
57#336—41no—
58#324—22no—
59#324Q814yes—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C2×M16—24no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—42no—
76C2×M16—42no—
77C22×C8—42no—
78C22×C8—42no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—81no—
89C2×C4⋊C4—81no—
90C2×C4⋊C4—81no—
91C2×C4⋊C4C2×Q812yes—
92C2×C4⋊C4#4112yes—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—22no—
99C2×C42—21no—
100C2×C42—41no—
101C2×C42—42no—
102C2×C42—42no—
103C22⋊C8—24no—
104C22⋊C8—42no—
105C4×C8—82no—
106#95—41no—
107#95—41no—
108#95—41no—
109#95—41no—
110#95—42no—
111#95—42no—
112#95—42no—
113#95—42no—
114#95—41no—
115#95—41no—
116#95—42no—
117#95—42no—
118#95—42no—
119#95—81no—
120#95—81no—
121#95—81no—
122C24C2×C22⋊C411yescharacteristic
123C22×C4—21no—
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—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
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×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4#9911yescharacteristic
182C22×C4#9911yescharacteristic
183C22×C4#10412yes—
184C22×C4C4×Q812yes—
185C22×C4C22⋊Q812yes—
186C22×C4#12411yescharacteristic
187C22×C4C4⋊Q811yescharacteristic
188C22×C4#13512yes—
189C22×C4#13512yes—
190M16—44no—
191M16—44no—
192M16—82no—
193M16—82no—
194C2×C8—44no—
195C2×C8—44no—
196C2×C8—82no—
197C2×C8—82no—
198C2×C8—82no—
199C2×C8—82no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—44no—
203C4⋊C4—84no—
204C4⋊C4—82no—
205C4⋊C4—82no—
206C42—42no—
207C42—42no—
208C42—42no—
209C42—42no—
210C42—42no—
211C42—81no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23#39111yescharacteristic
217C23C2×#9911yescharacteristic
218C23#41011yescharacteristic
219C23#41611yescharacteristic
220C23#41611yescharacteristic
221C23C2×#10411yescharacteristic
222C23#42111yescharacteristic
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—81no—
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—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4#35212yes—
292C2×C4#48812yes—
293C2×C4#49912yes—
294C8—84no—
295C8—84no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22#150011yescharacteristic
309C22#150111yescharacteristic
310C22#151611yescharacteristic
311C22#152111yescharacteristic
312C22#198611yescharacteristic
313C22#199111yescharacteristic
314C22#199611yescharacteristic
315C22C2×#35211yescharacteristic
316C22#212211yescharacteristic
317C22#213211yescharacteristic
318C22#213311yescharacteristic
319C4—22no—
320C4—22no—
321C4—22no—
322C4—41no—
323C4—42no—
324C4—42no—
325C4—42no—
326C4—82no—
327C4—81no—
328C4—81no—
329C4—161no—
330C4—161no—
331C4—161no—
332C2—21no—
333C2—21no—
334C2—21no—
335C2—21no—
336C2#988611yescharacteristic
337C2#1173511yescharacteristic
338C2#1173811yescharacteristic
339C2#1185011yescharacteristic
340C2#1185111yescharacteristic
341C2#1329511yescharacteristic
342C2#1330311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#150862
#108022
C4×#3242
#72512