Finite Group Databasebeta

← #126371

Subgroups — #126371

Order 512.

278 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 68 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#32682C211yesmaximal, characteristic
2#13537C211yesmaximal, characteristic
3#13505C211yesmaximal, characteristic
4#10617C211yesmaximal, characteristic
5#7128C211yesmaximal, characteristic
6#7120C211yesmaximal, characteristic
7#7120C211yesmaximal, characteristic
8#2546—21no—
9C2×#366—21no—
10#2086—21no—
11#2057—21no—
12#2057—21no—
13#1984C2211yescharacteristic
14#1942C2211yescharacteristic
15#1942C2211yescharacteristic
16#1918—21no—
17#1918—21no—
18#1845C2211yescharacteristic
19#1803—21no—
20#1380—21no—
21#1380—21no—
22#1380—21no—
23#1374C2211yescharacteristic
24#1374C2211yescharacteristic
25#1296—21no—
26#1277C2211yescharacteristic
27C2×C4⋊Q8—21no—
28C2×C4⋊Q8—41no—
29C2×C4×Q8D411yescharacteristic
30C22×C16—21no—
31C22×C16—21no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#107—21no—
35C2×#107D411yescharacteristic
36C2×#107D411yescharacteristic
37C2×#107D411yescharacteristic
38C2×#107D411yescharacteristic
39C2×C4⋊C8—21no—
40C2×C4⋊C8D411yescharacteristic
41C2×#103—41no—
42C2×C4×C8C2311yescharacteristic
43#398—41no—
44#398—41no—
45#391—41no—
46#389—21no—
47#389—21no—
48#389—21no—
49#389—21no—
50#384—21no—
51C4×C4⋊C4—21no—
52#366—42no—
53#340—21no—
54#340—21no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340—41no—
59#336—21no—
60#336—21no—
61C22×Q8—41no—
62C22×Q8D811yescharacteristic
63C22×C8—21no—
64C22×C8—41no—
65C22×C8C2×D411yescharacteristic
66C22×C8C2×D411yescharacteristic
67C4⋊Q8—81no—
68C4⋊Q8—81no—
69C4⋊Q8—82no—
70C4⋊Q8—82no—
71C4×Q8—41no—
72C4×Q8—41no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—41no—
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⋊C4D811yescharacteristic
87C2×C42—21no—
88C2×C42—41no—
89C2×C42C2×D411yescharacteristic
90C2×C16—22no—
91C2×C16—41no—
92C2×C16—41no—
93C2×C16—41no—
94C2×C16—41no—
95C2×C16—41no—
96#107—21no—
97#107—21no—
98#107—41no—
99#107—41no—
100#107—42no—
101#107—41no—
102#107—41no—
103#107—81no—
104#107SD1612yes—
105#107Q1612yes—
106C4⋊C8—41no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109#103—82no—
110C4×C8—41no—
111#95—41no—
112#95—41no—
113#95—41no—
114#95—41no—
115#95—41no—
116C2×Q8—21no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8—82no—
120C2×Q8—82no—
121C2×Q8—82no—
122C2×Q8D1612yes—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—81no—
130C22×C4#12011yescharacteristic
131C22×C4C2×D811yescharacteristic
132C22×C4AGL(1,ℤ/8)11yescharacteristic
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—81no—
141C2×C8—81no—
142C2×C8—81no—
143C2×C8C2×SD1611yescharacteristic
144C2×C8C2×Q1611yescharacteristic
145C2×C8#13711yescharacteristic
146C2×C8#13711yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—81no—
157C4⋊C4—82no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C4⋊C4—81no—
161C4⋊C4—81no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—161no—
169C42—21no—
170C42—21no—
171C42—41no—
172C42—41no—
173C42—82no—
174C16—42no—
175C16—81no—
176C23#44711yescharacteristic
177Q8—42no—
178Q8—81no—
179Q8—164no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—161no—
204C2×C4—161no—
205C2×C4—161no—
206C2×C4#44811yescharacteristic
207C2×C4#45111yescharacteristic
208C2×C4#45411yescharacteristic
209C2×C4#45511yescharacteristic
210C2×C4#45611yescharacteristic
211C2×C4#45811yescharacteristic
212C2×C4C2×D1611yescharacteristic
213C2×C4#50911yescharacteristic
214C8—21no—
215C8—21no—
216C8—21no—
217C8—21no—
218C8—81no—
219C8—161no—
220C22#160711yescharacteristic
221C22#162711yescharacteristic
222C22#164911yescharacteristic
223C22#218411yescharacteristic
224C22#219111yescharacteristic
225C22#219311yescharacteristic
226C22#219511yescharacteristic
227C4—21no—
228C4—21no—
229C4—42no—
230C4—41no—
231C4—82no—
232C4—82no—
233C4—81no—
234C4—161no—
235C4—161no—
236C4—321no—
237C4#161111yescharacteristic
238C4#161711yescharacteristic
239C4#164511yescharacteristic
240C4#166411yescharacteristic
241C2#779411yescharacteristic
242C2#1227811yescharacteristic
243C2#1228611yescharacteristic
244C2#1235211yescharacteristic
245C2#1236311yescharacteristic
246C2#1240411yescharacteristic
247C2#1241311yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
commutator, Frattini, cosocleC2×C4×C8C23
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1984 ◁ 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
#326822
#135372
#135052
#106172
#71282
#71202