Finite Group Databasebeta

← #145185

Subgroups — #145185

Order 512.

294 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 76 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#28297C211yesmaximal, characteristic
2#12910C211yesmaximal, characteristic
3#12819C211yesmaximal, characteristic
4#11612C211yesmaximal, characteristic
5C2×#1334C211yesmaximal, characteristic
6#8683C211yesmaximal, characteristic
7#7280C211yesmaximal, characteristic
8#3078—21no—
9#2960C2211yescharacteristic
10C2×C4×Q16C2211yescharacteristic
11C2×#368—22no—
12#2150C2211yescharacteristic
13C2×#362—21no—
14C2×#346C2211yescharacteristic
15#2064—22no—
16#1979—21no—
17#1952—21no—
18#1848C2211yescharacteristic
19#1848C2211yescharacteristic
20#1801—21no—
21C2×#334C2211yescharacteristic
22#1550—21no—
23#1387—21no—
24#1338—22no—
25#1334—22no—
26#1292—21no—
27C22×Q16D411yescharacteristic
28#533D411yescharacteristic
29C2×C4×Q8—21no—
30C2×M32—21no—
31C22×C16—21no—
32#479—42no—
33#477—41no—
34#475—41no—
35#446—21no—
36#446—21no—
37C8×Q8—21no—
38C8×Q8—21no—
39C4×Q16—21no—
40C4×Q16—21no—
41C4×Q16—21no—
42C4×Q16—21no—
43C2×#108—21no—
44C2×#108—21no—
45C2×#108—21no—
46C2×#108D412yes—
47#428—22no—
48C2×#107D411yescharacteristic
49#424—21no—
50#423—21no—
51#423—21no—
52#423D411yescharacteristic
53C2×C4⋊C8—21no—
54#419—22no—
55C2×#103—21no—
56C2×#101—22no—
57#405—21no—
58C2×C4×C8C2311yescharacteristic
59#368—44no—
60C4⋊C16—21no—
61C4⋊C16—21no—
62#362—42no—
63#346—21no—
64#346D412yes—
65#343—22no—
66#334—21no—
67#334—21no—
68#333—41no—
69#332—41no—
70C22×Q8—21no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×Q16—21no—
74C2×Q16—21no—
75C2×Q16—21no—
76C2×Q16Q1612yes—
77C2×M16—21no—
78C2×M16—22no—
79C2×M16—41no—
80C22×C8—41no—
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C4⋊Q8D811yescharacteristic
84C4⋊Q8D811yescharacteristic
85#125—21no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C4×Q8—21no—
89C4×Q8—21no—
90C4×Q8—41no—
91C4×Q8—41no—
92C4×Q8—41no—
93C4×Q8—41no—
94#117—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—41no—
97C2×C42—41no—
98C2×C42C2×D411yescharacteristic
99M32—41no—
100M32—41no—
101C2×C16—22no—
102C2×C16—22no—
103C2×C16—22no—
104C2×C16—41no—
105#108—42no—
106#108—42no—
107#108—41no—
108#108—41no—
109#108—41no—
110#108—41no—
111#107—21no—
112#107—21no—
113#107—42no—
114#106—42no—
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125#103—42no—
126#103—42no—
127#103—41no—
128#103—41no—
129#101—82no—
130#97—41no—
131C4×C8—41no—
132C4×C8C2×D411yescharacteristic
133C4×C8#4111yescharacteristic
134C4×C8#4111yescharacteristic
135C4×C8#4111yescharacteristic
136C2×Q8—22no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—41no—
140C2×Q8—41no—
141C2×Q8—41no—
142C2×Q8—41no—
143C2×Q8—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4#12011yescharacteristic
147C22×C4#13511yescharacteristic
148C22×C4#13711yescharacteristic
149Q16—22no—
150Q16—22no—
151Q16—41no—
152Q16—41no—
153M16—42no—
154M16—41no—
155M16—42no—
156M16—41no—
157M16—81no—
158M16—81no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—41no—
162C2×C8—42no—
163C2×C8—81no—
164C2×C8—81no—
165C2×C8—161no—
166C2×C8#12111yescharacteristic
167C2×C8#12111yescharacteristic
168C2×C8#12311yescharacteristic
169C2×C8#12411yescharacteristic
170C2×C8C2×Q1611yescharacteristic
171C2×C8#13511yescharacteristic
172C2×C8AGL(1,ℤ/8)11yescharacteristic
173C2×C8#13711yescharacteristic
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C22⋊C4—41no—
185C42—41no—
186C42—81no—
187C42—81no—
188C42#12111yescharacteristic
189C42#12311yescharacteristic
190C42C2×D811yescharacteristic
191C42AGL(1,ℤ/8)11yescharacteristic
192C16—24no—
193C16—81no—
194C23#45211yescharacteristic
195Q8—42no—
196Q8—42no—
197Q8—81no—
198Q8—81no—
199Q8—81no—
200Q8—81no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—161no—
210C2×C4#39411yescharacteristic
211C2×C4#44711yescharacteristic
212C2×C4#44911yescharacteristic
213C2×C4#45111yescharacteristic
214C2×C4#45311yescharacteristic
215C2×C4#45611yescharacteristic
216C2×C4#46611yescharacteristic
217C2×C4#46811yescharacteristic
218C2×C4#48011yescharacteristic
219C2×C4#48211yescharacteristic
220C8—21no—
221C8—21no—
222C8—82no—
223C8—81no—
224C8—161no—
225C8—161no—
226C8#46711yescharacteristic
227C8#46811yescharacteristic
228C8#48411yescharacteristic
229C8#48511yescharacteristic
230C22—21no—
231C22—21no—
232C22#163211yescharacteristic
233C22#165011yescharacteristic
234C22#203311yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—81no—
240C4—81no—
241C4—82no—
242C4—161no—
243C4#162111yescharacteristic
244C4#164911yescharacteristic
245C4#203611yescharacteristic
246C4#203911yescharacteristic
247C2—21no—
248C2—21no—
249C2#1043411yescharacteristic
250C2#1339011yescharacteristic
251C2#1339111yescharacteristic
subgroupG/H
center, socleC22#2033
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#452
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ 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
#282972
#129102
#128192
#116122
C2×#13342
#86832
#72802