Finite Group Databasebeta

← #101458

Subgroups — #101458

Order 512.

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

normal subgroups only

This group has 359 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#29277C211yesmaximal, characteristic
2#22311C211yesmaximal, characteristic
3#9874C211yesmaximal, characteristic
4#8591C211yesmaximal, characteristic
5#7653C211yesmaximal, characteristic
6#7040C211yesmaximal, characteristic
7#7018C211yesmaximal, characteristic
8C4×C4⋊Q8C2211yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10#2100C412yes—
11#2057C412yes—
12#1987—21no—
13#1985—22no—
14#1947—21no—
15#1939—21no—
16#1921—22no—
17#1917—22no—
18#1842C2211yescharacteristic
19#1831C2211yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1770—21no—
23#1744C2211yescharacteristic
24C2×#332—21no—
25#1296—21no—
26#1279—21no—
27#1279—21no—
28#1272—22no—
29C2×C4⋊Q8C2×C411yescharacteristic
30C2×C4⋊Q8D411yescharacteristic
31C2×C4×Q8—21no—
32C2×C4×Q8—21no—
33C2×#107—22no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—21no—
41C2×#97—22no—
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#391—21no—
45#391—21no—
46#389—21no—
47#389—21no—
48#389—21no—
49#389C2×C412yes—
50#384C2×C411yescharacteristic
51#384D411yescharacteristic
52C4×C4⋊C4—21no—
53C4×C4⋊C4—21no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4D412yes—
56C43C2311yescharacteristic
57#340—22no—
58#340—22no—
59#340—22no—
60#336—22no—
61#336—22no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336C2×C412yes—
69#336D412yes—
70#332—42no—
71C22×Q8—21no—
72C22×Q8—21no—
73C22×C8—21no—
74C22×C8—22no—
75C22×C8—21no—
76C22×C8—22no—
77C22×C8—41no—
78C22×C8—41no—
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81C4⋊Q8—21no—
82C4⋊Q8—21no—
83C4⋊Q8—42no—
84C4×Q8—81no—
85C4×Q8—81no—
86C4×Q8—82no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4C22⋊C412yes—
98C2×C4⋊C4C22⋊C412yes—
99C2×C42—22no—
100C2×C42—41no—
101C2×C42C22×C411yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C42#4111yescharacteristic
107C2×C42#4111yescharacteristic
108#107—44no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—81no—
120C4⋊C8—82no—
121#97—82no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8—21no—
125C4×C8Q1612yes—
126#95—22no—
127#95—22no—
128#95—41no—
129#95—41no—
130C2×Q8—22no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—81no—
135C2×Q8—81no—
136C2×Q8—81no—
137C2×Q8—81no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4C2×C22⋊C411yescharacteristic
143C22×C4C4×D411yescharacteristic
144C22×C4C4×D411yescharacteristic
145C22×C4#12011yescharacteristic
146C22×C4#12111yescharacteristic
147C22×C4#12111yescharacteristic
148C22×C4#12311yescharacteristic
149C2×C8—22no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—42no—
153C2×C8—41no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—41no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—41no—
160C2×C8—42no—
161C2×C8—81no—
162C2×C8—81no—
163C2×C8—81no—
164C2×C8—81no—
165C2×C8—81no—
166C2×C8—81no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—42no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—81no—
181C4⋊C4—82no—
182C4⋊C4—81no—
183C4⋊C4—82no—
184C42—21no—
185C42—21no—
186C42—41no—
187C42—41no—
188C42—41no—
189C42—41no—
190C42—42no—
191C42—42no—
192C42—81no—
193C42—81no—
194C42C2×Q1611yescharacteristic
195C42#13511yescharacteristic
196C42AGL(1,ℤ/8)11yescharacteristic
197C42#13711yescharacteristic
198C42#13711yescharacteristic
199C42#13711yescharacteristic
200C42#13711yescharacteristic
201C42#13711yescharacteristic
202C23#38611yescharacteristic
203Q8—82no—
204Q8—82no—
205Q8—82no—
206Q8—82no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4#41711yescharacteristic
234C2×C4#41911yescharacteristic
235C2×C4C4×Q1611yescharacteristic
236C2×C4#44111yescharacteristic
237C2×C4#45111yescharacteristic
238C2×C4#45211yescharacteristic
239C2×C4#46211yescharacteristic
240C2×C4#46411yescharacteristic
241C2×C4#46811yescharacteristic
242C2×C4#46811yescharacteristic
243C2×C4#47011yescharacteristic
244C2×C4#47011yescharacteristic
245C2×C4#47111yescharacteristic
246C2×C4#47211yescharacteristic
247C8—42no—
248C8—82no—
249C8—82no—
250C8—81no—
251C8—161no—
252C8—161no—
253C22#187811yescharacteristic
254C22#188111yescharacteristic
255C22#189111yescharacteristic
256C22#189411yescharacteristic
257C22#190411yescharacteristic
258C22#190711yescharacteristic
259C22#191211yescharacteristic
260C4—21no—
261C4—21no—
262C4—21no—
263C4—21no—
264C4—21no—
265C4—21no—
266C4—21no—
267C4—21no—
268C4—41no—
269C4—41no—
270C4—82no—
271C4—82no—
272C4—82no—
273C4—161no—
274C4#169211yescharacteristic
275C4#169611yescharacteristic
276C4#169811yescharacteristic
277C4#170411yescharacteristic
278C2#904811yescharacteristic
279C2#922411yescharacteristic
280C2#929811yescharacteristic
281C2#930711yescharacteristic
282C2#937211yescharacteristic
283C2#948811yescharacteristic
284C2#952911yescharacteristic
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
#292772
#223112
#98742
#85912
#76532
#70402
#70182