Finite Group Databasebeta

← #126348

Subgroups — #126348

Order 512.

285 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 66 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#31236C211yesmaximal, characteristic
2#13538C211yesmaximal, characteristic
3#13481C211yesmaximal, characteristic
4#10652C211yesmaximal, characteristic
5#7120C211yesmaximal, characteristic
6#7118C211yesmaximal, characteristic
7#7113C211yesmaximal, characteristic
8C2×#479—21no—
9#2453—21no—
10C2×#358—21no—
11#2068—21no—
12#2066—21no—
13#2059—21no—
14#1969C2211yescharacteristic
15#1948—21no—
16#1942C2211yescharacteristic
17#1939C2211yescharacteristic
18#1878—21no—
19#1845C2211yescharacteristic
20#1380—21no—
21#1380—21no—
22#1380—21no—
23#1374C2211yescharacteristic
24#1374C2211yescharacteristic
25#1296—21no—
26#1277C2211yescharacteristic
27C22×Q16D411yescharacteristic
28C2×#125—21no—
29C2×C4×Q8—21no—
30C22×C16—21no—
31C22×C16—21no—
32#479—42no—
33#479—42no—
34C2×#107—21no—
35C2×#107—21no—
36C2×#107D411yescharacteristic
37C2×#107D411yescharacteristic
38C2×#107D411yescharacteristic
39C2×#106—21no—
40C2×#106—41no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8D411yescharacteristic
44C2×#103—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×C4×C8C2311yescharacteristic
48#400—41no—
49#398—41no—
50#391—21no—
51#391—21no—
52#389—21no—
53#389—21no—
54#389—41no—
55#387—41no—
56#384—21no—
57C4×C4⋊C4D411yescharacteristic
58#358—42no—
59#340—21no—
60#340—21no—
61#340—41no—
62#336—21no—
63C22×Q8—21no—
64C2×Q16—21no—
65C2×Q16—41no—
66C2×Q16—41no—
67C2×Q16SD1612yes—
68C22×C8—21no—
69C22×C8—41no—
70C22×C8C2×D411yescharacteristic
71C22×C8C2×D411yescharacteristic
72#125—42no—
73#125—81no—
74C4×Q8—42no—
75C4×Q8—81no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4D811yescharacteristic
88C2×C4⋊C4D811yescharacteristic
89C2×C42—21no—
90C2×C42—41no—
91C2×C42C2×D411yescharacteristic
92C2×C16—41no—
93C2×C16—41no—
94C2×C16—41no—
95C2×C16—41no—
96C2×C16—41no—
97C2×C16—41no—
98#107—21no—
99#107—41no—
100#107—41no—
101#107—42no—
102#107—41no—
103#107—41no—
104#107SD1612yes—
105#106—42no—
106#106—82no—
107C4⋊C8—22no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112#103—42no—
113#103—41no—
114#103—41no—
115#103—81no—
116C4×C8—41no—
117#95—21no—
118#95—41no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124C2×Q8—22no—
125C2×Q8—41no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—81no—
137C22×C4#12011yescharacteristic
138C22×C4C2×D811yescharacteristic
139C22×C4AGL(1,ℤ/8)11yescharacteristic
140Q16—42no—
141Q16—81no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—42no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—81no—
151C2×C8—81no—
152C2×C8C2×SD1611yescharacteristic
153C2×C8C2×SD1611yescharacteristic
154C2×C8AGL(1,ℤ/8)11yescharacteristic
155C2×C8#13711yescharacteristic
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C4⋊C4—82no—
170C4⋊C4—82no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C4⋊C4—161no—
176C42—21no—
177C42—21no—
178C42—41no—
179C42—41no—
180C42—82no—
181C16—81no—
182C16—81no—
183C23#44711yescharacteristic
184Q8—42no—
185Q8—42no—
186Q8—81no—
187Q8—81no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—161no—
212C2×C4—161no—
213C2×C4—161no—
214C2×C4#44811yescharacteristic
215C2×C4#45011yescharacteristic
216C2×C4#45411yescharacteristic
217C2×C4#45511yescharacteristic
218C2×C4#45611yescharacteristic
219C2×C4#45811yescharacteristic
220C2×C4#50811yescharacteristic
221C2×C4#51011yescharacteristic
222C8—21no—
223C8—21no—
224C8—21no—
225C8—21no—
226C8—82no—
227C8—81no—
228C22#161011yescharacteristic
229C22#162711yescharacteristic
230C22#163911yescharacteristic
231C22#218911yescharacteristic
232C22#219011yescharacteristic
233C22#219311yescharacteristic
234C22#219511yescharacteristic
235C4—21no—
236C4—21no—
237C4—41no—
238C4—81no—
239C4—82no—
240C4—81no—
241C4—82no—
242C4—161no—
243C4—161no—
244C4—321no—
245C4#160811yescharacteristic
246C4#161611yescharacteristic
247C4#165111yescharacteristic
248C4#166411yescharacteristic
249C2#779111yescharacteristic
250C2#1229311yescharacteristic
251C2#1229611yescharacteristic
252C2#1235711yescharacteristic
253C2#1236111yescharacteristic
254C2#1237511yescharacteristic
255C2#1237911yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
commutator, Frattini, cosocleC2×C4×C8C23
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1939 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#312362
#135382
#134812
#106522
#71202
#71182
#71132