Finite Group Databasebeta

← #223133

Subgroups — #223133

Order 512.

821 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36165C211yesmaximal, characteristic
2#21108C212yesmaximal
3#9894C212yesmaximal, complemented
4#9125C212yesmaximal, complemented
5#9122C212yesmaximal, complemented
6#8718C212yesmaximal, complemented
7#8584C211yesmaximal, characteristic, complemented
8#8225C211yesmaximal, characteristic, complemented
9#8033C212yesmaximal, complemented
10C2×#432C2212yes—
11C2×C4×M16C2212yes—
12#2859C2211yescharacteristic
13#2851C2212yes—
14#2571C412yes—
15#2508—22no—
16#2505—22no—
17#2505—22no—
18#2442C412yes—
19#2442C412yes—
20#2349—22no—
21D4×C4⋊C4C412yes—
22C4×#125C2211yescharacteristic
23C2×#382C2212yes—
24C42×D4C2211yescharacteristic
25#1923—22no—
26#1916—22no—
27#1870—22no—
28#1851—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—
43C22×M16—22no—
44C2×#125C2×C411yescharacteristic
45C2×#125C2×C411yescharacteristic
46C2×#125C2311yescharacteristic
47C2×C4×D4—22no—
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4D412yes—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×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—
60C4×M16—24no—
61C4×M16—42no—
62C2×#97C2312yes—
63C2×#97C2312yes—
64C2×#97C2312yes—
65C2×#97C2312yes—
66C2×C4×C8—22no—
67#400—22no—
68#400—22no—
69#400—22no—
70#400—22no—
71#398—22no—
72#398—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389C2×C412yes—
78#389C2×C412yes—
79#389C2×C412yes—
80#387—22no—
81#387—22no—
82#387—22no—
83#385—22no—
84#385—22no—
85#385—22no—
86#385—22no—
87#385—22no—
88#385—22no—
89#384C2×C412yes—
90#382—22no—
91#382—22no—
92#382—22no—
93#382C2×C412yes—
94#382C2×C412yes—
95#382C2311yescharacteristic
96#382C2312yes—
97#380—22no—
98#380—22no—
99#380C2×C412yes—
100#380C2×C412yes—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2×C412yes—
103C4×C4⋊C4C2×C411yescharacteristic
104C4×C4⋊C4D412yes—
105C4×C22⋊C4C2×C412yes—
106C4×C22⋊C4D412yes—
107C4×C22⋊C4D412yes—
108C2×#95—22no—
109C2×#95—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×C4—22no—
121C23×C4—22no—
122C23×C4C22×C412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C2×M16—44no—
126C2×M16—44no—
127C2×M16—44no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132#125—22no—
133#125—21no—
134#125—21no—
135#125—81no—
136#125—81no—
137C4×D4—24no—
138C4×D4—44no—
139C4×D4—48no—
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—82no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C411yescharacteristic
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⋊C4C22⋊C412yes—
161C2×C22⋊C4C22⋊C412yes—
162C2×C22⋊C4C22⋊C412yes—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C412yes—
170C2×C42C22×C411yescharacteristic
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2411yescharacteristic
176C4⋊C8—44no—
177C4⋊C8—44no—
178C22⋊C8—44no—
179C22⋊C8—44no—
180C22⋊C8—44no—
181#97—22no—
182#97—22no—
183#97—22no—
184#97—22no—
185#97—22no—
186#97—22no—
187#97—22no—
188#97#4114yes—
189C4×C8—44no—
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#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—24no—
209C2×D4—22no—
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×C4—44no—
227C22×C4—44no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C23×C411yescharacteristic
239C22×C4C22×D412yes—
240M16—416no—
241C2×C8—24no—
242C2×C8—24no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—44no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4#10114yes—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—42no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—42no—
284C42—41no—
285C42—42no—
286C42—44no—
287C42C2×#4112yes—
288C42H(5,𝔽2)11yescharacteristic
289C422−1+411yescharacteristic
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23#42114yes—
297C23C22×C22⋊C411yescharacteristic
298D4—28no—
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×C4—164no—
336C2×C4—164no—
337C2×C4C2×#10112yes—
338C2×C4C2×#10114yes—
339C2×C4C4×#4111yescharacteristic
340C2×C4#52011yescharacteristic
341C2×C4#54612yes—
342C2×C4#54712yes—
343C8—44no—
344C8—44no—
345C8—82no—
346C8—82no—
347C22—24no—
348C22—24no—
349C22—24no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22#233911yescharacteristic
355C22C22×#10111yescharacteristic
356C22#288811yescharacteristic
357C22C2×#42112yes—
358C22#290112yes—
359C4—22no—
360C4—21no—
361C4—21no—
362C4—24no—
363C4—22no—
364C4—22no—
365C4—42no—
366C4—41no—
367C4—42no—
368C4—42no—
369C4—82no—
370C4—82no—
371C4—82no—
372C4—82no—
373C2—44no—
374C2#1766711yescharacteristic
375C2#1767912yes—
376C2#1769211yescharacteristic
377C2#2248111yescharacteristic
378C2#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. #9122C2semidirect
6. #8718C2semidirect
7. #8584C2semidirect
8. #8225C2semidirect
9. #8033C2semidirect

Maximal subgroups

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

subgroupindex
#361652
#211082
#98942
#91252
#91222
#87182
#85842
#82252
#80332