Finite Group Databasebeta

← #223128

Subgroups — #223128

Order 512.

787 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36196C211yesmaximal, characteristic
2#21107C212yesmaximal
3#9894C212yesmaximal, complemented
4#9125C212yesmaximal, complemented
5#9121C212yesmaximal, complemented
6#8720C212yesmaximal, complemented
7#8580C211yesmaximal, characteristic, complemented
8#8225C211yesmaximal, characteristic, complemented
9#8036C212yesmaximal, complemented
10C2×#432C2212yes—
11C22×#97C2212yes—
12#2859C2211yescharacteristic
13#2851C2212yes—
14#2573—22no—
15#2562C412yes—
16#2533—22no—
17#2497C412yes—
18#2496C412yes—
19#2466—22no—
20#2373—22no—
21#2370C412yes—
22C4×#125C2211yescharacteristic
23#2283C2212yes—
24C42×D4C2211yescharacteristic
25#1924—22no—
26#1917—22no—
27#1869—22no—
28#1850—22no—
29#1767C2212yes—
30#1749C2212yes—
31#1749C2212yes—
32#1745C2212yes—
33C2×#330C2212yes—
34C2×#330C2211yescharacteristic
35C2×#330C2211yescharacteristic
36#1295C2211yescharacteristic
37#1295C2212yes—
38#1295C2212yes—
39#1295C2212yes—
40#1295C2211yescharacteristic
41#1295C2212yes—
42#1295C2212yes—
43C23×C8—22no—
44C2×#126—22no—
45C2×#125C2×C412yes—
46C2×#125C2311yescharacteristic
47C2×C4×D4C2×C411yescharacteristic
48C2×C4×D4D412yes—
49C22×C42C2312yes—
50#432—24no—
51#432—42no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8—22no—
57C2×#97—24no—
58C2×#97—22no—
59C2×#97—42no—
60C2×#97C2312yes—
61C2×#97C2312yes—
62C2×#97C2312yes—
63C2×#97C2312yes—
64#400—22no—
65#400—22no—
66#400—22no—
67#397—22no—
68#396—22no—
69#394—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389C2×C412yes—
74#389C2×C412yes—
75#389C2×C412yes—
76#388—22no—
77#388—22no—
78#388—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#385—22no—
87#385—22no—
88#384C2×C411yescharacteristic
89#384C2×C411yescharacteristic
90#383—22no—
91#382C2311yescharacteristic
92#382C2312yes—
93#381—22no—
94#380C2×C412yes—
95#380C2×C412yes—
96#380C2×C412yes—
97#380C2×C412yes—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4C2×C412yes—
103C4×C4⋊C4C2×C411yescharacteristic
104C4×C4⋊C4D412yes—
105C4×C22⋊C4—22no—
106C4×C22⋊C4C2×C412yes—
107C4×C22⋊C4D412yes—
108C4×C22⋊C4D412yes—
109#376—22no—
110C43C2311yescharacteristic
111#336—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#330—24no—
117#330—41no—
118#330—41no—
119C22×D4C22⋊C411yescharacteristic
120C23×C4C22×C412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—44no—
128C22×C8—44no—
129C22×C8—44no—
130#126—48no—
131#125—22no—
132#125—21no—
133#125—21no—
134#125—44no—
135C4×D4—42no—
136C4×D4—44no—
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⋊C4C22⋊C411yescharacteristic
146C2×C4⋊C4C22⋊C412yes—
147C2×C4⋊C4C22×C412yes—
148C2×C4⋊C4C22×C412yes—
149C2×C4⋊C4C22×C412yes—
150C2×C4⋊C4C22×C412yes—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4C22⋊C412yes—
156C2×C22⋊C4C22⋊C412yes—
157C2×C22⋊C4C22⋊C412yes—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C412yes—
167C2×C42C22×C411yescharacteristic
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2411yescharacteristic
173C4⋊C8—44no—
174C4⋊C8—44no—
175C22⋊C8—44no—
176C22⋊C8—44no—
177C22⋊C8—44no—
178#97—22no—
179#97—22no—
180#97—22no—
181#97—22no—
182#97—22no—
183#97—22no—
184#97—22no—
185#97—44no—
186#97#4114yes—
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#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95C22⋊C412yes—
201#95C22⋊C412yes—
202#95C22⋊C411yescharacteristic
203#95C22⋊C411yescharacteristic
204C24C2×C22⋊C412yes—
205C2×D4—24no—
206C2×D4—41no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C23×C411yescharacteristic
230C22×C4C22×D412yes—
231C2×C8—24no—
232C2×C8—24no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—416no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—42no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—82no—
266C22⋊C4—82no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—42no—
275C42—41no—
276C42—42no—
277C42—44no—
278C42—82no—
279C42—82no—
280C42C2×#4112yes—
281C42H(5,𝔽2)11yescharacteristic
282C422−1+411yescharacteristic
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23#42114yes—
290C23C22×C22⋊C411yescharacteristic
291D4—44no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4#41312yes—
330C2×C4C4×#4111yescharacteristic
331C2×C4#52011yescharacteristic
332C2×C4#54612yes—
333C2×C4#54712yes—
334C8—44no—
335C8—44no—
336C8—82no—
337C8—82no—
338C22—24no—
339C22—24no—
340C22—24no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22#233911yescharacteristic
346C22C2×#41311yescharacteristic
347C22#288911yescharacteristic
348C22C2×#42112yes—
349C22#290112yes—
350C4—22no—
351C4—21no—
352C4—21no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—42no—
357C4—42no—
358C4—42no—
359C4—42no—
360C4—82no—
361C4—82no—
362C4—82no—
363C4—82no—
364C2—44no—
365C2#1767811yescharacteristic
366C2#1768012yes—
367C2#1769011yescharacteristic
368C2#2248411yescharacteristic
369C2#2250612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9894C2semidirect
4. #9125C2semidirect
5. #9121C2semidirect
6. #8720C2semidirect
7. #8580C2semidirect
8. #8225C2semidirect
9. #8036C2semidirect

Maximal subgroups

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

subgroupindex
#361962
#211072
#98942
#91252
#91212
#87202
#85802
#82252
#80362