Finite Group Databasebeta

← #136378

Subgroups — #136378

Order 512.

331 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 102 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
1Q8×#106C211yesmaximal, characteristic
2#13185C211yesmaximal, characteristic
3#12903C211yesmaximal, characteristic
4#11607C211yesmaximal, characteristic
5#8659C211yesmaximal, characteristic
6#7278C211yesmaximal, characteristic
7#7174C211yesmaximal, characteristic
8C2×C8×Q8C2211yescharacteristic
9Q8×C4⋊C4—21no—
10C2×#365—22no—
11C2×#365—21no—
12#2150C2211yescharacteristic
13C4×M32C2211yescharacteristic
14#1948—22no—
15#1948—21no—
16#1848C2211yescharacteristic
17#1844—21no—
18#1844C2211yescharacteristic
19C4×#106—21no—
20C4×#106C2211yescharacteristic
21#1564C2211yescharacteristic
22#1487—21no—
23#1383—21no—
24Q8⋊C16C414yescomplemented
25#1301—21no—
26C2×C4×Q8—41no—
27C2×C4×Q8D411yescharacteristic
28C2×M32—21no—
29C2×M32—21no—
30C8×Q8—21no—
31C8×Q8—21no—
32C8×Q8C2×C412yes—
33C8×Q8Q812yes—
34#431—21no—
35C2×#108—21no—
36C2×#108—21no—
37C2×#106—21no—
38C2×#106—22no—
39C2×#106D411yescharacteristic
40C2×#106D411yescharacteristic
41C2×#106D412yes—
42#423—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×C4×C8—21no—
46C2×C4×C8C2311yescharacteristic
47#389—42no—
48#389—41no—
49#384—22no—
50#384—21no—
51C4×C4⋊C4—22no—
52C4×C4⋊C4—21no—
53#365—44no—
54#365—42no—
55C4⋊C16—21no—
56C4⋊C16C2×C412yes—
57#346—21no—
58C4×C16C2×C412yes—
59#340—22no—
60#340—21no—
61#335—21no—
62#334—21no—
63Q8⋊C8—42no—
64C22×Q8SD1611yescharacteristic
65C2×M16—41no—
66C22×C8—21no—
67C22×C8C2×D411yescharacteristic
68C22×C8C2×D411yescharacteristic
69C4×Q8—21no—
70C4×Q8—21no—
71C4×Q8—84no—
72C4×Q8C4⋊C412yes—
73C4×Q8SD1612yes—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4SD1611yescharacteristic
83C2×C42—21no—
84C2×C42—42no—
85C2×C42—41no—
86C2×C42C2×D411yescharacteristic
87M32—22no—
88M32—22no—
89M32—41no—
90M32—41no—
91C2×C16—22no—
92C2×C16—22no—
93#108—42no—
94#108—41no—
95#108—41no—
96#106—21no—
97#106—21no—
98#106—21no—
99#106—41no—
100#106—41no—
101#106—42no—
102#106—82no—
103#106SD1612yes—
104C4⋊C8—22no—
105C4⋊C8—22no—
106C4⋊C8—21no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110C4⋊C8C4⋊C412yes—
111C22⋊C8—41no—
112C4×C8—22no—
113C4×C8—22no—
114C4×C8—41no—
115C4×C8C22×C411yescharacteristic
116C4×C8C2×Q811yescharacteristic
117C4×C8#4111yescharacteristic
118C4×C8#4111yescharacteristic
119#95—42no—
120#95—41no—
121C2×Q8—24no—
122C2×Q8—21no—
123C2×Q8#10612yes—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—41no—
127C22×C4—42no—
128C22×C4—41no—
129C22×C4#12011yescharacteristic
130C22×C4C2×SD1611yescharacteristic
131C22×C4AGL(1,ℤ/8)11yescharacteristic
132M16—81no—
133M16—81no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—22no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—81no—
140C2×C8—81no—
141C2×C8C4×D411yescharacteristic
142C2×C8C4×D411yescharacteristic
143C2×C8C22⋊Q811yescharacteristic
144C2×C8C22⋊Q811yescharacteristic
145C2×C8C2×SD1611yescharacteristic
146C2×C8#13511yescharacteristic
147C2×C8#13711yescharacteristic
148C2×C8#13711yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—41no—
158C4⋊C4—84no—
159C4⋊C4—82no—
160C4⋊C4—161no—
161C4⋊C4—162no—
162C4⋊C4#10612yes—
163C42—22no—
164C42—22no—
165C42—84no—
166C42—82no—
167C42C2×C4⋊C411yescharacteristic
168C42#12311yescharacteristic
169C42C2×SD1611yescharacteristic
170C42#13711yescharacteristic
171C16—24no—
172C16—42no—
173C23#45011yescharacteristic
174Q8—24no—
175Q8—22no—
176Q8#36514yescomplemented
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—82no—
187C2×C4—81no—
188C2×C4—82no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—82no—
192C2×C4#38511yescharacteristic
193C2×C4C2×#10611yescharacteristic
194C2×C4#42811yescharacteristic
195C2×C4#44811yescharacteristic
196C2×C4#44811yescharacteristic
197C2×C4#45211yescharacteristic
198C2×C4#45311yescharacteristic
199C2×C4#45611yescharacteristic
200C2×C4#48111yescharacteristic
201C2×C4#48311yescharacteristic
202C8—21no—
203C8—21no—
204C8—42no—
205C8—161no—
206C8—161no—
207C8C4×SD1611yescharacteristic
208C8#44111yescharacteristic
209C8#47511yescharacteristic
210C8#47911yescharacteristic
211C22—21no—
212C22—21no—
213C22#163111yescharacteristic
214C22#165111yescharacteristic
215C22#185311yescharacteristic
216C4—24no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—42no—
221C4—82no—
222C4—161no—
223C4—162no—
224C4#162311yescharacteristic
225C4#165311yescharacteristic
226C4#185911yescharacteristic
227C4#186511yescharacteristic
228C4C2×#36512yes—
229C2—21no—
230C2—21no—
231C2#888811yescharacteristic
232C2#1292911yescharacteristic
233C2#1293411yescharacteristic
subgroupG/H
center, socleC22#1853
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#450
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
24. Q8⋊C16C4semidirect
176. Q8#365semidirect

Maximal subgroups

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

subgroupindex
Q8×#1062
#131852
#129032
#116072
#86592
#72782
#71742