Finite Group Databasebeta

← #100743

Subgroups — #100743

Order 512.

345 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 72 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#29578C211yesmaximal, characteristic
2#22015C211yesmaximal, characteristic
3#9894C211yesmaximal, characteristic, complemented
4#7703C211yesmaximal, characteristic, complemented
5#7608C211yesmaximal, characteristic
6#7048C211yesmaximal, characteristic, complemented
7#7043C211yesmaximal, characteristic, complemented
8#2859C2211yescharacteristic
9#2293C2211yescharacteristic
10#2099C411yescharacteristic
11#2093C411yescharacteristic
12#2084C411yescharacteristic
13#2071C411yescharacteristic
14#1955—21no—
15#1944—21no—
16#1940—21no—
17#1936—21no—
18#1924—22no—
19#1921—21no—
20#1864—21no—
21#1839C2211yescharacteristic
22#1818—21no—
23#1798—21no—
24#1767C2211yescharacteristic
25#1745C2211yescharacteristic
26#1745C2211yescharacteristic
27#1745C2211yescharacteristic
28#1295—22no—
29#1271—21no—
30#1271—21no—
31#1271—21no—
32#1270—21no—
33C2×#125—21no—
34C2×C4×D4D411yescharacteristic
35C2×#117—21no—
36C2×#107—21no—
37C2×#106—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—22no—
40C2×#102—21no—
41C2×#102—21no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×#97—21no—
45C2×#97—21no—
46C2×#97—21no—
47C2×#97D412yes—
48#390C2×C411yescharacteristic
49#390D411yescharacteristic
50#389—21no—
51#389—21no—
52#389—21no—
53#389—21no—
54#389C2×C411yescharacteristic
55#389D411yescharacteristic
56#388—21no—
57#388—21no—
58#384C2×C411yescharacteristic
59#384C2×C411yescharacteristic
60#382—21no—
61C4×C4⋊C4D411yescharacteristic
62C4×C22⋊C4—21no—
63C43C2311yescharacteristic
64#340—21no—
65#340—21no—
66#340—21no—
67#340—21no—
68#336—22no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336C2×C411yescharacteristic
78#336C2×C411yescharacteristic
79#336D411yescharacteristic
80#336D411yescharacteristic
81C22×D4C22⋊C411yescharacteristic
82C23×C4—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—22no—
87C22×C8—21no—
88C22×C8—41no—
89C22×C8—41no—
90#125—82no—
91C4×D4—41no—
92C4×D4—41no—
93#117—82no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4C22⋊C411yescharacteristic
105C2×C4⋊C4C22⋊C411yescharacteristic
106C2×C4⋊C4C22⋊C411yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C42—21no—
113C2×C42C22×C411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42#4111yescharacteristic
119C2×C42#4111yescharacteristic
120#107—42no—
121#106—42no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—82no—
125#102—42no—
126#102—42no—
127C22⋊C8—82no—
128C22⋊C8—82no—
129#97—22no—
130#97—42no—
131#97—41no—
132#97—41no—
133#97—41no—
134#97—42no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—41no—
139#95—41no—
140C24—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—41no—
144C2×D4—41no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4C2×C22⋊C411yescharacteristic
155C22×C4C4×D411yescharacteristic
156C22×C4C4×D411yescharacteristic
157C22×C4#12011yescharacteristic
158C22×C4#12111yescharacteristic
159C22×C4#12111yescharacteristic
160C22×C4#12311yescharacteristic
161C2×C8—22no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—42no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—42no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—81no—
178C2×C8—81no—
179C2×C8—81no—
180C2×C8—81no—
181C2×C8—81no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—42no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4—81no—
196C4⋊C4—82no—
197C4⋊C4—82no—
198C22⋊C4—42no—
199C22⋊C4—81no—
200C22⋊C4—81no—
201C22⋊C4—81no—
202C22⋊C4—81no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—41no—
208C42—41no—
209C42—41no—
210C42—41no—
211C42—41no—
212C42—42no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23#38611yescharacteristic
221D4—42no—
222D4—81no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4#42111yescharacteristic
260C2×C4#42111yescharacteristic
261C2×C4#45511yescharacteristic
262C2×C4#45611yescharacteristic
263C8—42no—
264C8—81no—
265C8—82no—
266C8—81no—
267C8—81no—
268C8—82no—
269C8—161no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22#188111yescharacteristic
278C22#188211yescharacteristic
279C22#188411yescharacteristic
280C22#189711yescharacteristic
281C22#190311yescharacteristic
282C22#190411yescharacteristic
283C22#191511yescharacteristic
284C4—21no—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4—161no—
297C4—161no—
298C2—161no—
299C2#899811yescharacteristic
300C2#931311yescharacteristic
301C2#932711yescharacteristic
302C2#933511yescharacteristic
303C2#933811yescharacteristic
304C2#939011yescharacteristic
305C2#941711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #9894C2semidirect
4. #7703C2semidirect
6. #7048C2semidirect
7. #7043C2semidirect

Maximal subgroups

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

subgroupindex
#295782
#220152
#98942
#77032
#76082
#70482
#70432