Finite Group Databasebeta

← #100704

Subgroups — #100704

Order 512.

348 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 88 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#29989C211yesmaximal, characteristic
2#22737C211yesmaximal, characteristic
3#7720C211yesmaximal, characteristic
4#7683C211yesmaximal, characteristic
5#7581C211yesmaximal, characteristic
6#7041C211yesmaximal, characteristic
7#7018C211yesmaximal, characteristic
8C2×#446—21no—
9#2859C2211yescharacteristic
10#2295C2211yescharacteristic
11#2100C411yescharacteristic
12#2099C411yescharacteristic
13#2096C411yescharacteristic
14#2095C411yescharacteristic
15#2059—21no—
16#2026—21no—
17#1949—21no—
18#1940—21no—
19#1940—21no—
20#1939—21no—
21#1839C2211yescharacteristic
22#1832—21no—
23#1831C2211yescharacteristic
24#1831C2211yescharacteristic
25#1805—21no—
26#1768—21no—
27#1767C2211yescharacteristic
28#1744C2211yescharacteristic
29#1272—21no—
30#1271—21no—
31#1271—21no—
32#1271—21no—
33#1271—21no—
34#1271—21no—
35C2×#125—21no—
36C2×C4×Q8D411yescharacteristic
37#446—42no—
38#446—42no—
39C2×#107—21no—
40C2×#106—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×#103—21no—
49C2×#103—21no—
50C2×#97—21no—
51C2×#97—21no—
52C2×C4×C8—21no—
53C2×C4×C8—21no—
54#391C2×C411yescharacteristic
55#391D411yescharacteristic
56#389—21no—
57#389—21no—
58#389—21no—
59#389—21no—
60#389C2×C411yescharacteristic
61#389D411yescharacteristic
62#387—21no—
63#387—21no—
64#384C2×C411yescharacteristic
65#384C2×C411yescharacteristic
66#382—21no—
67C4×C4⋊C4—21no—
68C4×C4⋊C4D411yescharacteristic
69#376—21no—
70C43C2311yescharacteristic
71#340—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336C2×C411yescharacteristic
85#336C2×C411yescharacteristic
86#336D411yescharacteristic
87#336D411yescharacteristic
88#336D411yescharacteristic
89#336D411yescharacteristic
90C22×Q8C22⋊C411yescharacteristic
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—41no—
98C22×C8—41no—
99#125—81no—
100#125—81no—
101C4×Q8—21no—
102C4×Q8—21no—
103C4×Q8SD1612yes—
104C4×Q8Q1612yes—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C22⋊C411yescharacteristic
117C2×C4⋊C4C22⋊C411yescharacteristic
118C2×C4⋊C4C22⋊C411yescharacteristic
119C2×C42—21no—
120C2×C42—21no—
121C2×C42C22×C411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42#4111yescharacteristic
127C2×C42#4111yescharacteristic
128#107—81no—
129#106—81no—
130C4⋊C8—42no—
131C4⋊C8—42no—
132C4⋊C8—42no—
133C4⋊C8—42no—
134C4⋊C8—42no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139#103—81no—
140#103—81no—
141#97—42no—
142#97—41no—
143#97—41no—
144C4×C8—22no—
145C4×C8—41no—
146C4×C8—42no—
147#95—21no—
148#95—21no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—41no—
155#95—41no—
156C2×Q8—21no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8#10312yes—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4C2×C22⋊C411yescharacteristic
167C22×C4C4×D411yescharacteristic
168C22×C4C4×D411yescharacteristic
169C22×C4#12011yescharacteristic
170C22×C4#12111yescharacteristic
171C22×C4#12111yescharacteristic
172C22×C4#12311yescharacteristic
173C2×C8—22no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—42no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—81no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—81no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C4⋊C4#10312yes—
217C42—21no—
218C42—21no—
219C42—22no—
220C42—22no—
221C42—41no—
222C42—41no—
223C42—41no—
224C42—41no—
225C42—41no—
226C42—41no—
227C42—41no—
228C42C2×SD1611yescharacteristic
229C42C2×Q1611yescharacteristic
230C42AGL(1,ℤ/8)11yescharacteristic
231C42#13711yescharacteristic
232C23#38611yescharacteristic
233Q8—42no—
234Q8—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4C2×#10311yescharacteristic
271C2×C4#41711yescharacteristic
272C2×C4#44311yescharacteristic
273C2×C4#44411yescharacteristic
274C2×C4#45011yescharacteristic
275C2×C4#45111yescharacteristic
276C2×C4#48311yescharacteristic
277C2×C4#48411yescharacteristic
278C8—42no—
279C8—81no—
280C8—81no—
281C8—81no—
282C8—81no—
283C8—81no—
284C8—82no—
285C8—161no—
286C22#187211yescharacteristic
287C22#187611yescharacteristic
288C22#188111yescharacteristic
289C22#190511yescharacteristic
290C22#190611yescharacteristic
291C22#191011yescharacteristic
292C22#191511yescharacteristic
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—42no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—42no—
306C4—81no—
307C4—81no—
308C4—161no—
309C4—161no—
310C2#899311yescharacteristic
311C2#917611yescharacteristic
312C2#919311yescharacteristic
313C2#926611yescharacteristic
314C2#928211yescharacteristic
315C2#933611yescharacteristic
316C2#937011yescharacteristic
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
#299892
#227372
#77202
#76832
#75812
#70412
#70182