Finite Group Databasebeta

← #145307

Subgroups — #145307

Order 512.

286 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 70 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#12032C211yesmaximal, characteristic, complemented
3#11949C211yesmaximal, characteristic, complemented
4#11857C211yesmaximal, characteristic, complemented
5#11487C211yesmaximal, characteristic
6#8648C211yesmaximal, characteristic
7#7217C211yesmaximal, characteristic, complemented
8#3170C2211yescharacteristic
9#3077—21no—
10C2×C8×D4C2211yescharacteristic
11#2162C2211yescharacteristic
12#2162C2211yescharacteristic
13#1834C2211yescharacteristic
14#1802—21no—
15#1683—22no—
16#1675—21no—
17#1669—21no—
18#1567C2211yescharacteristic
19C2×#335C2211yescharacteristic
20#1550—21no—
21#1371—22no—
22#1365—21no—
23#1362—21no—
24#1348—22no—
25#1314—22no—
26#1296—21no—
27C23×C8—21no—
28#533—21no—
29C2×C4×D4D411yescharacteristic
30#500D411yescharacteristic
31#494D411yescharacteristic
32#487—21no—
33#478—42no—
34#476—22no—
35#474—22no—
36#467—21no—
37#467—21no—
38C4×Q16—21no—
39C4×Q16—21no—
40C8×D4—21no—
41C8×D4—21no—
42C8×D4—21no—
43C8×D4—21no—
44#428—41no—
45#423—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8D411yescharacteristic
48#418—41no—
49C2×#102—41no—
50C2×C22⋊C8—21no—
51C2×C4×C8C2311yescharacteristic
52#384—21no—
53C4⋊C16—21no—
54C4⋊C16—21no—
55C22⋊C16—21no—
56C22⋊C16D412yes—
57C4×C16—21no—
58C4×C16—21no—
59#340—21no—
60#340—21no—
61#340—41no—
62#335—21no—
63#335—21no—
64#335—21no—
65#335D412yes—
66#333—22no—
67#332—22no—
68C22×D4D811yescharacteristic
69C23×C4—21no—
70C2×Q16—21no—
71C2×Q16—21no—
72C2×M16—41no—
73C22×C8—21no—
74C22×C8—41no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8—82no—
78C22×C8C2×D411yescharacteristic
79C22×C8#4111yescharacteristic
80C4⋊Q8—21no—
81C4⋊Q8—21no—
82#125—41no—
83C22⋊Q8—41no—
84C22⋊Q8—41no—
85C4×Q8—41no—
86C4×Q8—41no—
87C4×D4—21no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4—21no—
91#117—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4D811yescharacteristic
96C2×C22⋊C4—21no—
97C2×C42C2×D411yescharacteristic
98C2×C16—22no—
99C2×C16—41no—
100#107—21no—
101#107—21no—
102#107—42no—
103#106—42no—
104C4⋊C8—21no—
105C4⋊C8—22no—
106C4⋊C8—21no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110#103—41no—
111#103—41no—
112#102—42no—
113#102—42no—
114#102—82no—
115C22⋊C8—41no—
116C22⋊C8—41no—
117C4×C8C2×D411yescharacteristic
118C4×C8C2×D411yescharacteristic
119C4×C8#4111yescharacteristic
120C4×C8#4111yescharacteristic
121C24—21no—
122C2×Q8—41no—
123C2×Q8—41no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4Q3212yes—
130C22×C4—21no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—81no—
134C22×C4—81no—
135C22×C4—81no—
136C22×C4#12111yescharacteristic
137C22×C4C2×D811yescharacteristic
138C22×C4#13511yescharacteristic
139Q16—81no—
140Q16—81no—
141M16—81no—
142M16—81no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—42no—
147C2×C8—82no—
148C2×C8—82no—
149C2×C8—81no—
150C2×C8—81no—
151C2×C8—81no—
152C2×C8—81no—
153C2×C8#12111yescharacteristic
154C2×C8#12111yescharacteristic
155C2×C8#12311yescharacteristic
156C2×C8#12411yescharacteristic
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—161no—
170C22⋊C4—41no—
171C22⋊C4—41no—
172C22⋊C4—81no—
173C42—81no—
174C42#12011yescharacteristic
175C42#12311yescharacteristic
176C42AGL(1,ℤ/8)11yescharacteristic
177C42#13711yescharacteristic
178C16—42no—
179C16—81no—
180C23—22no—
181C23—41no—
182C23—41no—
183C23—81no—
184C23—81no—
185C23#46611yescharacteristic
186Q8—81no—
187Q8—81no—
188Q8—161no—
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—82no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—161no—
204C2×C4—161no—
205C2×C4—161no—
206C2×C4#39411yescharacteristic
207C2×C4#44711yescharacteristic
208C2×C4#45211yescharacteristic
209C2×C4#46811yescharacteristic
210C2×C4#48011yescharacteristic
211C2×C4#48211yescharacteristic
212C2×C4C2×Q3211yescharacteristic
213C2×C4#50811yescharacteristic
214C2×C4#50911yescharacteristic
215C2×C4#51011yescharacteristic
216C8—21no—
217C8—21no—
218C8—21no—
219C8—21no—
220C8—81no—
221C8—82no—
222C8—161no—
223C22—21no—
224C22—21no—
225C22—42no—
226C22—82no—
227C22—82no—
228C22—81no—
229C22—81no—
230C22#203311yescharacteristic
231C22#221611yescharacteristic
232C22#222011yescharacteristic
233C4—21no—
234C4—21no—
235C4—21no—
236C4—81no—
237C4—161no—
238C4—161no—
239C4—321no—
240C4#204811yescharacteristic
241C4#205311yescharacteristic
242C4#218611yescharacteristic
243C4#218811yescharacteristic
244C4#223611yescharacteristic
245C4#223711yescharacteristic
246C2—21no—
247C2—21no—
248C2—82no—
249C2#1046411yescharacteristic
250C2#1338311yescharacteristic
251C2#1339111yescharacteristic
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. #12032C2semidirect
3. #11949C2semidirect
4. #11857C2semidirect
7. #7217C2semidirect

Maximal subgroups

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

subgroupindex
#192562
#120322
#119492
#118572
#114872
#86482
#72172