Finite Group Databasebeta

← #99822

Subgroups — #99822

Order 512.

530 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 112 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#21912C211yesmaximal, characteristic
2#14320C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic
4#7575C212yesmaximal
5#7024C212yesmaximal
6C2×#395C411yescharacteristic
7C2×#391C2211yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9#1916—24no—
10#1817—21no—
11#1797C2212yes—
12C4×C22⋊C8—21no—
13C4×C22⋊C8C2212yes—
14C2×#324—21no—
15#1446C411yescharacteristic
16#1445—21no—
17#1442—21no—
18#1442C411yescharacteristic
19#1441C411yescharacteristic
20#1436—21no—
21#1434—21no—
22#1434C2211yescharacteristic
23#1280—21no—
24#1272—24no—
25C23×Q8C2×C411yescharacteristic
26C22×C4⋊C4—21no—
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×C4×C8—22no—
37C2×C4×C8—22no—
38#395—22no—
39#395C814yescomplemented
40#391—22no—
41#391—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—24no—
60#336—22no—
61#336—22no—
62#336D414yes—
63#324—24no—
64C22×Q8—21no—
65C22×Q8—22no—
66C22×Q8—22no—
67C22×Q8—81no—
68C22×Q8—81no—
69C22×Q8C2×C812yes—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C22×C8—24no—
82C22×C8—42no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—81no—
91C2×C4⋊C4—81no—
92C2×C4⋊C4—81no—
93C2×C4⋊C4C2×C812yes—
94C2×C4⋊C4M1612yes—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—21no—
101C2×C42—81no—
102C2×C42—81no—
103C2×C42C2×D412yes—
104C2×C42#4112yes—
105C4⋊C8—82no—
106C22⋊C8—24no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110C4×C8—82no—
111C4×C8—82no—
112#95—24no—
113#95—22no—
114#95—22no—
115#95—21no—
116#95—22no—
117#95—42no—
118#95—42no—
119#95—41no—
120#95—42no—
121#95—41no—
122#95—42no—
123#95—81no—
124#95—81no—
125#95—81no—
126#95—81no—
127#95—81no—
128#95C2×C812yes—
129C24C2×C22⋊C411yescharacteristic
130C2×Q8—24no—
131C2×Q8—42no—
132C2×Q8—82no—
133C2×Q8—82no—
134C2×Q8—82no—
135C2×Q8—82no—
136C2×Q8—84no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
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—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4C22⋊C812yes—
193C22×C4C22⋊C812yes—
194C22×C4#10412yes—
195C22×C4C4×D412yes—
196C22×C4#12011yescharacteristic
197C22×C4#12112yes—
198C22×C4#12311yescharacteristic
199C22×C4C22×C811yescharacteristic
200C22×C4C2×M1611yescharacteristic
201C22×C4#13112yes—
202C22×C4#13512yes—
203C22×C4#13712yes—
204C2×C8—44no—
205C2×C8—44no—
206C2×C8—44no—
207C2×C8—44no—
208C2×C8—82no—
209C2×C8—82no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—44no—
213C4⋊C4—82no—
214C4⋊C4—84no—
215C42—42no—
216C42—42no—
217C42—42no—
218C42—42no—
219C42—42no—
220C42—81no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23#38611yescharacteristic
226C23C2×C22⋊C811yescharacteristic
227C23#40811yescharacteristic
228C23#41611yescharacteristic
229C23#41911yescharacteristic
230C23C2×#10411yescharacteristic
231C23#42111yescharacteristic
232Q8—84no—
233Q8—84no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
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—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4C8×D412yes—
301C2×C4#43512yes—
302C2×C4#45212yes—
303C2×C4#48512yes—
304C8—84no—
305C8—84no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22#147611yescharacteristic
319C22#147911yescharacteristic
320C22#148211yescharacteristic
321C22#148511yescharacteristic
322C22#186911yescharacteristic
323C22#187111yescharacteristic
324C22#187311yescharacteristic
325C22#190911yescharacteristic
326C22#191211yescharacteristic
327C22#191311yescharacteristic
328C22#191411yescharacteristic
329C4—22no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—42no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—82no—
338C4—82no—
339C4—81no—
340C4—161no—
341C4—161no—
342C2—21no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2#893811yescharacteristic
347C2#910011yescharacteristic
348C2#910711yescharacteristic
349C2#918311yescharacteristic
350C2#918811yescharacteristic
351C2#919211yescharacteristic
352C2#920211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
39. #395C8semidirect

Maximal subgroups

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

subgroupindex
#219122
#143202
C4×#3242
#75752
#70242