Finite Group Databasebeta

← #491306

Subgroups — #491306

Order 512.

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

normal subgroups only

This group has 417 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#33772C211yesmaximal, characteristic
2#33500C211yesmaximal, characteristic, complemented
3#19344C211yesmaximal, characteristic
4#19204C211yesmaximal, characteristic, complemented
5#12481C212yesmaximal, complemented
6#12477C212yesmaximal, complemented
7#12444C212yesmaximal, complemented
8#12441C212yesmaximal, complemented
9#12036C211yesmaximal, characteristic, complemented
10#11962C211yesmaximal, characteristic, complemented
11#11948C211yesmaximal, characteristic
12#3254—21no—
13#3229—21no—
14#3169C2211yescharacteristic
15C2×#494C2211yescharacteristic
16#2960C2211yescharacteristic
17#2248C2212yes—
18#2241C2212yes—
19#2208—21no—
20#2208—21no—
21#2185—22no—
22#2183—21no—
23#2183C2212yes—
24Q8×C16—21no—
25Q8×C16C2212yes—
26#2164—22no—
27#2162C2211yescharacteristic
28#2162C2211yescharacteristic
29#1698—22no—
30#1690—22no—
31#1676C2212yes—
32#1670C2212yes—
33#1670C2212yes—
34#1668C2212yes—
35#1567C2211yescharacteristic
36#1564C2211yescharacteristic
37#1560—21no—
38#1546—21no—
39#1372C2212yes—
40#1371C2212yes—
41#1365C2211yescharacteristic
42#1365C2211yescharacteristic
43#1362C2212yes—
44#1362C2212yes—
45#1361C2211yescharacteristic
46#1361C2211yescharacteristic
47C22×Q16—21no—
48C22×Q16—21no—
49#533D411yescharacteristic
50#532—21no—
51C2×C4⋊Q8—21no—
52C2×SD32—22no—
53C2×SD32—22no—
54#500C2311yescharacteristic
55#498—41no—
56#494—21no—
57#494—21no—
58#494—21no—
59#494C2312yes—
60#493C2311yescharacteristic
61#487—41no—
62#486—21no—
63#483—42no—
64#477—22no—
65#475—22no—
66#475—22no—
67#474—41no—
68#468—42no—
69#466—21no—
70#466—21no—
71#461—41no—
72#460—22no—
73#446—21no—
74#446D412yes—
75C8×Q8—21no—
76C8×Q8C2312yes—
77C4×D8—21no—
78C4×D8—21no—
79#431—21no—
80#424—21no—
81#423—21no—
82#423D411yescharacteristic
83#405—21no—
84C2×C4×C8C2311yescharacteristic
85C4⋊C16—22no—
86C4⋊C16—22no—
87C4⋊C16D411yescharacteristic
88C4⋊C16D411yescharacteristic
89C4⋊C16C2311yescharacteristic
90C4⋊C16C2311yescharacteristic
91#358—22no—
92#358—22no—
93C22⋊C16—21no—
94C22⋊C16D412yes—
95#346—22no—
96C4×C16—22no—
97C4×C16—21no—
98C4×C16C2312yes—
99#335—21no—
100#335C2311yescharacteristic
101#335C2311yescharacteristic
102#335C2312yes—
103#334—21no—
104#333—22no—
105#332—22no—
106#332—22no—
107#331—22no—
108C2×#41—41no—
109C22×Q8—41no—
110C2×Q16—22no—
111C2×Q16—22no—
112C2×Q16—41no—
113C2×Q16—41no—
114C2×Q16—81no—
115C2×Q16—81no—
116C2×Q16—82no—
117C2×SD16—42no—
118C2×D8—21no—
119C2×D8—21no—
120C2×M16—21no—
121C2×M16—41no—
122C22×C8C2×D411yescharacteristic
123C22×C8#4111yescharacteristic
124C4⋊Q8—22no—
125C4⋊Q8—21no—
126C4⋊Q8—41no—
127C4⋊Q8—41no—
128C4⋊Q8—41no—
129C4⋊Q8D811yescharacteristic
130C4⋊Q8D811yescharacteristic
131#127—21no—
132#125—21no—
133#124—41no—
134C22⋊Q8—21no—
135C22⋊Q8D812yes—
136#121—41no—
137#121—41no—
138C4×Q8—21no—
139C4×Q8C2×D412yes—
140C4×D4—41no—
141C4×D4—41no—
142#117—21no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C42C2×D411yescharacteristic
146SD32—82no—
147SD32—82no—
148C2×C16—21no—
149C2×C16—21no—
150C2×C16C2×D412yes—
151C2×C16C2×D412yes—
152#107—21no—
153#107—21no—
154#107—42no—
155#106—42no—
156#106—42no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—21no—
160C4⋊C8—22no—
161C4⋊C8—21no—
162C4⋊C8—21no—
163C4⋊C8—41no—
164C4⋊C8C2×D412yes—
165#103—42no—
166#103—42no—
167#103—42no—
168#102—41no—
169#102—41no—
170#102—42no—
171C22⋊C8—41no—
172#97—22no—
173C4×C8—22no—
174C4×C8—41no—
175C4×C8C2×D411yescharacteristic
176C4×C8C2×D411yescharacteristic
177C4×C8#4111yescharacteristic
178C4×C8C2411yescharacteristic
179#41—162no—
180C2×Q8—21no—
181C2×Q8—42no—
182C2×Q8—41no—
183C2×Q8—82no—
184C2×Q8—82no—
185C2×Q8—81no—
186C2×Q8C2×D812yes—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4#12111yescharacteristic
194C22×C4C2×D811yescharacteristic
195C22×C4#13511yescharacteristic
196Q16—42no—
197Q16—42no—
198Q16—82no—
199Q16—82no—
200Q16—82no—
201SD16—162no—
202D8—41no—
203D8—41no—
204D8—81no—
205M16—81no—
206M16—161no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—22no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8#12111yescharacteristic
215C2×C8#12111yescharacteristic
216C2×C8C22×D411yescharacteristic
217C2×C8C2×#4111yescharacteristic
218C4⋊C4—22no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—22no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—42no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4C2×D812yes—
230C22⋊C4—22no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C42—22no—
234C42#12111yescharacteristic
235C42C2×D811yescharacteristic
236C42#13511yescharacteristic
237C42C22×D411yescharacteristic
238C16—22no—
239C16—22no—
240C16—41no—
241C16—41no—
242C23—81no—
243C23#46611yescharacteristic
244Q8—22no—
245Q8—22no—
246Q8—42no—
247Q8—82no—
248Q8—82no—
249Q8—161no—
250Q8—162no—
251D4—81no—
252D4—81no—
253D4—81no—
254D4—81no—
255D4—161no—
256D4—162no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—81no—
265C2×C4—82no—
266C2×C4—162no—
267C2×C4—162no—
268C2×C4#46611yescharacteristic
269C2×C4#46611yescharacteristic
270C2×C4#46611yescharacteristic
271C2×C4C2×#12111yescharacteristic
272C2×C4C22×D811yescharacteristic
273C2×C4C2×#13511yescharacteristic
274C8—21no—
275C8—21no—
276C8—21no—
277C8—42no—
278C8—81no—
279C8—82no—
280C8—81no—
281C8#57711yescharacteristic
282C8#57711yescharacteristic
283C22—21no—
284C22—21no—
285C22—21no—
286C22—161no—
287C22—161no—
288C22—161no—
289C22C2×#46611yescharacteristic
290C4—21no—
291C4—21no—
292C4—21no—
293C4—42no—
294C4—42no—
295C4—161no—
296C4—162no—
297C4#222312yes—
298C4#304311yescharacteristic
299C4#306811yescharacteristic
300C4#341611yescharacteristic
301C4#341711yescharacteristic
302C2—41no—
303C2—321no—
304C2#1939811yescharacteristic
305C2#3361311yescharacteristic
306C2C2×#222311yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radical#3169C22
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33500C2semidirect
4. #19204C2semidirect
5. #12481C2semidirect
6. #12477C2semidirect
7. #12444C2semidirect
8. #12441C2semidirect
9. #12036C2semidirect
10. #11962C2semidirect

Maximal subgroups

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

subgroupindex
#337722
#335002
#193442
#192042
#124812
#124772
#124442
#124412
#120362
#119622
#119482