Finite Group Databasebeta

← #100805

Subgroups — #100805

Order 512.

335 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 80 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#29211C211yesmaximal, characteristic
2#23366C211yesmaximal, characteristic
3#8593C211yesmaximal, characteristic
4#7722C211yesmaximal, characteristic
5#7722C211yesmaximal, characteristic
6#7041C211yesmaximal, characteristic
7#7041C211yesmaximal, characteristic
8#2862C2211yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10#2058C412yes—
11#2057C412yes—
12#1993—21no—
13#1985—22no—
14#1949—22no—
15#1917—22no—
16#1917—22no—
17#1847—21no—
18#1839C2211yescharacteristic
19#1839C2211yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1767C2211yescharacteristic
23C2×#332—21no—
24#1296—21no—
25#1272—22no—
26#1272—22no—
27C2×C4⋊Q8C2×C411yescharacteristic
28C2×C4⋊Q8C2×C411yescharacteristic
29C2×C4⋊Q8D411yescharacteristic
30C2×C4×Q8—21no—
31C2×#107—22no—
32C2×#106—22no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—22no—
37C2×#97—21no—
38C2×#97—22no—
39#391—21no—
40#391—21no—
41#391—21no—
42#391—21no—
43#391—21no—
44#389—21no—
45#389C2×C412yes—
46#384D411yescharacteristic
47#382—21no—
48#382—21no—
49C4×C4⋊C4—21no—
50C4×C4⋊C4D412yes—
51C43C2311yescharacteristic
52#340—22no—
53#340—22no—
54#336—24no—
55#336—21no—
56#336—21no—
57#336—21no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336C2×C412yes—
64#336D412yes—
65#336D412yes—
66#332—42no—
67C22×Q8—21no—
68C22×Q8—21no—
69C22×Q8—21no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—41no—
74C22×C8—41no—
75C4⋊Q8—21no—
76C4⋊Q8—21no—
77C4⋊Q8—21no—
78C4⋊Q8—21no—
79C4⋊Q8—81no—
80C4⋊Q8—81no—
81C4×Q8—81no—
82C4×Q8—81no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4C22⋊C412yes—
92C2×C4⋊C4C22⋊C412yes—
93C2×C42—22no—
94C2×C42C22×C411yescharacteristic
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D411yescharacteristic
97C2×C42C2×D411yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C42#4111yescharacteristic
100C2×C42#4111yescharacteristic
101#107—82no—
102#106—82no—
103C4⋊C8—24no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108C4⋊C8—82no—
109#97—41no—
110#97—41no—
111#97—82no—
112#95—22no—
113#95—22no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×Q8—81no—
123C2×Q8—81no—
124C2×Q8—81no—
125C2×Q8—81no—
126C2×Q8—81no—
127C2×Q8—81no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4C4×D411yescharacteristic
134C22×C4C4×D411yescharacteristic
135C22×C4#12011yescharacteristic
136C22×C4#12111yescharacteristic
137C22×C4#12111yescharacteristic
138C22×C4#12311yescharacteristic
139C2×C8—24no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8—81no—
151C2×C8—81no—
152C2×C8—81no—
153C2×C8—81no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—81no—
162C4⋊C4—81no—
163C4⋊C4—82no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C42—22no—
167C42—41no—
168C42—41no—
169C42—41no—
170C42—41no—
171C42—41no—
172C42—42no—
173C42—42no—
174C42AGL(1,ℤ/8)11yescharacteristic
175C42#13711yescharacteristic
176C42#13711yescharacteristic
177C42#13711yescharacteristic
178C23#38611yescharacteristic
179Q8—82no—
180Q8—82no—
181Q8—162no—
182Q8—162no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4#41711yescharacteristic
213C2×C4#41911yescharacteristic
214C2×C4#45612yes—
215C2×C4#46811yescharacteristic
216C2×C4#46811yescharacteristic
217C2×C4#47011yescharacteristic
218C2×C4#47011yescharacteristic
219C8—44no—
220C8—82no—
221C8—82no—
222C8—161no—
223C8—161no—
224C22#188111yescharacteristic
225C22#188111yescharacteristic
226C22#188611yescharacteristic
227C22#189411yescharacteristic
228C22#190411yescharacteristic
229C22#190511yescharacteristic
230C22#191411yescharacteristic
231C4—21no—
232C4—21no—
233C4—21no—
234C4—21no—
235C4—21no—
236C4—21no—
237C4—22no—
238C4—41no—
239C4—41no—
240C4—41no—
241C4—82no—
242C4—82no—
243C4—161no—
244C4—161no—
245C2#900611yescharacteristic
246C2#930711yescharacteristic
247C2#930811yescharacteristic
248C2#933311yescharacteristic
249C2#933411yescharacteristic
250C2#940011yescharacteristic
251C2#942711yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#292112
#233662
#85932
#77222
#70412