Finite Group Databasebeta

← #223137

Subgroups — #223137

Order 512.

809 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36192C211yesmaximal, characteristic
2#21107C212yesmaximal
3#9894C212yesmaximal, complemented
4#9126C212yesmaximal, complemented
5#9121C212yesmaximal, complemented
6#8721C212yesmaximal, complemented
7#8584C211yesmaximal, characteristic, complemented
8#8225C211yesmaximal, characteristic, complemented
9#8034C212yesmaximal, complemented
10C2×#432C2212yes—
11C22×#97C2212yes—
12#2859C2211yescharacteristic
13#2851C2212yes—
14#2552C412yes—
15#2522—22no—
16#2522—22no—
17#2454—22no—
18#2441C412yes—
19#2441C412yes—
20#2370—22no—
21#2354C412yes—
22C4×#125C2211yescharacteristic
23#2285C2212yes—
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×#125—22no—
45C2×#125C2×C411yescharacteristic
46C2×#125C2×C411yescharacteristic
47C2×#125C2311yescharacteristic
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4D412yes—
50C2×#117C2×C412yes—
51C2×#117C2×C412yes—
52C22×C42C2312yes—
53#432—24no—
54#432—42no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×#97—24no—
61C2×#97—22no—
62C2×#97—42no—
63C2×#97C2312yes—
64C2×#97C2312yes—
65C2×#97C2312yes—
66C2×#97C2312yes—
67#399—22no—
68#399—22no—
69#398—22no—
70#398—22no—
71#396—22no—
72#396—22no—
73#390—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2×C412yes—
82#388—22no—
83#388—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#385—22no—
90#385—22no—
91#384C2×C412yes—
92#384C2×C412yes—
93#384C2×C412yes—
94#382C2311yescharacteristic
95#382C2312yes—
96#380—22no—
97#380—22no—
98#380C2×C412yes—
99#380C2×C412yes—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2×C411yescharacteristic
106C4×C4⋊C4D412yes—
107C4×C22⋊C4C2×C412yes—
108C4×C22⋊C4D412yes—
109C4×C22⋊C4D412yes—
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#125—22no—
131#125—21no—
132#125—21no—
133#125—48no—
134#125—81no—
135#125—81no—
136C4×D4—24no—
137C4×D4—44no—
138#117—82no—
139#117—82no—
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⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4C22⋊C411yescharacteristic
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4C22⋊C412yes—
163C2×C22⋊C4C22⋊C412yes—
164C2×C22⋊C4C22⋊C412yes—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C412yes—
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2411yescharacteristic
180C4⋊C8—44no—
181C4⋊C8—44no—
182C22⋊C8—44no—
183C22⋊C8—44no—
184C22⋊C8—44no—
185#97—22no—
186#97—22no—
187#97—22no—
188#97—22no—
189#97—22no—
190#97—22no—
191#97—22no—
192#97—44no—
193#97#4114yes—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95C22⋊C412yes—
204#95C22⋊C412yes—
205#95C22⋊C411yescharacteristic
206#95C22⋊C411yescharacteristic
207C24C2×C22⋊C412yes—
208C2×D4—42no—
209C2×D4#10114yes—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C412yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D412yes—
234C2×C8—24no—
235C2×C8—24no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—416no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C4⋊C4—22no—
248C4⋊C4—24no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—42no—
270C22⋊C4—82no—
271C22⋊C4—82no—
272C22⋊C4—82no—
273C22⋊C4—82no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—42no—
282C42—41no—
283C42—42no—
284C42—44no—
285C42—82no—
286C42—82no—
287C42C2×#4112yes—
288C42H(5,𝔽2)11yescharacteristic
289C422−1+411yescharacteristic
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23C2×#10114yes—
297C23C22×C22⋊C411yescharacteristic
298D4—44no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4C2×#10112yes—
336C2×C4#42114yes—
337C2×C4C4×#4111yescharacteristic
338C2×C4#52011yescharacteristic
339C2×C4#54612yes—
340C2×C4#54712yes—
341C8—44no—
342C8—44no—
343C8—82no—
344C8—82no—
345C22—24no—
346C22—24no—
347C22—24no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22#233911yescharacteristic
353C22C22×#10111yescharacteristic
354C22#288811yescharacteristic
355C22C2×#42112yes—
356C22#290112yes—
357C4—22no—
358C4—21no—
359C4—21no—
360C4—22no—
361C4—22no—
362C4—24no—
363C4—42no—
364C4—41no—
365C4—42no—
366C4—42no—
367C4—82no—
368C4—82no—
369C4—82no—
370C4—82no—
371C2—44no—
372C2#1767611yescharacteristic
373C2#1768412yes—
374C2#1768911yescharacteristic
375C2#2248111yescharacteristic
376C2#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. #9126C2semidirect
5. #9121C2semidirect
6. #8721C2semidirect
7. #8584C2semidirect
8. #8225C2semidirect
9. #8034C2semidirect

Maximal subgroups

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

subgroupindex
#361922
#211072
#98942
#91262
#91212
#87212
#85842
#82252
#80342