Finite Group Databasebeta

← #138206

Subgroups — #138206

Order 512.

454 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 114 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#33172C211yesmaximal, characteristic
2C2×#1568C211yesmaximal, characteristic
3#12783C211yesmaximal, characteristic
4C2×#1362C212yesmaximal, complemented
5#7274C212yesmaximal, complemented
6C2×#494C2211yescharacteristic
7C2×#467—21no—
8C2×#467C412yes—
9C2×#432—21no—
10C22×C4×C8C2211yescharacteristic
11#2857—21no—
12C2×C4⋊C16C2212yes—
13C2×C22⋊C16—21no—
14#1969—21no—
15#1969C412yes—
16#1956—21no—
17#1845C2211yescharacteristic
18#1568—24no—
19C2×#335C2212yes—
20C2×#334—21no—
21#1374—21no—
22#1362—24no—
23C22×Q16—21no—
24C22×Q16C2×C412yes—
25C23×C8C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C4⋊Q8—21no—
28C2×C22⋊Q8—22no—
29C22×C42D411yescharacteristic
30C22×C16—22no—
31#494—22no—
32#494—22no—
33#467—44no—
34#467—44no—
35#432—24no—
36#432—42no—
37C2×#107—21no—
38C2×#107C2×C412yes—
39C2×C4⋊C8—22no—
40C2×#103—22no—
41C2×#103—22no—
42C2×C22⋊C8—41no—
43C2×#97—41no—
44C2×C4×C8—22no—
45C2×C4×C8—22no—
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8C2311yescharacteristic
50#391—22no—
51#389—41no—
52#389—41no—
53#384—21no—
54#380—41no—
55#380—41no—
56C4⋊C16—22no—
57C4⋊C16—22no—
58C22⋊C16—42no—
59#335—22no—
60#335D414yes—
61#334—24no—
62C22×Q8—22no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4D811yescharacteristic
65C23×C4SD1611yescharacteristic
66C2×Q16—24no—
67C2×Q16—42no—
68C2×Q16—42no—
69C2×Q16—42no—
70C2×Q16—41no—
71C2×Q16—42no—
72C2×Q16—41no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—21no—
77C22×C8—22no—
78C22×C8—21no—
79C22×C8—41no—
80C22×C8—42no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22×C411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C4⋊Q8—41no—
86C4⋊Q8—41no—
87C4⋊Q8—41no—
88C4⋊Q8—41no—
89C22⋊Q8—44no—
90C22⋊Q8—44no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C22⋊C4—42no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C22⋊C411yescharacteristic
103C2×C42D811yescharacteristic
104C2×C42SD1611yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C16—42no—
107C2×C16—42no—
108C2×C16—42no—
109#107—42no—
110#107—42no—
111C4⋊C8—24no—
112C4⋊C8—42no—
113#103—44no—
114#103—44no—
115C22⋊C8—44no—
116#97—44no—
117C4×C8—22no—
118C4×C8—22no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8C2×D412yes—
122C4×C8#4112yes—
123#95—42no—
124#95—42no—
125C24#10211yescharacteristic
126C2×Q8—42no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4#10211yescharacteristic
149C22×C4#10211yescharacteristic
150C22×C4#10211yescharacteristic
151C22×C4C2×C22⋊C411yescharacteristic
152C22×C4C2×D811yescharacteristic
153C22×C4C2×SD1611yescharacteristic
154Q16—84no—
155Q16—82no—
156Q16—82no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—44no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—44no—
168C2×C8—82no—
169C2×C8—82no—
170C2×C8#10012yes—
171C2×C8C4×D412yes—
172C2×C8#12112yes—
173C2×C8AGL(1,ℤ/8)12yes—
174C2×C8AGL(1,ℤ/8)11yescharacteristic
175C2×C8#13711yescharacteristic
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—82no—
179C4⋊C4—84no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C4⋊C4—82no—
185C22⋊C4—84no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42#12011yescharacteristic
193C42#12311yescharacteristic
194C42AGL(1,ℤ/8)11yescharacteristic
195C42#13711yescharacteristic
196C16—82no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23#36212yes—
204C23C2×#10211yescharacteristic
205Q8—82no—
206Q8—82no—
207Q8—82no—
208Q8—82no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4#38611yescharacteristic
235C2×C4C2×#10011yescharacteristic
236C2×C4#41311yescharacteristic
237C2×C4#41711yescharacteristic
238C2×C4#41711yescharacteristic
239C2×C4#41811yescharacteristic
240C2×C4#44711yescharacteristic
241C2×C4#44811yescharacteristic
242C2×C4#48011yescharacteristic
243C2×C4#48111yescharacteristic
244C2×C4#50812yes—
245C2×C4#51012yes—
246C8—22no—
247C8—22no—
248C8—41no—
249C8—84no—
250C8#46812yes—
251C8#46912yes—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22#154511yescharacteristic
260C22#154711yescharacteristic
261C22#187411yescharacteristic
262C22#213811yescharacteristic
263C22#214111yescharacteristic
264C22C2×#36211yescharacteristic
265C22#214811yescharacteristic
266C4—21no—
267C4—21no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—162no—
272C4—162no—
273C4#155711yescharacteristic
274C4#156011yescharacteristic
275C4#189511yescharacteristic
276C4#190011yescharacteristic
277C4#218912yes—
278C4#223712yes—
279C2—42no—
280C2#914111yescharacteristic
281C2#1196311yescharacteristic
282C2#1196511yescharacteristic
283C2#1198411yescharacteristic
284C2#1198711yescharacteristic
285C2#1299411yescharacteristic
286C2#1300311yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1362C2semidirect
5. #7274C2semidirect

Maximal subgroups

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

subgroupindex
#331722
C2×#15682
#127832
C2×#13622
#72742