Finite Group Databasebeta

← #134525

Subgroups — #134525

Order 512.

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

normal subgroups only

This group has 408 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#21537C211yesmaximal, characteristic
2#12786C211yesmaximal, characteristic
3C2×#1342C212yesmaximal
4#10867C211yesmaximal, characteristic
5#7255C212yesmaximal
6C2×#470C2211yescharacteristic
7C2×C4×Q16C411yescharacteristic
8C22×#108—21no—
9C2×#405C2211yescharacteristic
10#2323—21no—
11#2150—21no—
12C2×#349C2212yes—
13#1971C411yescharacteristic
14#1944C411yescharacteristic
15#1933—21no—
16#1892—21no—
17#1809C2211yescharacteristic
18C4×#106C411yescharacteristic
19#1763—21no—
20#1742—21no—
21C2×#338C2212yes—
22#1377—21no—
23#1342—24no—
24C22×Q16C2×C411yescharacteristic
25C22×M16—22no—
26C22×M16C2311yescharacteristic
27C23×C8D411yescharacteristic
28C2×C22⋊Q8—21no—
29C2×C4×Q8—21no—
30C2×#117D411yescharacteristic
31C2×M32—22no—
32#470—21no—
33#470—21no—
34#470D412yes—
35#470Q812yes—
36C4×Q16—42no—
37C2×#108—24no—
38C2×#108—42no—
39C2×#108—42no—
40C2×#107C2×C411yescharacteristic
41C2×#106C2×C411yescharacteristic
42C2×#106C2×C411yescharacteristic
43C2×#103—21no—
44C2×#103—21no—
45C2×#103—21no—
46#405—42no—
47C2×#97C2×C411yescharacteristic
48C2×#97D411yescharacteristic
49C2×C4×C8C2×C411yescharacteristic
50C2×C4×C8D411yescharacteristic
51#391—21no—
52#389—21no—
53#389—41no—
54#388—41no—
55#388—41no—
56#387—41no—
57#380—21no—
58C4×C4⋊C4—21no—
59C4⋊C16—42no—
60#349—22no—
61#349—22no—
62#343—41no—
63#343—41no—
64#340—21no—
65#340—21no—
66#340—21no—
67#338—22no—
68#338—22no—
69C22×Q8—21no—
70C23×C4C2×D411yescharacteristic
71C2×Q16—21no—
72C2×Q16—41no—
73C2×Q16—41no—
74C2×Q16C4⋊C412yes—
75C2×M16—24no—
76C2×M16—41no—
77C2×M16—41no—
78C2×M16—42no—
79C2×M16—42no—
80C2×M16—42no—
81C2×M16—42no—
82C2×M16—42no—
83C2×M16C2×D411yescharacteristic
84C2×M16C2×Q811yescharacteristic
85C2×M16#4111yescharacteristic
86C2×M16#4111yescharacteristic
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—42no—
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22×C411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22⋊Q8—22no—
97C22⋊Q8—22no—
98C22⋊Q8—41no—
99C22⋊Q8—41no—
100C4×Q8—42no—
101C4×Q8—42no—
102#117—41no—
103#117—41no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4D811yescharacteristic
111C2×C4⋊C4D811yescharacteristic
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4SD1611yescharacteristic
115C2×C22⋊C4SD1611yescharacteristic
116C2×C42—41no—
117C2×C42—41no—
118C2×C42C22⋊C411yescharacteristic
119C2×C42C22⋊C411yescharacteristic
120M32—42no—
121M32—42no—
122C2×C16—42no—
123#108—48no—
124#107—41no—
125#106—21no—
126#106—41no—
127#106C4⋊C412yes—
128#103—22no—
129#103—22no—
130#103—42no—
131#103—42no—
132#97—41no—
133#97—41no—
134C4×C8—22no—
135C4×C8—22no—
136#95—41no—
137#95—41no—
138#95—41no—
139#95—41no—
140#95—41no—
141C24C2×SD1611yescharacteristic
142C2×Q8—22no—
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—41no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4#10211yescharacteristic
157C22×C4#10211yescharacteristic
158C22×C4#10211yescharacteristic
159C22×C4#10211yescharacteristic
160C22×C4C2×C22⋊C411yescharacteristic
161C22×C4#12111yescharacteristic
162C22×C4C22⋊Q811yescharacteristic
163C22×C4C2×D811yescharacteristic
164C22×C4AGL(1,ℤ/8)11yescharacteristic
165C22×C4#13711yescharacteristic
166Q16—42no—
167Q16—42no—
168M16—41no—
169M16—41no—
170M16—41no—
171M16—41no—
172M16—44no—
173M16—44no—
174M16—82no—
175M16—82no—
176C2×C8—24no—
177C2×C8—24no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—44no—
187C2×C8—82no—
188C2×C8—82no—
189C2×C8#9912yes—
190C2×C8#10412yes—
191C2×C8C2×C4⋊C411yescharacteristic
192C2×C8#11711yescharacteristic
193C2×C8#12111yescharacteristic
194C2×C8C22⋊Q811yescharacteristic
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—82no—
206C4⋊C4—161no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—161no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—82no—
217C42—161no—
218C16—82no—
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223C23C2×#10211yescharacteristic
224C23#46111yescharacteristic
225C23#47611yescharacteristic
226Q8—42no—
227Q8—42no—
228Q8—81no—
229Q8—81no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4#38011yescharacteristic
261C2×C4C2×#9911yescharacteristic
262C2×C4#41011yescharacteristic
263C2×C4#41711yescharacteristic
264C2×C4C2×#10411yescharacteristic
265C2×C4#42111yescharacteristic
266C2×C4#45911yescharacteristic
267C2×C4#47411yescharacteristic
268C8—21no—
269C8—21no—
270C8—21no—
271C8—21no—
272C8—22no—
273C8—22no—
274C8—84no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—81no—
284C22#150211yescharacteristic
285C22#151911yescharacteristic
286C22#180211yescharacteristic
287C4—21no—
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—82no—
294C4—82no—
295C4—161no—
296C4—161no—
297C4#150411yescharacteristic
298C4#151211yescharacteristic
299C4#179911yescharacteristic
300C4#180611yescharacteristic
301C2—21no—
302C2—21no—
303C2—41no—
304C2—41no—
305C2#857711yescharacteristic
306C2#1187111yescharacteristic
307C2#1187211yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#215372
#127862
C2×#13422
#108672
#72552