Finite Group Databasebeta

← #510057

Subgroups — #510057

Order 512.

492 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 136 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#33631C212yesmaximal, complemented
2#26501C212yesmaximal, complemented
3#13478C212yesmaximal, complemented
4#13470C212yesmaximal, complemented
5#13456C212yesmaximal, complemented
6#13239C211yesmaximal, characteristic
7#13170C211yesmaximal, characteristic, complemented
8#13162C211yesmaximal, characteristic, complemented
9#13017C212yesmaximal
10C22×D16C2211yescharacteristic
11#3164C2212yes—
12#3081—22no—
13#2948C2212yes—
14#2228—22no—
15#2228—22no—
16#2226—22no—
17#2224—22no—
18C2×#368C2211yescharacteristic
19#2159C2212yes—
20#2158C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#364C2211yescharacteristic
23#2151C2212yes—
24C2×#361C2212yes—
25#2140C2212yes—
26C2×#357C2212yes—
27C2×#346C2211yescharacteristic
28#2063—22no—
29#2027—22no—
30#1978C2212yes—
31#1970C2212yes—
32#1952C2212yes—
33#1945C2212yes—
34#1856—22no—
35#1772C2211yescharacteristic
36#1387C2212yes—
37#1382C2212yes—
38#1379C2212yes—
39C22×D8C2312yes—
40#544—22no—
41#518—22no—
42C2×D16—41no—
43C2×D16—41no—
44C2×D16Q814yescomplemented
45C2×M32C2312yes—
46C22×C16C2311yescharacteristic
47C22×C16C2311yescharacteristic
48#501—42no—
49#500—22no—
50#499—22no—
51#478—42no—
52#478—42no—
53#476—42no—
54#474—42no—
55#442—42no—
56C4×D8—22no—
57C4×D8—22no—
58C2×#108D411yescharacteristic
59C2×#108C2312yes—
60#428—22no—
61#428—22no—
62#427D411yescharacteristic
63#427C2312yes—
64C2×#107C2312yes—
65#424—22no—
66#418—22no—
67#418—22no—
68C2×#102—22no—
69C2×#100—22no—
70#405C2312yes—
71C2×C4×C8C2311yescharacteristic
72#390—22no—
73#368—41no—
74#367—22no—
75#367—41no—
76#367D411yescharacteristic
77#367D411yescharacteristic
78#366—22no—
79#366D412yes—
80#364—41no—
81C4⋊C16—22no—
82C4⋊C16—22no—
83#361—42no—
84#357—22no—
85#357—22no—
86#357—22no—
87#357—22no—
88#346—41no—
89#343—22no—
90#339—22no—
91#339D412yes—
92C22×D4—22no—
93C2×D8—22no—
94C2×D8—22no—
95C2×D8—22no—
96C2×D8—22no—
97C2×D8—22no—
98C2×D8C2×Q814yes—
99C2×M16—22no—
100C2×M16C2×D412yes—
101C22×C8C2×D411yescharacteristic
102C22×C8C2411yescharacteristic
103C4⋊Q8—42no—
104C4⋊Q8—42no—
105#125—42no—
106#125—42no—
107C22⋊Q8—82no—
108C4×D4—42no—
109C4×D4—42no—
110C4×D4—42no—
111C4×D4—42no—
112#117—22no—
113#117C2×D412yes—
114C2×C4⋊C4—22no—
115C2×C22⋊C4—82no—
116C2×C22⋊C4—82no—
117C2×C42C2×D411yescharacteristic
118D16—44no—
119M32—22no—
120M32—22no—
121C2×C16—21no—
122C2×C16—21no—
123C2×C16—21no—
124C2×C16—21no—
125C2×C16—22no—
126C2×C16C2×D411yescharacteristic
127C2×C16C2×D411yescharacteristic
128C2×C16C2×Q812yes—
129#108—22no—
130#108—22no—
131#108—41no—
132#107—22no—
133#107—22no—
134#107—42no—
135#107—42no—
136#107D812yes—
137#107C2×D412yes—
138#107C2×D412yes—
139#106—22no—
140#106—42no—
141#106—42no—
142#106D811yescharacteristic
143#106D811yescharacteristic
144C4⋊C8—42no—
145C4⋊C8—42no—
146#102—42no—
147#102—42no—
148#102—42no—
149#102—42no—
150#102—42no—
151#102—42no—
152#100—44no—
153#97—22no—
154C4×C8—22no—
155C4×C8—21no—
156C4×C8—21no—
157C24—42no—
158C2×Q8—82no—
159C2×D4—24no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C22×C4—21no—
166C22×C4—42no—
167C22×C4—82no—
168C22×C4—82no—
169C22×C4C22×D411yescharacteristic
170D8—24no—
171D8—24no—
172D8—42no—
173D8—42no—
174M16—22no—
175M16—22no—
176M16—42no—
177M16—42no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—22no—
181C2×C8—42no—
182C2×C8C2×D811yescharacteristic
183C2×C8C2×D811yescharacteristic
184C2×C8C22×D411yescharacteristic
185C2×C8C22×Q811yescharacteristic
186C2×C8H(5,𝔽2)11yescharacteristic
187C2×C8H(5,𝔽2)11yescharacteristic
188C2×C82−1+411yescharacteristic
189C2×C82−1+411yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C4⋊C4C2×D812yes—
198C4⋊C4C2×D812yes—
199C22⋊C4—22no—
200C22⋊C4—42no—
201C22⋊C4—82no—
202C22⋊C4—82no—
203C22⋊C4—164no—
204C42—22no—
205C42—21no—
206C42—21no—
207C42—42no—
208C16—21no—
209C16—21no—
210C16—22no—
211C16—42no—
212C23—44no—
213C23—82no—
214C23—82no—
215C23—162no—
216C23—162no—
217C23#57611yescharacteristic
218Q8—322no—
219D4—44no—
220D4—44no—
221D4—82no—
222D4—82no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—82no—
230C2×C4—162no—
231C2×C4—322no—
232C2×C4—322no—
233C2×C4#53711yescharacteristic
234C2×C4D4211yescharacteristic
235C2×C4D4×Q811yescharacteristic
236C2×C4C22×D811yescharacteristic
237C2×C4#57511yescharacteristic
238C2×C4#57811yescharacteristic
239C8—21no—
240C8—21no—
241C8—42no—
242C8—82no—
243C8#53711yescharacteristic
244C8#54311yescharacteristic
245C8D4×Q811yescharacteristic
246C8#56311yescharacteristic
247C22—22no—
248C22—84no—
249C22—164no—
250C22—162no—
251C22—162no—
252C22—322no—
253C22#326211yescharacteristic
254C22D4×D811yescharacteristic
255C22#338911yescharacteristic
256C4—42no—
257C4—42no—
258C4—81no—
259C4—82no—
260C4—82no—
261C4—162no—
262C4#268311yescharacteristic
263C4#324511yescharacteristic
264C4#327711yescharacteristic
265C4Q8×D811yescharacteristic
266C2—21no—
267C2—21no—
268C2—164no—
269C2#3123411yescharacteristic
270C2#3385412yes—
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1772 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33631C2semidirect
2. #26501C2semidirect
3. #13478C2semidirect
4. #13470C2semidirect
5. #13456C2semidirect
7. #13170C2semidirect
8. #13162C2semidirect
44. C2×D16Q8semidirect

Maximal subgroups

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

subgroupindex
#336312
#265012
#134782
#134702
#134562
#132392
#131702
#131622
#130172