Finite Group Databasebeta

← #492400

Subgroups — #492400

Order 512.

394 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 126 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
1Q8×Q32C211yesmaximal, characteristic, complemented
2#33905C211yesmaximal, characteristic
3#19930C212yesmaximal
4#12521C211yesmaximal, characteristic, complemented
5#12500C212yesmaximal, complemented
6#12458C211yesmaximal, characteristic, complemented
7#12452C211yesmaximal, characteristic
8#12410C212yesmaximal, complemented
9#12398C212yesmaximal
10#12083C211yesmaximal, characteristic, complemented
11C16×SD16C211yesmaximal, characteristic
12#3385—22no—
13Q8×Q16C2212yes—
14#3338C2212yes—
15#2252—21no—
16#2252C2211yescharacteristic
17#2252C2211yescharacteristic
18#2252C2212yes—
19#2236—22no—
20#2227—22no—
21#2227C2212yes—
22#2227C2212yes—
23#2216—21no—
24Q8×C16C2211yescharacteristic
25C4×Q32—21no—
26C4×Q32C2211yescharacteristic
27D4×C16C2211yescharacteristic
28C2×#366—21no—
29#1713C2212yes—
30#1696—22no—
31#1676C2212yes—
32#1675C2212yes—
33#1633—22no—
34#1632—22no—
35#1591C2212yes—
36C8×SD16C2211yescharacteristic
37#1371C2211yescharacteristic
38#1367C2211yescharacteristic
39#1365C2212yes—
40#1365C2212yes—
41Q8⋊C16C2211yescharacteristic
42#1332C2212yes—
43D4⋊C16C2211yescharacteristic
44C8×C16C2211yescharacteristic
45Q82—22no—
46#554—22no—
47C2×Q32D411yescharacteristic
48C22×C16—21no—
49#501—42no—
50#500—22no—
51#500—22no—
52#500C2312yes—
53#500C2312yes—
54#494—22no—
55#484—42no—
56#479—42no—
57#477—22no—
58#477—22no—
59#477—22no—
60#477—42no—
61#476—22no—
62#474—42no—
63#467—22no—
64#446—22no—
65C8×Q8C2311yescharacteristic
66#440—42no—
67C4×Q16—22no—
68C4×Q16C2312yes—
69C4×SD16D411yescharacteristic
70C8×D4C2311yescharacteristic
71C2×#107—22no—
72#366—21no—
73#366—21no—
74#366—21no—
75#366—21no—
76#366—22no—
77#366—22no—
78#366—42no—
79#366D411yescharacteristic
80C4⋊C16—21no—
81C4⋊C16C2311yescharacteristic
82C4⋊C16C2311yescharacteristic
83#358—22no—
84#358—22no—
85#358D412yes—
86C22⋊C16—21no—
87C4×C16—21no—
88C4×C16C2311yescharacteristic
89C4×C16C2311yescharacteristic
90#335C2312yes—
91#334D411yescharacteristic
92#333—22no—
93#333—22no—
94Q8⋊C8—22no—
95Q8⋊C8D411yescharacteristic
96D4⋊C8D411yescharacteristic
97C82C2311yescharacteristic
98C2×Q16—82no—
99C2×Q16C2×D412yes—
100C2×SD16D811yescharacteristic
101C22×C8—21no—
102C4⋊Q8—22no—
103C4⋊Q8—22no—
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106C4⋊Q8—82no—
107#125—42no—
108C22⋊Q8—42no—
109C22⋊Q8—82no—
110C4×Q8—22no—
111C4×Q8—42no—
112C4×Q8—42no—
113C4×Q8C2×D411yescharacteristic
114C4×D4—42no—
115C4×D4C2×D411yescharacteristic
116C2×C4⋊C4—42no—
117Q32—22no—
118C2×C16—21no—
119C2×C16—21no—
120C2×C16—21no—
121C2×C16—42no—
122C2×C16C2×D411yescharacteristic
123C2×C16C2×D411yescharacteristic
124#107—22no—
125#107—22no—
126#107—22no—
127#107—22no—
128#107—42no—
129#107—42no—
130#107—42no—
131#107—42no—
132#107—42no—
133#107C2×D412yes—
134#106—42no—
135#106D811yescharacteristic
136C4⋊C8—21no—
137C4⋊C8—22no—
138C4⋊C8—42no—
139C4⋊C8C2×D411yescharacteristic
140C4⋊C8C2×D411yescharacteristic
141#103—22no—
142#103—42no—
143#103—42no—
144#103—42no—
145#103D811yescharacteristic
146#102—42no—
147#102D811yescharacteristic
148C22⋊C8—21no—
149#97—42no—
150C4×C8—21no—
151C4×C8C2×D411yescharacteristic
152C4×C8C2×D411yescharacteristic
153C4×C8C2411yescharacteristic
154C2×Q8—22no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—82no—
158C2×Q8C2×D811yescharacteristic
159C2×D4C2×D811yescharacteristic
160C22×C4—21no—
161C22×C4—82no—
162Q16—22no—
163Q16—22no—
164Q16—164no—
165SD16Q3214yescomplemented
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—42no—
172C2×C8C2×D811yescharacteristic
173C2×C8C2×D811yescharacteristic
174C2×C8C22×D411yescharacteristic
175C2×C8H(5,𝔽2)11yescharacteristic
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—82no—
184C4⋊C4—82no—
185C4⋊C4—82no—
186C4⋊C4—82no—
187C4⋊C4C2×D811yescharacteristic
188C4⋊C4C2×D811yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—82no—
191C42—21no—
192C42—42no—
193C42—82no—
194C42C22×D411yescharacteristic
195C16—22no—
196C16—21no—
197C16—41no—
198C16—41no—
199C16#13712yes—
200C23—21no—
201Q8—21no—
202Q8—42no—
203Q8—42no—
204Q8—162no—
205Q8—162no—
206Q8—322no—
207Q8C2×Q3212yes—
208D4—21no—
209D4C2×Q3212yes—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—322no—
220C2×C4D4211yescharacteristic
221C2×C4C22×D811yescharacteristic
222C2×C4#57611yescharacteristic
223C8—22no—
224C8—21no—
225C8—21no—
226C8—41no—
227C8—41no—
228C8—82no—
229C8C2×Q3212yes—
230C8C2×#13711yescharacteristic
231C8#57811yescharacteristic
232C22—22no—
233C22—41no—
234C22—41no—
235C22D4×D811yescharacteristic
236C4—22no—
237C4—21no—
238C4—41no—
239C4—41no—
240C4—82no—
241C4—82no—
242C4—162no—
243C4—162no—
244C4#327611yescharacteristic
245C4#329311yescharacteristic
246C4C22×Q3211yescharacteristic
247C4#341711yescharacteristic
248C2—42no—
249C2#1988711yescharacteristic
250C2D4×Q3211yescharacteristic
251C2#3385811yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radicalC2×D4C2×D8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×SD16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. Q8×Q32C2semidirect
4. #12521C2semidirect
5. #12500C2semidirect
6. #12458C2semidirect
8. #12410C2semidirect
10. #12083C2semidirect
165. SD16Q32semidirect

Maximal subgroups

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

subgroupindex
Q8×Q322
#339052
#199302
#125212
#125002
#124582
#124522
#124102
#123982
#120832
C16×SD162