Finite Group Databasebeta

← #140824

Subgroups — #140824

Order 512.

279 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 76 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#17902C211yesmaximal, characteristic
2#12014C211yesmaximal, characteristic, complemented
3#11643C211yesmaximal, characteristic, complemented
4#11631C211yesmaximal, characteristic
5#11498C211yesmaximal, characteristic, complemented
6#11488C211yesmaximal, characteristic
7#11416C211yesmaximal, characteristic, complemented
8#3101C411yescharacteristic
9#2960C2211yescharacteristic
10C2×C8×D4C2211yescharacteristic
11#2163C2211yescharacteristic
12#2163C2211yescharacteristic
13#2163C2211yescharacteristic
14#2162C2211yescharacteristic
15#1582—21no—
16#1582—21no—
17C8×SD16—21no—
18C8×SD16—21no—
19#1570C411yescharacteristic
20#1486C411yescharacteristic
21#1481C411yescharacteristic
22#1450C2211yescharacteristic
23#1370—21no—
24#1367—21no—
25#1338—22no—
26D4⋊C16—22no—
27#1323—22no—
28#1314—22no—
29#1313—21no—
30#1313—21no—
31C23×C8—21no—
32#533C2×C411yescharacteristic
33C2×C4×D4C2×C411yescharacteristic
34#483—21no—
35#476—21no—
36#461—21no—
37#450—21no—
38#446—21no—
39#446—21no—
40C8×Q8—21no—
41C8×Q8—21no—
42C4×SD16—21no—
43C4×SD16—21no—
44C8×D4—21no—
45C8×D4—21no—
46C8×D4—21no—
47C8×D4—21no—
48#431C2×C411yescharacteristic
49#431C2×C411yescharacteristic
50#424—21no—
51#423C2×C411yescharacteristic
52C2×C4⋊C8C2×C411yescharacteristic
53C2×C22⋊C8—21no—
54#405—21no—
55C2×C4×C8C2311yescharacteristic
56C4⋊C16—21no—
57C4⋊C16—21no—
58C4⋊C16—21no—
59C4⋊C16D412yes—
60C22⋊C16—21no—
61C22⋊C16—21no—
62C22⋊C16—21no—
63C22⋊C16D412yes—
64#346—21no—
65#346—21no—
66#346—21no—
67C4×C16—21no—
68#334—21no—
69#334—21no—
70Q8⋊C8—21no—
71Q8⋊C8—21no—
72D4⋊C8—21no—
73D4⋊C8—21no—
74#322D411yescharacteristic
75#322D411yescharacteristic
76C82D411yescharacteristic
77C82D411yescharacteristic
78C22×D4M1611yescharacteristic
79C23×C4—21no—
80C2×SD16—21no—
81C2×SD16—21no—
82C2×M16—21no—
83C22×C8—21no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8—82no—
87C22×C8C2×D411yescharacteristic
88C22×C8#4111yescharacteristic
89C4⋊Q8M1611yescharacteristic
90C4⋊Q8M1611yescharacteristic
91#125—21no—
92C22⋊Q8—21no—
93C22⋊Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100#117—21no—
101C2×C4⋊C4M1611yescharacteristic
102C2×C22⋊C4—21no—
103C2×C42C22×C411yescharacteristic
104C2×C16—22no—
105C2×C16—22no—
106C2×C16—41no—
107C2×C16—41no—
108#106—21no—
109#106—21no—
110C4⋊C8—21no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8—21no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118#103—21no—
119#103—21no—
120#102—21no—
121#102—21no—
122C22⋊C8—21no—
123C22⋊C8—21no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126#97—41no—
127C4×C8—41no—
128C4×C8C22⋊C411yescharacteristic
129C4×C8C22⋊C411yescharacteristic
130C4×C8C22⋊C411yescharacteristic
131C4×C8C22⋊C411yescharacteristic
132C4×C8C2×D411yescharacteristic
133C4×C8C2×D411yescharacteristic
134C4×C8C2×D411yescharacteristic
135C4×C8#4111yescharacteristic
136C24—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—42no—
142C22×C4—21no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—81no—
146C22×C4—81no—
147C22×C4C4×D411yescharacteristic
148C22×C4C2×M1611yescharacteristic
149C22×C4#13111yescharacteristic
150SD16—42no—
151SD16—42no—
152M16—81no—
153C2×C8—22no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—82no—
162C2×C8—82no—
163C2×C8—81no—
164C2×C8#12111yescharacteristic
165C2×C8#12111yescharacteristic
166C2×C8#12411yescharacteristic
167C2×C8#12711yescharacteristic
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C22⋊C4—41no—
177C22⋊C4—41no—
178C22⋊C4—41no—
179C42—41no—
180C42C2×C22⋊C411yescharacteristic
181C42C4×D411yescharacteristic
182C42C2×M1611yescharacteristic
183C42#13111yescharacteristic
184C16—42no—
185C16—42no—
186C16—81no—
187C16—81no—
188C23—22no—
189C23—41no—
190C23—41no—
191C23—81no—
192C23—81no—
193C23#43511yescharacteristic
194Q8—41no—
195Q8—41no—
196Q8—81no—
197D4—22no—
198D4—22no—
199D4—41no—
200D4—41no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—82no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4#39011yescharacteristic
213C2×C4#40711yescharacteristic
214C2×C4#40811yescharacteristic
215C2×C4#43511yescharacteristic
216C2×C4#43511yescharacteristic
217C2×C4#43511yescharacteristic
218C8—21no—
219C8—21no—
220C8—21no—
221C8—41no—
222C8—41no—
223C8—41no—
224C8—41no—
225C8—81no—
226C8—81no—
227C8#47112yes—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—42no—
232C22—82no—
233C22—82no—
234C22—81no—
235C22—81no—
236C22#195511yescharacteristic
237C4—21no—
238C4—21no—
239C4—21no—
240C4—21no—
241C4—21no—
242C4—81no—
243C4—81no—
244C4—81no—
245C4#197611yescharacteristic
246C4#197711yescharacteristic
247C2—41no—
248C2—82no—
249C2#967211yescharacteristic
250C2#1319911yescharacteristic
251C2#1320011yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1955
prime radicalC22×D4M16
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×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
2. #12014C2semidirect
3. #11643C2semidirect
5. #11498C2semidirect
7. #11416C2semidirect

Maximal subgroups

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

subgroupindex
#179022
#120142
#116432
#116312
#114982
#114882
#114162