Finite Group Databasebeta

← #353095

Subgroups — #353095

Order 512.

590 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 218 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#33297C211yesmaximal, characteristic
2#33263C211yesmaximal, characteristic
3#30307C212yesmaximal
4#30304C212yesmaximal
5#8821C212yesmaximal
6#8821C212yesmaximal
7#8820C212yesmaximal
8#8816C212yesmaximal
9#7844C211yesmaximal, characteristic
10#2868C2211yescharacteristic
11#2859C2211yescharacteristic
12#2859C2212yes—
13#2085—24no—
14#2083—24no—
15#2059—24no—
16#2057—24no—
17#1993—22no—
18#1992—22no—
19#1944—22no—
20#1943—22no—
21#1845C2212yes—
22#1845C2212yes—
23#1845C2212yes—
24#1845C2211yescharacteristic
25#1844C2212yes—
26#1844C2211yescharacteristic
27#1834C2212yes—
28#1834C2212yes—
29#1834C2212yes—
30#1834C2212yes—
31#1834C2212yes—
32#1834C2212yes—
33#1834C2212yes—
34#1744C2212yes—
35C4×#97C2211yescharacteristic
36C2×#322C2212yes—
37C2×#322C2211yescharacteristic
38C2×C82C2211yescharacteristic
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—22no—
41C2×#125—21no—
42C2×#125—22no—
43C2×#107—22no—
44C2×#107—24no—
45C2×#107—24no—
46C2×#107—24no—
47C2×#106—22no—
48C2×#106—24no—
49C2×#97D412yes—
50C2×#97D412yes—
51C2×C4×C8C2311yescharacteristic
52C2×C4×C8C2311yescharacteristic
53C2×C4×C8C2312yes—
54C2×C4×C8C2312yes—
55#389—22no—
56#389—22no—
57#389—22no—
58#389—22no—
59#389—22no—
60#389—22no—
61#384—22no—
62#384—21no—
63#384C2312yes—
64#384C2312yes—
65#384C2312yes—
66#384C2312yes—
67#382—22no—
68#382—21no—
69C43C2311yescharacteristic
70#340—24no—
71#340—24no—
72#340—24no—
73#340—24no—
74#340—24no—
75#340—24no—
76#340—24no—
77#336D414yes—
78#336D414yes—
79#322Q818yescomplemented
80#322Q814yescomplemented
81C82Q814yescomplemented
82C22×Q8—41no—
83C22×C8C2×D414yes—
84C22×C8C2×D414yes—
85C22×C8C2×D412yes—
86C22×C8C2×D412yes—
87C4⋊Q8—44no—
88C4⋊Q8—84no—
89#125—44no—
90#125—84no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—42no—
108C2×C42C2×D412yes—
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D412yes—
112C2×C42C2411yescharacteristic
113#107—44no—
114#107—84no—
115#107—84no—
116#107—84no—
117#106—44no—
118#106—84no—
119#97—42no—
120#97—42no—
121C4×C8C2×Q812yes—
122C4×C8C2×Q814yes—
123C4×C8C2×Q812yes—
124C4×C8C2×Q814yes—
125C4×C8C2×Q814yes—
126C4×C8C2×Q814yes—
127C4×C8C2×Q814yes—
128#95—42no—
129#95—41no—
130#95—42no—
131#95—41no—
132C2×Q8—82no—
133C2×Q8—84no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4#12711yescharacteristic
139C22×C4#12712yes—
140C22×C4#12711yescharacteristic
141C22×C4C22×D411yescharacteristic
142C22×C4C22×D412yes—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—24no—
148C2×C8—24no—
149C2×C8—24no—
150C2×C8—24no—
151C2×C8C4⋊Q814yes—
152C2×C8C4⋊Q818yes—
153C2×C8C4⋊Q818yes—
154C2×C8C4⋊Q814yes—
155C4⋊C4—44no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—84no—
168C4⋊C4—84no—
169C4⋊C4—84no—
170C4⋊C4—82no—
171C4⋊C4—84no—
172C42—22no—
173C42—22no—
174C42—24no—
175C42—24no—
176C42C22×Q811yescharacteristic
177C42C22×Q811yescharacteristic
178C42C22×Q812yes—
179C23C2×#12711yescharacteristic
180Q8—164no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4C2×C4⋊Q811yescharacteristic
201C2×C4C2×C4⋊Q812yes—
202C2×C4C2×C4⋊Q812yes—
203C2×C4C2×C4⋊Q811yescharacteristic
204C2×C4#57512yes—
205C2×C4#57512yes—
206C2×C4#57614yes—
207C2×C4#57814yes—
208C8—24no—
209C8—28no—
210C8—28no—
211C8—24no—
212C22#286811yescharacteristic
213C22#315311yescharacteristic
214C22#315311yescharacteristic
215C22#315512yes—
216C22#315612yes—
217C4—41no—
218C4—42no—
219C4—41no—
220C4—162no—
221C4—162no—
222C4—162no—
223C4—162no—
224C4#316212yes—
225C4#316414yes—
226C4#316414yes—
227C4#316612yes—
228C2#1972911yescharacteristic
229C2#1973012yes—
230C2#1973211yescharacteristic
231C2#3331711yescharacteristic
232C2#3331912yes—
subgroupG/H
center, socle, prime radicalC23C2×#127
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
79. #322Q8semidirect
80. #322Q8semidirect
81. C82Q8semidirect

Maximal subgroups

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

subgroupindex
#332972
#332632
#303072
#303042
#88212
#88202
#88162
#78442