Finite Group Databasebeta

← #134408

Subgroups — #134408

Order 512.

354 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 114 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#21276C211yesmaximal, characteristic
2#12653C211yesmaximal, characteristic
3C2×#1334C212yesmaximal
4#8805C211yesmaximal, characteristic
5#7178C212yesmaximal
6C2×C4×Q16C2211yescharacteristic
7#2295—21no—
8C2×C4⋊C16—21no—
9C2×#346C2212yes—
10#1969C411yescharacteristic
11#1969C411yescharacteristic
12#1942C411yescharacteristic
13#1942C411yescharacteristic
14#1933—21no—
15#1933—21no—
16#1842—21no—
17C4×#107C2211yescharacteristic
18C4×#103—21no—
19C42×C8C2211yescharacteristic
20C2×#334—21no—
21C2×#334C2212yes—
22#1374—21no—
23#1334—24no—
24C22×Q16C2×C411yescharacteristic
25C2×C4×Q8—21no—
26C22×C16—22no—
27C4×Q16—21no—
28C4×Q16—21no—
29C4×Q16D412yes—
30C4×Q16Q812yes—
31C2×#107C2×C411yescharacteristic
32C2×#107C2×C411yescharacteristic
33C2×#107C2×C411yescharacteristic
34C2×C4⋊C8—22no—
35C2×C4⋊C8—41no—
36C2×C4⋊C8—41no—
37C2×#103—21no—
38C2×#103—21no—
39C2×#103—21no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46#391—21no—
47#391—21no—
48#389—21no—
49#389—21no—
50#387—41no—
51#387—41no—
52C4×C4⋊C4—21no—
53C4×C4⋊C4—41no—
54#376—41no—
55C43D411yescharacteristic
56C4⋊C16—24no—
57#346—22no—
58#346—22no—
59#340—21no—
60#340—21no—
61#340—21no—
62#334—22no—
63#334—22no—
64#334—42no—
65C22×Q8—21no—
66C2×Q16—21no—
67C2×Q16—22no—
68C2×Q16—22no—
69C2×Q16C4⋊C412yes—
70C22×C8—42no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22×C411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C4×Q8—22no—
76C4×Q8—22no—
77C4×Q8—41no—
78C4×Q8—41no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C42—41no—
86C2×C42—41no—
87C2×C42C22⋊C411yescharacteristic
88C2×C42C22⋊C411yescharacteristic
89C2×C42D811yescharacteristic
90C2×C42D811yescharacteristic
91C2×C42SD1611yescharacteristic
92C2×C42SD1611yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C16—24no—
95C2×C16—42no—
96C2×C16—42no—
97#107—21no—
98#107—22no—
99#107—22no—
100#107C4⋊C412yes—
101C4⋊C8—42no—
102C4⋊C8—42no—
103C4⋊C8—82no—
104C4⋊C8—82no—
105#103—22no—
106#103—22no—
107#103—42no—
108#103—42no—
109C4×C8—22no—
110C4×C8—22no—
111C4×C8—22no—
112C4×C8—21no—
113C4×C8—21no—
114C4×C8—41no—
115C4×C8C2×D411yescharacteristic
116C4×C8C2×Q811yescharacteristic
117C4×C8#4111yescharacteristic
118C4×C8#4111yescharacteristic
119#95—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124#95—41no—
125#95—41no—
126C2×Q8—22no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4#10211yescharacteristic
137C22×C4#10211yescharacteristic
138C22×C4#10211yescharacteristic
139C22×C4#10211yescharacteristic
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4C2×D811yescharacteristic
142C22×C4C2×SD1611yescharacteristic
143Q16—24no—
144Q16—42no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—82no—
153C2×C8—82no—
154C2×C8—82no—
155C2×C8#10412yes—
156C2×C8C2×C4⋊C411yescharacteristic
157C2×C8#11711yescharacteristic
158C2×C8#12111yescharacteristic
159C2×C8C22⋊Q811yescharacteristic
160C2×C8#13112yes—
161C2×C8#13112yes—
162C4⋊C4—22no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C42—22no—
173C42—22no—
174C42—22no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—41no—
180C42—42no—
181C42—42no—
182C42—82no—
183C42#12111yescharacteristic
184C42C22⋊Q811yescharacteristic
185C42AGL(1,ℤ/8)11yescharacteristic
186C42#13711yescharacteristic
187C16—44no—
188C23C2×#10211yescharacteristic
189Q8—42no—
190Q8—42no—
191Q8—81no—
192Q8—81no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4#36012yes—
219C2×C4#36012yes—
220C2×C4#36212yes—
221C2×C4#38011yescharacteristic
222C2×C4#40811yescharacteristic
223C2×C4#40811yescharacteristic
224C2×C4#41711yescharacteristic
225C2×C4C2×#10411yescharacteristic
226C2×C4#42111yescharacteristic
227C2×C4#45911yescharacteristic
228C2×C4#46111yescharacteristic
229C2×C4#47411yescharacteristic
230C2×C4#47611yescharacteristic
231C8—21no—
232C8—21no—
233C8—22no—
234C8—41no—
235C8—162no—
236C8#42412yes—
237C8#43312yes—
238C22#148011yescharacteristic
239C22#149211yescharacteristic
240C22#180211yescharacteristic
241C22C2×#36011yescharacteristic
242C22C2×#36011yescharacteristic
243C22C2×#36211yescharacteristic
244C22#214811yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—21no—
249C4—22no—
250C4—22no—
251C4—22no—
252C4—41no—
253C4—82no—
254C4—82no—
255C4—161no—
256C4—161no—
257C4#148311yescharacteristic
258C4#149511yescharacteristic
259C4#179811yescharacteristic
260C4#180611yescharacteristic
261C2#856711yescharacteristic
262C2#1165111yescharacteristic
263C2#1165511yescharacteristic
264C2#1167711yescharacteristic
265C2#1168111yescharacteristic
266C2#1281611yescharacteristic
267C2#1281911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ 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
#212762
#126532
C2×#13342
#88052
#71782