Finite Group Databasebeta

← #145266

Subgroups — #145266

Order 512.

287 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 72 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#19256C211yesmaximal, characteristic
2#12023C211yesmaximal, characteristic, complemented
3#11983C211yesmaximal, characteristic, complemented
4#11672C211yesmaximal, characteristic, complemented
5C2×D4⋊C16C211yesmaximal, characteristic
6#8616C211yesmaximal, characteristic
7#7241C211yesmaximal, characteristic, complemented
8#3170C2211yescharacteristic
9#3077—21no—
10C2×C8×D4C2211yescharacteristic
11C2×C4⋊C16C2211yescharacteristic
12C2×C4×C16C2211yescharacteristic
13#1800—21no—
14#1683—22no—
15#1675—21no—
16#1669—21no—
17#1567C2211yescharacteristic
18#1567C2211yescharacteristic
19#1567C2211yescharacteristic
20#1550—21no—
21#1372—22no—
22#1365—21no—
23#1362—21no—
24#1332—21no—
25#1332—21no—
26D4⋊C16—21no—
27D4⋊C16—21no—
28#1292—21no—
29C23×C8—21no—
30#533—21no—
31C2×C4×D4D411yescharacteristic
32C22×C16—21no—
33C22×C16—21no—
34#500D411yescharacteristic
35#494D411yescharacteristic
36#487—21no—
37#478—42no—
38#476—22no—
39#474—22no—
40#467—21no—
41#467—21no—
42C4×Q16—21no—
43C4×Q16—21no—
44C8×D4—21no—
45C8×D4—21no—
46C8×D4—21no—
47C8×D4—21no—
48#428—41no—
49#423—21no—
50#423—21no—
51#423—21no—
52C2×C4⋊C8D411yescharacteristic
53#418—41no—
54C2×#100—41no—
55C2×C22⋊C8—21no—
56C2×C4×C8C2311yescharacteristic
57C4⋊C16—21no—
58C4⋊C16—21no—
59C4×C16—21no—
60C4×C16D411yescharacteristic
61C4×C16D411yescharacteristic
62#343—41no—
63#335—21no—
64#335—21no—
65#335—21no—
66#335—21no—
67#335—21no—
68#335D412yes—
69#333—22no—
70#332—22no—
71C22×D4D811yescharacteristic
72C23×C4—21no—
73C2×Q16—21no—
74C2×Q16—21no—
75C2×M16—41no—
76C2×M16—41no—
77C2×M16—41no—
78C22×C8—21no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—81no—
82C22×C8—81no—
83C22×C8#4111yescharacteristic
84C22×C8#4111yescharacteristic
85C4⋊Q8—21no—
86C4⋊Q8—21no—
87#125—41no—
88C22⋊Q8—41no—
89C22⋊Q8—41no—
90C4×Q8—41no—
91C4×Q8—41no—
92C4×D4—21no—
93C4×D4—21no—
94C4×D4—21no—
95C4×D4—21no—
96#117—41no—
97C2×C4⋊C4D811yescharacteristic
98C2×C22⋊C4—21no—
99C2×C42C2×D411yescharacteristic
100C2×C16—21no—
101C2×C16—21no—
102C2×C16—42no—
103C2×C16—41no—
104C2×C16—81no—
105#107—21no—
106#107—21no—
107#107—42no—
108#106—42no—
109C4⋊C8—21no—
110C4⋊C8—22no—
111C4⋊C8—21no—
112C4⋊C8—41no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117#103—41no—
118#103—41no—
119#102—42no—
120#102—42no—
121#100—82no—
122C22⋊C8—41no—
123C22⋊C8—41no—
124C4×C8C2×D411yescharacteristic
125C4×C8C2×D411yescharacteristic
126C4×C8C2×D411yescharacteristic
127C4×C8#4111yescharacteristic
128C24—21no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4SD3212yes—
137C22×C4—21no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—81no—
141C22×C4—81no—
142C22×C4#12311yescharacteristic
143C22×C4C2×D811yescharacteristic
144C22×C4#13711yescharacteristic
145Q16—41no—
146Q16—41no—
147Q16—81no—
148M16—81no—
149M16—81no—
150M16—81no—
151M16—81no—
152M16—81no—
153M16—81no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—42no—
159C2×C8—82no—
160C2×C8—81no—
161C2×C8—81no—
162C2×C8—81no—
163C2×C8—81no—
164C2×C8—81no—
165C2×C8#12111yescharacteristic
166C2×C8#12111yescharacteristic
167C2×C8#12311yescharacteristic
168C2×C8#12411yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C22⋊C4—41no—
178C22⋊C4—41no—
179C22⋊C4—81no—
180C42—81no—
181C42#12011yescharacteristic
182C42#12111yescharacteristic
183C42#13511yescharacteristic
184C42AGL(1,ℤ/8)11yescharacteristic
185C16—22no—
186C16—41no—
187C16—81no—
188C23—22no—
189C23—41no—
190C23—41no—
191C23—81no—
192C23—81no—
193C23#48011yescharacteristic
194Q8—81no—
195Q8—81no—
196Q8—161no—
197D4—22no—
198D4—22no—
199D4—41no—
200D4—41no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—82no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4#39411yescharacteristic
212C2×C4#44711yescharacteristic
213C2×C4#45211yescharacteristic
214C2×C4#46611yescharacteristic
215C2×C4#46811yescharacteristic
216C2×C4#48211yescharacteristic
217C2×C4C2×SD3211yescharacteristic
218C2×C4#50811yescharacteristic
219C2×C4#51011yescharacteristic
220C2×C4#51011yescharacteristic
221C8—21no—
222C8—21no—
223C8—21no—
224C8—81no—
225C8—82no—
226C8—161no—
227C8—161no—
228C8#47111yescharacteristic
229C8#47211yescharacteristic
230C22—21no—
231C22—21no—
232C22—42no—
233C22—82no—
234C22—82no—
235C22—81no—
236C22—81no—
237C22#203311yescharacteristic
238C22#223511yescharacteristic
239C22#223711yescharacteristic
240C4—21no—
241C4—21no—
242C4—21no—
243C4—81no—
244C4—161no—
245C4—161no—
246C4#205011yescharacteristic
247C4#205111yescharacteristic
248C4#218511yescharacteristic
249C4#218911yescharacteristic
250C4#221711yescharacteristic
251C4#221911yescharacteristic
252C2—21no—
253C2—21no—
254C2—82no—
255C2#1045611yescharacteristic
256C2#1338211yescharacteristic
257C2#1339211yescharacteristic
subgroupG/H
center, socleC22#2033
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4D8
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12023C2semidirect
3. #11983C2semidirect
4. #11672C2semidirect
7. #7241C2semidirect

Maximal subgroups

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

subgroupindex
#192562
#120232
#119832
#116722
C2×D4⋊C162
#86162
#72412