Finite Group Databasebeta

← #511652

Subgroups — #511652

Order 512.

502 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 128 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#33216C211yesmaximal, characteristic
2#33180C211yesmaximal, characteristic
3#13531C211yesmaximal, characteristic, complemented
4#13530C211yesmaximal, characteristic, complemented
5#13507C212yesmaximal, complemented
6#13409C212yesmaximal, complemented
7#13383C212yesmaximal
8#13382C212yesmaximal
9#13366C211yesmaximal, characteristic, complemented
10#13366C211yesmaximal, characteristic, complemented
11#12845C211yesmaximal, characteristic
12C2×#499C2211yescharacteristic
13C2×#494C2211yescharacteristic
14C2×#470—21no—
15C2×#467C2211yescharacteristic
16C2×#467C2211yescharacteristic
17C2×#405C2211yescharacteristic
18#2629—21no—
19#2621—21no—
20C2×#367C2212yes—
21C2×#366C2212yes—
22C2×#358C2212yes—
23C2×#358C2212yes—
24C2×#358C2212yes—
25C2×#358C2212yes—
26C2×C22⋊C16C2212yes—
27#1992—21no—
28#1971—21no—
29#1969C2211yescharacteristic
30#1969C2211yescharacteristic
31#1960—21no—
32#1943—21no—
33#1942C2212yes—
34#1846C2211yescharacteristic
35#1845C2211yescharacteristic
36#1809—21no—
37#1808C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1374C2211yescharacteristic
41#1374C2211yescharacteristic
42#1374C2212yes—
43C22×Q16—21no—
44C22×Q16C2311yescharacteristic
45C22×Q16C2311yescharacteristic
46C22×M16D411yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4⋊Q8—21no—
49C2×#125—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×#117D411yescharacteristic
53C22×C16C2312yes—
54C22×C16C2312yes—
55#499—42no—
56#494—42no—
57#470—44no—
58#467—42no—
59#467—42no—
60C2×#107—21no—
61C2×#107C2312yes—
62C2×#107C2312yes—
63C2×#107C2311yescharacteristic
64C2×#107C2311yescharacteristic
65C2×#106—21no—
66C2×#106—21no—
67C2×#103—21no—
68C2×#103—21no—
69C2×#103—21no—
70C2×#103—21no—
71#405—42no—
72C2×#97D411yescharacteristic
73C2×#97D411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#398—42no—
77#398—42no—
78#397—42no—
79#396—42no—
80#391—21no—
81#391—21no—
82#389—22no—
83#384—21no—
84#382—21no—
85#380—22no—
86#367—42no—
87#366—42no—
88#358—42no—
89#358—42no—
90#358—42no—
91#358—42no—
92C22⋊C16—42no—
93#340—22no—
94#340—22no—
95C22×Q8—21no—
96C22×Q8—21no—
97C23×C4C2×D411yescharacteristic
98C2×Q16—22no—
99C2×Q16—22no—
100C2×Q16—22no—
101C2×Q16—22no—
102C2×Q16—21no—
103C2×Q16—21no—
104C2×Q16—21no—
105C2×Q16—21no—
106C2×Q16—42no—
107C2×Q16—42no—
108C2×Q16—42no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—41no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C22×C8C2×D411yescharacteristic
119C22×C8C2411yescharacteristic
120C4⋊Q8—42no—
121C4⋊Q8—42no—
122#125—42no—
123#125—42no—
124C22⋊Q8—42no—
125C22⋊Q8—42no—
126C22⋊Q8—42no—
127C22⋊Q8—42no—
128#117—41no—
129#117—41no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4D812yes—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4D812yes—
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C16—22no—
148C2×C16—22no—
149C2×C16—22no—
150C2×C16—22no—
151C2×C16—22no—
152C2×C16—22no—
153#107—22no—
154#107—22no—
155#107—22no—
156#107—22no—
157#107—22no—
158#107—22no—
159#107—42no—
160#106—42no—
161#106—42no—
162#103—42no—
163#103—42no—
164#103—42no—
165#103—42no—
166#97—41no—
167#97—41no—
168C4×C8—22no—
169C4×C8—22no—
170#95—42no—
171#95—42no—
172#95—42no—
173#95—42no—
174#95—42no—
175C24C2×D811yescharacteristic
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4C2×D812yes—
201C22×C4C2×D812yes—
202C22×C4C2×D811yescharacteristic
203C22×C4C22×D411yescharacteristic
204Q16—42no—
205Q16—42no—
206Q16—42no—
207Q16—42no—
208Q16—84no—
209M16—44no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8AGL(1,ℤ/8)12yes—
220C2×C8#13712yes—
221C2×C8H(5,𝔽2)12yes—
222C2×C8H(5,𝔽2)12yes—
223C2×C82−1+411yescharacteristic
224C2×C82−1+411yescharacteristic
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—82no—
235C4⋊C4—82no—
236C4⋊C4—84no—
237C4⋊C4—84no—
238C4⋊C4—162no—
239C4⋊C4—162no—
240C22⋊C4—42no—
241C22⋊C4—82no—
242C22⋊C4—82no—
243C22⋊C4—162no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C16—42no—
249C16—42no—
250C23—22no—
251C23—22no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23C22×D811yescharacteristic
256Q8—81no—
257Q8—81no—
258Q8—81no—
259Q8—81no—
260Q8—81no—
261Q8—81no—
262Q8—81no—
263Q8—81no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4#53411yescharacteristic
294C2×C4#53511yescharacteristic
295C2×C4#53611yescharacteristic
296C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
297C2×C4C2×#13711yescharacteristic
298C2×C4#57511yescharacteristic
299C8—22no—
300C8—22no—
301C8—41no—
302C8#56212yes—
303C8#56212yes—
304C22—41no—
305C22—42no—
306C22—42no—
307C22—41no—
308C22—41no—
309C22#318611yescharacteristic
310C22#320511yescharacteristic
311C22#322611yescharacteristic
312C22#341612yes—
313C22#341712yes—
314C4—41no—
315C4—42no—
316C4—42no—
317C4—161no—
318C4—161no—
319C4—162no—
320C4—162no—
321C4—161no—
322C4—161no—
323C4#281611yescharacteristic
324C4#319311yescharacteristic
325C4#321611yescharacteristic
326C4#323511yescharacteristic
327C2—81no—
328C2#3277011yescharacteristic
329C2#3376712yes—
330C2#3378711yescharacteristic
331C2#3378911yescharacteristic
332C2#3380812yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13531C2semidirect
4. #13530C2semidirect
5. #13507C2semidirect
6. #13409C2semidirect
9. #13366C2semidirect
10. #13366C2semidirect

Maximal subgroups

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

subgroupindex
#332162
#331802
#135312
#135302
#135072
#134092
#133832
#133822
#133662
#128452