Finite Group Databasebeta

← #122549

Subgroups — #122549

Order 512.

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

normal subgroups only

This group has 336 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#29876C211yesmaximal, characteristic
2#11460C211yesmaximal, characteristic
3#7980C211yesmaximal, characteristic
4#7079C214yesmaximal
5C2×#500C411yescharacteristic, complemented
6C2×#431C2211yescharacteristic
7#2857—21no—
8#2283—21no—
9#2163—21no—
10#2086—21no—
11#2086C412yescomplemented
12#2075—21no—
13#2075C2212yes—
14#1990—21no—
15#1834C411yescharacteristic, complemented
16#1322—21no—
17#1314C2212yes—
18#1289—21no—
19#1289C2212yes—
20C2×C4⋊Q8—21no—
21C2×C22⋊Q8—41no—
22C22×C42D411yescharacteristic
23#500—41no—
24#500—41no—
25#431—41no—
26#431—41no—
27C2×#107—21no—
28C2×#107C2×C412yes—
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8D411yescharacteristic
31C2×C22⋊C8D411yescharacteristic
32C2×C22⋊C8C2311yescharacteristic
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8D411yescharacteristic
35#391—41no—
36#389—22no—
37#389—41no—
38#384—21no—
39#383—22no—
40#382—41no—
41#381—41no—
42#380—22no—
43#380—22no—
44#380—41no—
45C4×C4⋊C4—41no—
46C4×C22⋊C4—41no—
47#376—22no—
48C4⋊C16—41no—
49C22⋊C16—22no—
50#346—41no—
51#340—21no—
52#340—22no—
53#340—22no—
54#340C2×C412yes—
55C22×Q8—41no—
56C23×C4D811yescharacteristic
57C23×C4D811yescharacteristic
58C23×C4C2×D411yescharacteristic
59C22×C8C22⋊C411yescharacteristic
60C22×C8C22⋊C411yescharacteristic
61C22×C8C22×C411yescharacteristic
62C22×C8C2×D411yescharacteristic
63C4⋊Q8—41no—
64C4⋊Q8—41no—
65C4⋊Q8—41no—
66C4⋊Q8—41no—
67C22⋊Q8—84no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C22⋊C4—42no—
76C2×C22⋊C4—42no—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—42no—
82C2×C42C22⋊C411yescharacteristic
83C2×C42C22⋊C411yescharacteristic
84C2×C42SD1611yescharacteristic
85C2×C42SD1611yescharacteristic
86C2×C16—42no—
87#107—42no—
88#107—42no—
89C4⋊C8—41no—
90C4⋊C8—41no—
91C22⋊C8—41no—
92C22⋊C8—41no—
93C4×C8—21no—
94C4×C8—21no—
95C4×C8—21no—
96C4×C8—21no—
97#95—24no—
98#95—42no—
99#95—42no—
100#95—42no—
101#95—42no—
102#95—42no—
103C24C2×D811yescharacteristic
104C2×Q8—82no—
105C2×Q8—82no—
106C2×Q8—81no—
107C2×Q8—81no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—41no—
116C22×C4—42no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—44no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4#10211yescharacteristic
123C22×C4#10211yescharacteristic
124C22×C4#10211yescharacteristic
125C22×C4#10211yescharacteristic
126C22×C4C2×C22⋊C411yescharacteristic
127C22×C4C2×SD1611yescharacteristic
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—82no—
141C4⋊C4—82no—
142C4⋊C4—84no—
143C4⋊C4—82no—
144C4⋊C4—82no—
145C4⋊C4—81no—
146C4⋊C4—81no—
147C22⋊C4—82no—
148C22⋊C4—82no—
149C22⋊C4—84no—
150C42—21no—
151C42—21no—
152C42—21no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—84no—
157C42#9911yescharacteristic
158C42#9911yescharacteristic
159C42#10411yescharacteristic
160C42#10411yescharacteristic
161C16—82no—
162C23—21no—
163C23—21no—
164C23—41no—
165C23—41no—
166C23C2×#10211yescharacteristic
167C23#50912yes—
168Q8—162no—
169Q8—162no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—84no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—84no—
193C2×C4—84no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4#32711yescharacteristic
199C2×C4#32711yescharacteristic
200C2×C4#32711yescharacteristic
201C2×C4#32711yescharacteristic
202C2×C4C2×#9911yescharacteristic
203C2×C4C2×#10411yescharacteristic
204C8—41no—
205C8—41no—
206C8—41no—
207C8—41no—
208C22—21no—
209C22—21no—
210C22—22no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—81no—
215C22C2×#32711yescharacteristic
216C22#214011yescharacteristic
217C22#214511yescharacteristic
218C4—21no—
219C4—21no—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—162no—
226C4—164no—
227C4—162no—
228C4#134012yes—
229C4#212311yescharacteristic
230C4#212911yescharacteristic
231C2—21no—
232C2—21no—
233C2—42no—
234C2C2×#134011yescharacteristic
235C2#1169411yescharacteristic
236C2#1170511yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×C22⋊C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#500C4semidirect
11. #2086C4semidirect
15. #1834C4semidirect

Maximal subgroups

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

subgroupindex
#298762
#114602
#79802
#70792