Finite Group Databasebeta

← #100980

Subgroups — #100980

Order 512.

482 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 84 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#21916C211yesmaximal, characteristic
2#15085C211yesmaximal, characteristic
3#8718C212yesmaximal
4#8153C211yesmaximal, characteristic
5#7036C212yesmaximal
6C2×#400C411yescharacteristic
7C2×#398C411yescharacteristic
8C2×#382C2211yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1917—24no—
11#1817—21no—
12#1797—21no—
13#1749C2212yes—
14#1749C2212yes—
15C2×#324—21no—
16#1445C411yescharacteristic
17#1445C411yescharacteristic
18#1443—21no—
19#1441—21no—
20#1436—21no—
21#1434—21no—
22#1434C2211yescharacteristic
23#1295—24no—
24#1280—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C42D411yescharacteristic
31C22×C42D411yescharacteristic
32C22×C42C2311yescharacteristic
33C2×C4⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×#97—22no—
37C2×#97D414yes—
38#400—42no—
39#398—42no—
40#382—22no—
41#382—22no—
42C4×C4⋊C4—22no—
43C4×C4⋊C4—22no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95D411yescharacteristic
59#336—22no—
60#336—24no—
61#336—22no—
62#336—22no—
63#324—24no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C22×C8—24no—
76C22×C8—42no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—42no—
97C2×C42—41no—
98C2×C42—41no—
99C2×C42—42no—
100C2×C42C2×D412yes—
101C2×C42#4112yes—
102C4⋊C8—82no—
103C22⋊C8—24no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107#97—44no—
108#97—44no—
109#95—22no—
110#95—22no—
111#95—41no—
112#95—41no—
113#95—42no—
114#95—42no—
115#95—42no—
116#95—42no—
117#95—42no—
118#95—42no—
119#95—42no—
120#95—42no—
121#95—42no—
122#95—41no—
123#95—42no—
124#95—41no—
125C24C2×C22⋊C411yescharacteristic
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—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
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—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4#10012yes—
187C22×C4#10412yes—
188C22×C4C4×D412yes—
189C22×C4#12011yescharacteristic
190C22×C4#12112yes—
191C22×C4#12311yescharacteristic
192C2×C8—44no—
193C2×C8—44no—
194C2×C8—44no—
195C2×C8—44no—
196C2×C8—82no—
197C2×C8—82no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—82no—
201C4⋊C4—82no—
202C4⋊C4—84no—
203C42—24no—
204C42—42no—
205C42—42no—
206C42—42no—
207C42—42no—
208C42—81no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23#38611yescharacteristic
214C23C2×#10011yescharacteristic
215C23#41311yescharacteristic
216C23C2×#10411yescharacteristic
217C23#42111yescharacteristic
218C23#42111yescharacteristic
219C23#42111yescharacteristic
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
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—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
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#45812yes—
288C8—84no—
289C8—84no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22#152911yescharacteristic
303C22#153011yescharacteristic
304C22#153311yescharacteristic
305C22#153511yescharacteristic
306C22#187111yescharacteristic
307C22#188411yescharacteristic
308C22#189011yescharacteristic
309C22#189711yescharacteristic
310C22#190011yescharacteristic
311C22#191211yescharacteristic
312C22#191511yescharacteristic
313C4—22no—
314C4—41no—
315C4—42no—
316C4—42no—
317C4—42no—
318C4—42no—
319C4—82no—
320C4—81no—
321C4—81no—
322C4—161no—
323C4—161no—
324C4—161no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2#901111yescharacteristic
330C2#912411yescharacteristic
331C2#919011yescharacteristic
332C2#939011yescharacteristic
333C2#944811yescharacteristic
334C2#948211yescharacteristic
335C2#950811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#219162
#150852
#87182
#81532
#70362