Finite Group Databasebeta

← #144530

Subgroups — #144530

Order 512.

319 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 96 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#28814C211yesmaximal, characteristic
2#13282C211yesmaximal, characteristic
3#13276C211yesmaximal, characteristic
4#9730C211yesmaximal, characteristic
5#7274C211yesmaximal, characteristic
6#7171C211yesmaximal, characteristic
7#7167C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9C2×#499—21no—
10#2552—21no—
11C2×C4⋊C16C2211yescharacteristic
12C2×C4×C16C2211yescharacteristic
13#2059—22no—
14#1987—21no—
15#1948—21no—
16#1942—21no—
17#1845C2211yescharacteristic
18C4×#107C2211yescharacteristic
19C8×C4⋊C4C2211yescharacteristic
20C2×#335C2211yescharacteristic
21C2×#333—21no—
22#1380—21no—
23#1380—21no—
24#1380—22no—
25C2×C4⋊Q8—21no—
26C2×#125—22no—
27C22×C16—21no—
28C22×C16—21no—
29#500—21no—
30#500—21no—
31#500—21no—
32#500Q812yes—
33#499—42no—
34#499—42no—
35C2×#107—21no—
36C2×#107—21no—
37C2×#107—21no—
38C2×#107D411yescharacteristic
39C2×#107D411yescharacteristic
40C2×#107D412yes—
41C2×#106—22no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8D411yescharacteristic
44C2×C4×C8—21no—
45C2×C4×C8C2311yescharacteristic
46#398—42no—
47#389—22no—
48#389—41no—
49#384—21no—
50#384—41no—
51C4×C4⋊C4—21no—
52C4×C4⋊C4D411yescharacteristic
53C4⋊C16—21no—
54C4⋊C16—21no—
55C4×C16—21no—
56C4×C16—21no—
57#340—22no—
58#340—21no—
59#340—41no—
60#340—41no—
61#336—21no—
62#335—21no—
63#335—21no—
64#333—42no—
65C22×Q8—41no—
66C22×C8—21no—
67C22×C8—41no—
68C22×C8C2×D411yescharacteristic
69C22×C8C2×D411yescharacteristic
70C4⋊Q8—22no—
71C4⋊Q8—41no—
72C4⋊Q8—41no—
73C4⋊Q8—41no—
74#125—44no—
75#125—82no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4D811yescharacteristic
87C2×C4⋊C4D811yescharacteristic
88C2×C42—21no—
89C2×C42—41no—
90C2×C42C2×D411yescharacteristic
91C2×C16—22no—
92C2×C16—41no—
93C2×C16—41no—
94C2×C16—41no—
95C2×C16—41no—
96C2×C16—41no—
97#107—21no—
98#107—21no—
99#107—22no—
100#107—22no—
101#107—41no—
102#107—41no—
103#107—41no—
104#107—41no—
105#107—42no—
106#107SD1612yes—
107#107Q1612yes—
108#106—44no—
109C4⋊C8—21no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112C4⋊C8SD1612yes—
113C4×C8—41no—
114C4×C8—41no—
115C4×C8C2×Q811yescharacteristic
116C4×C8#4111yescharacteristic
117C4×C8#4111yescharacteristic
118C4×C8#4111yescharacteristic
119#95—21no—
120#95—42no—
121#95—41no—
122C2×Q8—42no—
123C2×Q8—81no—
124C2×Q8—161no—
125C2×Q8—161no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—42no—
131C22×C4—81no—
132C22×C4#12011yescharacteristic
133C22×C4C2×D811yescharacteristic
134C22×C4AGL(1,ℤ/8)11yescharacteristic
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—22no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—81no—
141C2×C8—81no—
142C2×C8—81no—
143C2×C8C22⋊Q811yescharacteristic
144C2×C8C22⋊Q811yescharacteristic
145C2×C8#12411yescharacteristic
146C2×C8#12411yescharacteristic
147C2×C8C2×SD1611yescharacteristic
148C2×C8C2×Q1611yescharacteristic
149C2×C8AGL(1,ℤ/8)11yescharacteristic
150C2×C8#13711yescharacteristic
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C4⋊C4—81no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—161no—
170C4⋊C4—162no—
171C42—41no—
172C42—41no—
173C42—81no—
174C42—81no—
175C42C22⋊Q811yescharacteristic
176C42#12411yescharacteristic
177C42C2×SD1611yescharacteristic
178C42#13711yescharacteristic
179C16—42no—
180C16—81no—
181C23#44711yescharacteristic
182Q8—162no—
183Q8—162no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—161no—
203C2×C4—161no—
204C2×C4#39311yescharacteristic
205C2×C4#44811yescharacteristic
206C2×C4#45011yescharacteristic
207C2×C4#45111yescharacteristic
208C2×C4#45511yescharacteristic
209C2×C4#45611yescharacteristic
210C2×C4#47411yescharacteristic
211C2×C4#47611yescharacteristic
212C2×C4#48611yescharacteristic
213C2×C4#48911yescharacteristic
214C2×C4#50811yescharacteristic
215C2×C4#50811yescharacteristic
216C2×C4#51011yescharacteristic
217C2×C4#51011yescharacteristic
218C8—21no—
219C8—21no—
220C8—42no—
221C8—161no—
222C8#47511yescharacteristic
223C8#47711yescharacteristic
224C8#48711yescharacteristic
225C8#48911yescharacteristic
226C22#163911yescharacteristic
227C22#164911yescharacteristic
228C22#202311yescharacteristic
229C22#218911yescharacteristic
230C22#218911yescharacteristic
231C22#219211yescharacteristic
232C22#219311yescharacteristic
233C4—21no—
234C4—21no—
235C4—81no—
236C4—81no—
237C4—82no—
238C4—82no—
239C4—162no—
240C4—162no—
241C4#164511yescharacteristic
242C4#165111yescharacteristic
243C4#202411yescharacteristic
244C4#202811yescharacteristic
245C4#222911yescharacteristic
246C4#222911yescharacteristic
247C4#224211yescharacteristic
248C4#224511yescharacteristic
249C2#1035011yescharacteristic
250C2#1237311yescharacteristic
251C2#1237511yescharacteristic
252C2#1240911yescharacteristic
253C2#1241111yescharacteristic
254C2#1336611yescharacteristic
255C2#1336911yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
commutator, Frattini, cosocleC2×C4×C8C23
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ 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
#288142
#132822
#132762
#97302
#72742
#71712
#71672