Finite Group Databasebeta

← #224634

Subgroups — #224634

Order 512.

699 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36600C211yesmaximal, characteristic
2#21345C212yesmaximal
3#9690C212yesmaximal
4#9690C212yesmaximal
5#9686C212yesmaximal
6#8584C211yesmaximal, characteristic
7#8582C212yesmaximal
8#8227C212yesmaximal
9#8225C211yesmaximal, characteristic
10#2859C2212yes—
11#2859C2211yescharacteristic
12#2585—22no—
13#2585—22no—
14#2570C412yes—
15#2570C412yes—
16#2552C412yes—
17#2453—22no—
18#2370—22no—
19#2355C412yes—
20C4×#125C2211yescharacteristic
21C4×#125C2212yes—
22C42×Q8C2211yescharacteristic
23#1984—22no—
24#1939—22no—
25#1917—22no—
26#1917—22no—
27#1767C2212yes—
28#1767C2212yes—
29#1767C2212yes—
30#1745C2212yes—
31#1745C2212yes—
32C4×#97C2212yes—
33C2×#330C2212yes—
34C2×#330C2212yes—
35C2×#330C2212yes—
36C2×#330C2212yes—
37C2×#330C2212yes—
38C2×#330C2211yescharacteristic
39C2×#330C2211yescharacteristic
40#1295C2211yescharacteristic
41#1295C2211yescharacteristic
42#1295C2212yes—
43C2×#125—22no—
44C2×#125C2311yescharacteristic
45C2×#125C2312yes—
46C2×C4×Q8C2×C411yescharacteristic
47C2×C4×Q8D412yes—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×#97—22no—
52C2×#97C2312yes—
53C2×#97C2312yes—
54C2×#97C2312yes—
55C2×#97C2312yes—
56C2×C4×C8—22no—
57#400—22no—
58#400—22no—
59#398—22no—
60#398—22no—
61#398—22no—
62#398—22no—
63#391—22no—
64#389—22no—
65#389—22no—
66#389—22no—
67#389—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389C2×C412yes—
72#389C2×C412yes—
73#389C2×C412yes—
74#387—22no—
75#387—22no—
76#387—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#384C2×C412yes—
84#384C2×C412yes—
85#384C2×C412yes—
86#382C2×C411yescharacteristic
87#382C2×C411yescharacteristic
88#382C2×C412yes—
89#382C2×C412yes—
90#382C2311yescharacteristic
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4C2×C412yes—
98C4×C4⋊C4C2×C412yes—
99C4×C4⋊C4C2×C411yescharacteristic
100C4×C4⋊C4D412yes—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C43C2312yes—
104C43C2311yescharacteristic
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#330—24no—
113#330—24no—
114#330—24no—
115#330—42no—
116#330—42no—
117#330—41no—
118#330—41no—
119C22×Q8C22⋊C411yescharacteristic
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124#125—22no—
125#125—22no—
126#125—22no—
127#125—21no—
128#125—22no—
129#125—21no—
130#125—22no—
131#125—48no—
132C4×Q8—24no—
133C4×Q8—44no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C22⋊C411yescharacteristic
147C2×C4⋊C4C22⋊C412yes—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C22×C412yes—
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C22×C412yes—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42C22×C412yes—
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C411yescharacteristic
161C2×C42C22×C412yes—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C2×C42C2411yescharacteristic
169C4⋊C8—44no—
170C4⋊C8—44no—
171C4⋊C8—44no—
172#97—22no—
173#97—22no—
174#97—22no—
175#97—22no—
176#97—22no—
177#97—22no—
178#97—22no—
179#97—44no—
180#97#4114yes—
181C4×C8—44no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95C22⋊C412yes—
196#95C22⋊C412yes—
197#95C22⋊C411yescharacteristic
198#95C22⋊C411yescharacteristic
199C2×Q8—42no—
200C2×Q8#10114yes—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4C2×C22⋊C412yes—
206C22×C4C2×C22⋊C412yes—
207C22×C4C2×C22⋊C411yescharacteristic
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4C2×C22⋊C412yes—
212C22×C4C23×C411yescharacteristic
213C22×C4C22×D412yes—
214C2×C8—24no—
215C2×C8—24no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C42—24no—
254C42—22no—
255C42—24no—
256C42—42no—
257C42—42no—
258C42—42no—
259C42—42no—
260C42—41no—
261C42—42no—
262C42—42no—
263C42—44no—
264C42—82no—
265C42—82no—
266C42C2×#4112yes—
267C42H(5,𝔽2)11yescharacteristic
268C422−1+411yescharacteristic
269C23C22×C22⋊C411yescharacteristic
270Q8—44no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4C2×#10114yes—
304C2×C4C2×#10112yes—
305C2×C4#42114yes—
306C2×C4C4×#4111yescharacteristic
307C2×C4#52011yescharacteristic
308C2×C4#54812yes—
309C2×C4#55012yes—
310C8—44no—
311C8—44no—
312C8—82no—
313C8—82no—
314C22#234111yescharacteristic
315C22C22×#10111yescharacteristic
316C22#288911yescharacteristic
317C22C2×#42112yes—
318C22#290112yes—
319C4—21no—
320C4—21no—
321C4—24no—
322C4—22no—
323C4—24no—
324C4—22no—
325C4—22no—
326C4—42no—
327C4—42no—
328C4—42no—
329C4—41no—
330C4—82no—
331C4—82no—
332C4—82no—
333C4—82no—
334C2#1767611yescharacteristic
335C2#1768012yes—
336C2#1769211yescharacteristic
337C2#2253811yescharacteristic
338C2#2255112yes—
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#366002
#213452
#96902
#96862
#85842
#85822
#82272
#82252