Finite Group Databasebeta

← #241325

Subgroups — #241325

Order 512.

840 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37643C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21981C211yesmaximal, characteristic
4#20800C211yesmaximal, characteristic
5#10580C214yesmaximal, complemented
6#10579C214yesmaximal, complemented
7#8038C211yesmaximal, characteristic
8#7896C211yesmaximal, characteristic
9#7896C211yesmaximal, characteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#428—22no—
13C2×C4×M16C2211yescharacteristic
14#2845—22no—
15#2629—22no—
16#2629—22no—
17#2629—22no—
18#2552—21no—
19#2552—21no—
20#2552—21no—
21#2552—21no—
22#2552C2212yes—
23#2552C2212yes—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28#1948C2214yes—
29#1947C2214yes—
30#1853—22no—
31#1852—22no—
32#1847—22no—
33#1845—21no—
34#1845C2212yes—
35#1844—21no—
36#1844C2212yes—
37#1834—21no—
38#1834—21no—
39#1296C2212yes—
40#1296C2212yes—
41#1279C2214yes—
42#1279C2214yes—
43C22×M16—21no—
44C2×#125—22no—
45C2×#125—22no—
46C2×#125—22no—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117D412yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C42C2311yescharacteristic
54#431—22no—
55#431—22no—
56#431Q814yes—
57#431Q814yes—
58#428—48no—
59C2×#107—24no—
60C2×#107—24no—
61C2×#106—24no—
62C2×#106—24no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C4×M16—22no—
68C4×M16Q814yes—
69C2×#97—21no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8C2312yes—
73#398—24no—
74#398—24no—
75#398—24no—
76#398—24no—
77#398—24no—
78#398—24no—
79#396—24no—
80#396—24no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#384—22no—
88#384—22no—
89#384—22no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384D411yescharacteristic
94#384D411yescharacteristic
95#384D411yescharacteristic
96#384D411yescharacteristic
97#384C2312yes—
98#384C2311yescharacteristic
99#384C2311yescharacteristic
100#382—21no—
101#382—21no—
102#382D412yes—
103#380—21no—
104#380—21no—
105#380—21no—
106#380—21no—
107#380D412yes—
108#380D412yes—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4C2312yes—
114C4×C4⋊C4C2312yes—
115#340—24no—
116#340—24no—
117C23×C4—41no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—24no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127#125—84no—
128#125—84no—
129#125—84no—
130#117—44no—
131#117—82no—
132#117—82no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2×D412yes—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×Q812yes—
176C2×C42C2×Q812yes—
177C2×C42C2×Q812yes—
178C2×C42#4112yes—
179C2×C42C2411yescharacteristic
180#107—48no—
181#107—84no—
182#106—48no—
183#106—84no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8C2×Q814yes—
187C4⋊C8C2×Q814yes—
188C22⋊C8—24no—
189C22⋊C8—24no—
190#97—24no—
191C4×C8—24no—
192C4×C8—24no—
193C4×C8—22no—
194C4×C8C2×Q814yes—
195#95—24no—
196#95—24no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203C24#12011yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
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—42no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4#12011yescharacteristic
221C22×C4#12011yescharacteristic
222C22×C4#12011yescharacteristic
223C22×C4C22⋊Q812yes—
224C22×C4C22⋊Q812yes—
225C22×C4C22⋊Q812yes—
226C22×C4C4⋊Q812yes—
227C22×C4C4⋊Q812yes—
228C22×C4C4⋊Q812yes—
229C22×C4C22×D411yescharacteristic
230C22×C4C22×D411yescharacteristic
231C22×C4C22×D411yescharacteristic
232M16—216no—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—24no—
236C2×C8—24no—
237C2×C8—24no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—81no—
260C4⋊C4—81no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—84no—
266C4⋊C4—84no—
267C4⋊C4—84no—
268C4⋊C4—84no—
269C4⋊C4—84no—
270C4⋊C4—84no—
271C4⋊C4—82no—
272C22⋊C4—82no—
273C22⋊C4—82no—
274C22⋊C4—82no—
275C42—44no—
276C42—44no—
277C42—44no—
278C42C22⋊Q812yes—
279C42C22⋊Q812yes—
280C42C22⋊Q812yes—
281C42C4⋊Q812yes—
282C42C4⋊Q812yes—
283C42C4⋊Q812yes—
284C42C22×Q811yescharacteristic
285C42C22×Q811yescharacteristic
286C42C22×Q811yescharacteristic
287C42C2×#4111yescharacteristic
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#39512yes—
295C23C2×#12011yescharacteristic
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
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—42no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—164no—
330C2×C4#39512yes—
331C2×C4#39512yes—
332C2×C4#39512yes—
333C2×C4#45312yes—
334C2×C4#45612yes—
335C2×C4C2×C22⋊Q811yescharacteristic
336C2×C4C2×C22⋊Q811yescharacteristic
337C2×C4C2×C22⋊Q811yescharacteristic
338C2×C4C2×C4⋊Q811yescharacteristic
339C2×C4C2×C4⋊Q811yescharacteristic
340C2×C4C2×C4⋊Q811yescharacteristic
341C2×C4#57612yes—
342C2×C4#57712yes—
343C2×C4#57712yes—
344C2×C4#57812yes—
345C8—28no—
346C8—44no—
347C8—44no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22C2×#39511yescharacteristic
356C22#300811yescharacteristic
357C22#300911yescharacteristic
358C22#301011yescharacteristic
359C22#301111yescharacteristic
360C22C2×#45311yescharacteristic
361C22C2×#45611yescharacteristic
362C4—22no—
363C4—22no—
364C4—21no—
365C4—21no—
366C4—82no—
367C4—82no—
368C4—82no—
369C4—82no—
370C4—82no—
371C4—82no—
372C4—162no—
373C4#206014yes—
374C4#308312yes—
375C4#308412yes—
376C4#316312yes—
377C4#316412yes—
378C2—42no—
379C2#1871611yescharacteristic
380C2#1871711yescharacteristic
381C2#1873411yescharacteristic
382C2#1873711yescharacteristic
383C2#2355511yescharacteristic
384C2#2355611yescharacteristic
385C2C2×#206011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10580C2semidirect
6. #10579C2semidirect

Maximal subgroups

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

subgroupindex
#376432
#219822
#219812
#208002
#105802
#105792
#80382
#78962