Finite Group Databasebeta

← #137839

Subgroups — #137839

Order 512.

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

normal subgroups only

This group has 275 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#33501C211yesmaximal, characteristic
2#17961C211yesmaximal, characteristic
3#11643C211yesmaximal, characteristic
4#11643C211yesmaximal, characteristic
5#11497C211yesmaximal, characteristic
6#7196C211yesmaximal, characteristic
7#7146C211yesmaximal, characteristic
8#3170C411yescharacteristic
9#2960C2211yescharacteristic
10#2960C2211yescharacteristic
11#2183—22no—
12#2183—22no—
13#2164—22no—
14#2163C2211yescharacteristic
15#2163C2211yescharacteristic
16#2163C2211yescharacteristic
17#2163C2211yescharacteristic
18#1593—22no—
19#1579—22no—
20#1566C411yescharacteristic
21#1493C411yescharacteristic
22#1493C411yescharacteristic
23C2×#322C2211yescharacteristic
24#1338—22no—
25#1338—22no—
26#1323—22no—
27#533C2×C411yescharacteristic
28#533C2×C411yescharacteristic
29#500C811yescharacteristic
30#494C811yescharacteristic
31#487C811yescharacteristic
32#487C811yescharacteristic
33#467—22no—
34#446—21no—
35#446—21no—
36#446—21no—
37#446D412yes—
38C8×Q8—21no—
39C8×Q8—21no—
40C8×Q8—21no—
41C8×Q8D412yes—
42C4×Q16—22no—
43#424—21no—
44#424—21no—
45#423C2×C411yescharacteristic
46#423C2×C411yescharacteristic
47#405—21no—
48#405—21no—
49C2×C4×C8C2×C411yescharacteristic
50C2×C4×C8C2×C411yescharacteristic
51C2×C4×C8C2311yescharacteristic
52C4⋊C16—24no—
53C4⋊C16—21no—
54C4⋊C16—21no—
55C4⋊C16—21no—
56C4⋊C16—21no—
57C22⋊C16—21no—
58C22⋊C16—21no—
59C22⋊C16—21no—
60C22⋊C16D412yes—
61#346—22no—
62#346—21no—
63#346—21no—
64#346—21no—
65#346D412yes—
66C4×C16—22no—
67#334—22no—
68Q8⋊C8—22no—
69Q8⋊C8—22no—
70#322—21no—
71#322—21no—
72C2×Q16—22no—
73C2×M16—21no—
74C2×M16—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8C2×D411yescharacteristic
78C22×C8#4111yescharacteristic
79C4⋊Q8C2×C811yescharacteristic
80C4⋊Q8C2×C811yescharacteristic
81C4⋊Q8C2×C811yescharacteristic
82C4⋊Q8C2×C811yescharacteristic
83#125—21no—
84#125—21no—
85C22⋊Q8—21no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8M1612yes—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8C22⋊C412yes—
93#117—21no—
94#117—21no—
95C2×C42C22×C411yescharacteristic
96C2×C16—22no—
97C2×C16—22no—
98C2×C16—41no—
99C2×C16—41no—
100#107—22no—
101C4⋊C8—22no—
102C4⋊C8—22no—
103C4⋊C8—21no—
104C4⋊C8—21no—
105C4⋊C8—21no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108C4⋊C8C22⋊C412yes—
109#103—22no—
110#103—22no—
111#97—22no—
112#97—41no—
113C4×C8—22no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—41no—
118C4×C8C2×C811yescharacteristic
119C4×C8C2×C811yescharacteristic
120C4×C8C2×D411yescharacteristic
121C4×C8C2×D411yescharacteristic
122C4×C8C2×D411yescharacteristic
123C4×C8#4111yescharacteristic
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8C22⋊C812yes—
128C22×C4C4×D411yescharacteristic
129C22×C4C2×M1611yescharacteristic
130C22×C4#13111yescharacteristic
131Q16—44no—
132M16—81no—
133M16—81no—
134C2×C8—21no—
135C2×C8—22no—
136C2×C8—21no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—42no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—81no—
143C2×C8#12011yescharacteristic
144C2×C8#12111yescharacteristic
145C2×C8#12111yescharacteristic
146C2×C8#12311yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4C22⋊C812yes—
159C22⋊C4—22no—
160C22⋊C4—41no—
161C42—22no—
162C42—41no—
163C42C2×C22⋊C411yescharacteristic
164C42C4×D411yescharacteristic
165C42C22×C811yescharacteristic
166C42#13111yescharacteristic
167C16—42no—
168C16—42no—
169C16—81no—
170C16—81no—
171C23#43511yescharacteristic
172Q8—42no—
173Q8—42no—
174Q8—81no—
175Q8—81no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4#38611yescharacteristic
185C2×C4C2×C22⋊C811yescharacteristic
186C2×C4#40811yescharacteristic
187C2×C4C8×D411yescharacteristic
188C2×C4C8×D411yescharacteristic
189C2×C4#43511yescharacteristic
190C8—22no—
191C8—21no—
192C8—21no—
193C8—21no—
194C8—21no—
195C8—41no—
196C8—42no—
197C8—81no—
198C8—81no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22#186911yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—42no—
209C4—42no—
210C4—81no—
211C4—81no—
212C4#190211yescharacteristic
213C4#190511yescharacteristic
214C2—41no—
215C2#908811yescharacteristic
216C2#1298211yescharacteristic
217C2#1298611yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC23#435
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#335012
#179612
#116432
#114972
#71962
#71462