Finite Group Databasebeta

← #139028

Subgroups — #139028

Order 512.

352 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 86 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#33774C211yesmaximal, characteristic
2#18068C211yesmaximal, characteristic, complemented
3#11983C211yesmaximal, characteristic, complemented
4C2×#1361C211yesmaximal, characteristic
5#9802C211yesmaximal, characteristic
6#7288C211yesmaximal, characteristic, complemented
7#7209C211yesmaximal, characteristic, complemented
8#3170C2211yescharacteristic
9C2×#493C2211yescharacteristic
10#2962—21no—
11#2248—21no—
12#2240—21no—
13#2207—22no—
14#2187C414yes—
15#2162C2211yescharacteristic
16C2×C4⋊C16C2211yescharacteristic
17#1965—21no—
18#1591—21no—
19#1584—22no—
20C8×Q16—21no—
21#1567C2211yescharacteristic
22C2×#335C2211yescharacteristic
23#1545—21no—
24#1481—21no—
25C8×M16C2211yescharacteristic
26#1365—21no—
27#1362—21no—
28#1361—21no—
29#1361—21no—
30C22×D8—21no—
31C22×D8C2×C412yes—
32#533—21no—
33C2×#127—21no—
34C2×SD32—24no—
35C22×C16—21no—
36#500D411yescharacteristic
37#494D411yescharacteristic
38#493—21no—
39#493—21no—
40#493—21no—
41#493—21no—
42#487—21no—
43#467—21no—
44#467C2×C412yes—
45#446—22no—
46#446—42no—
47C8×Q8—22no—
48C4×Q16—21no—
49C4×Q16D412yes—
50#431D411yescharacteristic
51#424—41no—
52#423—21no—
53C2×C4⋊C8—21no—
54C2×#100—41no—
55C2×#100—41no—
56#405—41no—
57C4×M16D411yescharacteristic
58C2×C4×C8C2311yescharacteristic
59C4⋊C16—21no—
60C4⋊C16—21no—
61C4⋊C16—21no—
62#357—24no—
63C22⋊C16C2×C412yes—
64C4×C16—21no—
65#335—21no—
66#335—21no—
67#335—21no—
68#335D411yescharacteristic
69#335D411yescharacteristic
70#331—42no—
71Q8⋊C8—22no—
72Q8⋊C8—22no—
73#322—21no—
74C82—21no—
75C22×D4—22no—
76C22×D4—41no—
77C22×D4—41no—
78C2×Q16—21no—
79C2×Q16C22⋊C412yes—
80C2×D8—22no—
81C2×D8—22no—
82C2×D8—41no—
83C2×D8—41no—
84C2×D8—81no—
85C2×D8—81no—
86C2×D8—82no—
87C2×M16—41no—
88C2×M16D811yescharacteristic
89C2×M16D811yescharacteristic
90C22×C8—41no—
91C22×C8C22×C411yescharacteristic
92C22×C8#4111yescharacteristic
93C4⋊Q8—21no—
94C4⋊Q8—21no—
95#127—41no—
96#127—41no—
97#127—41no—
98#127—41no—
99#125—41no—
100C22⋊Q8—22no—
101C22⋊Q8—41no—
102C4×Q8—22no—
103C4×Q8—41no—
104#117—41no—
105C2×C42C2×D411yescharacteristic
106SD32—84no—
107C2×C16—22no—
108C2×C16—41no—
109C2×C16—81no—
110#107—21no—
111#107C22⋊C412yes—
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120#103—22no—
121#103—22no—
122#100—82no—
123#100—82no—
124C22⋊C8—21no—
125#97—21no—
126#97—42no—
127C4×C8—21no—
128C4×C8—21no—
129C4×C8—42no—
130C4×C8C2×D411yescharacteristic
131C4×C8C2×D411yescharacteristic
132C4×C8C2×D411yescharacteristic
133C4×C8#4111yescharacteristic
134C24—42no—
135C2×Q8—22no—
136C2×Q8—41no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—82no—
140C2×D4—82no—
141C2×D4—81no—
142C2×D4—81no—
143C2×D4—81no—
144C2×D4—81no—
145C2×D4—81no—
146C2×D4—81no—
147C2×D4—81no—
148C2×D4—81no—
149C2×D4—162no—
150C22×C4C4×D411yescharacteristic
151C22×C4C2×D811yescharacteristic
152C22×C4#13511yescharacteristic
153Q16—42no—
154Q16—41no—
155Q16—41no—
156D8—44no—
157D8—82no—
158D8—82no—
159D8—81no—
160D8—81no—
161M16—21no—
162M16—21no—
163M16—21no—
164M16—81no—
165M16—81no—
166M16SD3212yescomplemented
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—81no—
176C2×C8—81no—
177C2×C8—81no—
178C2×C8C2×C22⋊C411yescharacteristic
179C2×C8C4×D411yescharacteristic
180C2×C8#12111yescharacteristic
181C2×C8#12311yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C22⋊C4—42no—
189C42—42no—
190C42#12011yescharacteristic
191C42#12111yescharacteristic
192C42AGL(1,ℤ/8)11yescharacteristic
193C42AGL(1,ℤ/8)11yescharacteristic
194C16—42no—
195C16—81no—
196C23—82no—
197C23—82no—
198C23—82no—
199C23—162no—
200C23—162no—
201C23C4×D811yescharacteristic
202Q8—42no—
203Q8—42no—
204Q8—161no—
205D4—82no—
206D4—82no—
207D4—162no—
208D4—162no—
209D4—162no—
210D4—161no—
211D4—161no—
212D4—161no—
213D4—161no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4#38611yescharacteristic
220C2×C4#44211yescharacteristic
221C2×C4#44711yescharacteristic
222C2×C4#44911yescharacteristic
223C2×C4#45911yescharacteristic
224C2×C4#46311yescharacteristic
225C2×C4C2×SD3211yescharacteristic
226C2×C4#50811yescharacteristic
227C2×C4#50911yescharacteristic
228C2×C4#51011yescharacteristic
229C8—21no—
230C8—21no—
231C8—21no—
232C8—22no—
233C8—41no—
234C8—81no—
235C8—81no—
236C8—81no—
237C8—161no—
238C8#47111yescharacteristic
239C8#47211yescharacteristic
240C22—21no—
241C22—21no—
242C22—162no—
243C22—162no—
244C22—162no—
245C22—322no—
246C22—322no—
247C22#187911yescharacteristic
248C22C4×SD3211yescharacteristic
249C22#217511yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—82no—
254C4—82no—
255C4#190711yescharacteristic
256C4#191411yescharacteristic
257C4#218511yescharacteristic
258C4#218611yescharacteristic
259C4#220711yescharacteristic
260C4#221111yescharacteristic
261C2—21no—
262C2—21no—
263C2—322no—
264C2#927311yescharacteristic
265C2#1301311yescharacteristic
266C2#1303211yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#493C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18068C2semidirect
3. #11983C2semidirect
6. #7288C2semidirect
7. #7209C2semidirect
166. M16SD32semidirect

Maximal subgroups

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

subgroupindex
#337742
#180682
#119832
C2×#13612
#98022
#72882
#72092