Finite Group Databasebeta

← #141484

Subgroups — #141484

Order 512.

338 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 78 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#28737C211yesmaximal, characteristic
2#13301C212yesmaximal
3#11907C212yesmaximal
4#8615C211yesmaximal, characteristic
5#8165C211yesmaximal, characteristic
6C2×#494C411yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2568C2212yes—
9#2056—21no—
10#2056—21no—
11#1981C412yes—
12#1969—21no—
13#1969—21no—
14#1801C2212yes—
15C4×#107C411yescharacteristic
16C4×#101C2211yescharacteristic
17#1566—21no—
18#1461C2211yescharacteristic
19#1451—21no—
20#1394—24no—
21#1353—24no—
22#1292—21no—
23#1283—21no—
24C22×Q16—21no—
25C22×Q16—21no—
26C2×C4⋊Q8C2×C412yes—
27C2×C4×Q8C2311yescharacteristic
28#494—41no—
29#494—41no—
30C8×Q8—41no—
31C8×Q8—41no—
32#431—21no—
33C2×#107—21no—
34#423—22no—
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8D411yescharacteristic
37C2×C4⋊C8D411yescharacteristic
38C2×#103—22no—
39C2×#103—22no—
40C2×#101—22no—
41C2×#101D412yes—
42C4×M16—21no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#400—22no—
47#400—22no—
48#391—22no—
49#391—22no—
50#391—22no—
51#382—22no—
52C4×C4⋊C4C2×C412yes—
53#343—22no—
54#343—21no—
55#340—21no—
56#334—41no—
57#334—41no—
58#324—21no—
59#324D412yes—
60#322—41no—
61#322—41no—
62C22×Q8—22no—
63C22×Q8C2×D411yescharacteristic
64C2×Q16—42no—
65C2×Q16—42no—
66C2×Q16—42no—
67C2×Q16—41no—
68C2×Q16—41no—
69C2×Q16—82no—
70C2×M16—22no—
71C2×M16—42no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C4⋊Q8—42no—
77C4⋊Q8—42no—
78C4×Q8—41no—
79C4×Q8—41no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4C2×D411yescharacteristic
84C2×C4⋊C4#4111yescharacteristic
85C2×C4⋊C4#4111yescharacteristic
86C2×C42—22no—
87C2×C42C22×C411yescharacteristic
88C2×C42C2×D411yescharacteristic
89C2×C42C2×D411yescharacteristic
90#107—42no—
91C4⋊C8—41no—
92C4⋊C8—41no—
93C4⋊C8—41no—
94C4⋊C8—41no—
95C4⋊C8—42no—
96C4⋊C8—42no—
97#103—82no—
98#103—82no—
99#101—24no—
100#101—44no—
101C22⋊C8—22no—
102#97—41no—
103C4×C8—21no—
104C4×C8—21no—
105C4×C8—21no—
106C4×C8—21no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8—41no—
110C4×C8—41no—
111#95—22no—
112#95—42no—
113#95—42no—
114#95—42no—
115#95—42no—
116C2×Q8—41no—
117C2×Q8—41no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8#13714yes—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4C2×C22⋊C411yescharacteristic
128C22×C4C4×D411yescharacteristic
129C22×C4C4×D411yescharacteristic
130C22×C4#12111yescharacteristic
131C22×C4#12111yescharacteristic
132C22×C4#12411yescharacteristic
133C22×C4#12711yescharacteristic
134Q16—82no—
135Q16—84no—
136Q16—82no—
137M16—42no—
138M16—42no—
139M16—82no—
140M16—82no—
141C2×C8—22no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—82no—
151C2×C8#9911yescharacteristic
152C2×C8#9911yescharacteristic
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—82no—
167C23#39011yescharacteristic
168Q8—44no—
169Q8—82no—
170Q8—82no—
171Q8—82no—
172Q8—82no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4C2×#9911yescharacteristic
194C2×C4#41011yescharacteristic
195C2×C4#44112yes—
196C2×C4#47012yes—
197C2×C4#49712yes—
198C8—21no—
199C8—41no—
200C8—41no—
201C8—41no—
202C8—82no—
203C8—82no—
204C8—162no—
205C8#35512yes—
206C22—21no—
207C22—21no—
208C22#196411yescharacteristic
209C22#197111yescharacteristic
210C22#197611yescharacteristic
211C4—22no—
212C4—22no—
213C4—22no—
214C4—21no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—162no—
219C4—162no—
220C4C2×#35511yescharacteristic
221C4#213411yescharacteristic
222C2—21no—
223C2—21no—
224C2#978411yescharacteristic
225C2#1322611yescharacteristic
226C2#1323311yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×Q8C23
prime radicalC23#390
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#287372
#133012
#119072
#86152
#81652