Finite Group Databasebeta

← #229270

Subgroups — #229270

Order 512.

652 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23184C211yesmaximal, characteristic
2#23062C212yesmaximal
3#22727C211yesmaximal, characteristic
4C2×#1670C211yesmaximal, characteristic
5#9825C211yesmaximal, characteristic
6#9615C211yesmaximal, characteristic
7#9565C212yesmaximal
8#8982C212yesmaximal
9#8925C212yesmaximal
10C8×#107C211yesmaximal, characteristic
11C8×#103C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×#475C2212yes—
14C2×C8×Q8C2211yescharacteristic
15C2×C4×Q16—21no—
16#2365C2212yes—
17#2355C2212yes—
18#1990C2212yes—
19#1990C2212yes—
20#1981C2212yes—
21#1969—21no—
22#1949—22no—
23#1948C2212yes—
24#1948C2212yes—
25#1933—22no—
26#1927—22no—
27#1919—22no—
28#1878—22no—
29#1846—21no—
30#1845C2211yescharacteristic
31C4×#107—21no—
32C4×#107C2211yescharacteristic
33C4×#106—22no—
34C8×C4⋊C4C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36C4×#103C2211yescharacteristic
37#1670C418yescomplemented
38C2×#335C2211yescharacteristic
39C2×#332C2212yes—
40C2×Q8⋊C8C2211yescharacteristic
41C2×C82C2211yescharacteristic
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1279C2212yes—
45#1277C2211yescharacteristic
46C22×Q16—21no—
47C22×Q16—21no—
48C2×C4⋊Q8C2312yes—
49C2×C4×Q8—22no—
50C2×C4×Q8C2311yescharacteristic
51#494—22no—
52#494C2×C414yes—
53#475—24no—
54#475C2×C418yes—
55C8×Q8—22no—
56C8×Q8C2×C414yes—
57C4×Q16—44no—
58C2×#107—21no—
59C2×#107—21no—
60C2×#107D411yescharacteristic
61C2×#107D411yescharacteristic
62C2×#106—22no—
63C2×#106—22no—
64C2×#106—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2311yescharacteristic
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103D411yescharacteristic
73C2×#103D411yescharacteristic
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—22no—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80#391—22no—
81#391—22no—
82#389—22no—
83#387—42no—
84#387—42no—
85#384C2312yes—
86#382—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4C2312yes—
91C4×C4⋊C4C2311yescharacteristic
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—42no—
97#340D412yes—
98#340D411yescharacteristic
99#340D411yescharacteristic
100#336—21no—
101#336—22no—
102#335C2×C414yes—
103#332C2×C418yes—
104Q8⋊C8—22no—
105C82—22no—
106C22×Q8—22no—
107C22×Q8C2×D411yescharacteristic
108C2×Q16—24no—
109C2×Q16—24no—
110C2×Q16—42no—
111C2×Q16—42no—
112C2×Q16—81no—
113C2×Q16—81no—
114C22×C8—22no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8C2×D411yescharacteristic
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8C22×C414yes—
126C4×Q8—21no—
127C4×Q8—21no—
128C4×Q8—44no—
129C4×Q8—82no—
130C4×Q8C22×C412yes—
131C4×Q8#4112yes—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4C2×D411yescharacteristic
140C2×C4⋊C4C2×D412yes—
141C2×C4⋊C4C2×D411yescharacteristic
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×D411yescharacteristic
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—22no—
147C2×C42—42no—
148C2×C42—42no—
149C2×C42C2411yescharacteristic
150#107—22no—
151#107—42no—
152#107—81no—
153#107SD1614yes—
154#106—28no—
155#106—44no—
156#106—82no—
157C4⋊C8—24no—
158C4⋊C8—21no—
159C4⋊C8—22no—
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8C22×C412yes—
163C4⋊C8C22×C414yes—
164#103—28no—
165#103—22no—
166#103—22no—
167#103—44no—
168#103—82no—
169C4×C8—24no—
170C4×C8—21no—
171C4×C8—21no—
172C4×C8—21no—
173C4×C8—21no—
174C4×C8—42no—
175C4×C8—44no—
176C4×C8C22×C412yes—
177C4×C8#4112yes—
178#95—21no—
179#95—22no—
180#95—42no—
181#95—42no—
182#95—42no—
183#95—42no—
184#95—42no—
185C2×Q8—21no—
186C2×Q8—24no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8C4×D412yes—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207C22×C4H(5,𝔽2)11yescharacteristic
208Q16—48no—
209Q16—84no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—24no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8C4×D412yes—
230C2×C8C4×D412yes—
231C2×C8C2×SD1612yes—
232C2×C8#13512yes—
233C2×C8#13512yes—
234C4⋊C4—22no—
235C4⋊C4—21no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—21no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—21no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—21no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—84no—
252C4⋊C4—82no—
253C4⋊C4C4×D412yes—
254C4⋊C4C2×SD1614yes—
255C4⋊C4#13512yes—
256C42—22no—
257C42—22no—
258C42—41no—
259C42—42no—
260C42—41no—
261C42—42no—
262C42—84no—
263C42—162no—
264C42C23×C411yescharacteristic
265C42C2×#4111yescharacteristic
266C42C2×#4111yescharacteristic
267C422−1+411yescharacteristic
268C23D4211yescharacteristic
269Q8—22no—
270Q8—22no—
271Q8—41no—
272Q8—41no—
273Q8—44no—
274Q8—44no—
275Q8—82no—
276Q8—82no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4C2×C4×D411yescharacteristic
304C2×C4C2×C4×D411yescharacteristic
305C2×C4#52011yescharacteristic
306C2×C4#54711yescharacteristic
307C2×C4#54811yescharacteristic
308C2×C4#55311yescharacteristic
309C2×C4C22×SD1611yescharacteristic
310C2×C4C2×#13511yescharacteristic
311C2×C4C2×#13511yescharacteristic
312C2×C4C2×#13511yescharacteristic
313C2×C4#57611yescharacteristic
314C2×C4#57711yescharacteristic
315C2×C4#57811yescharacteristic
316C2×C4#57811yescharacteristic
317C8—22no—
318C8—22no—
319C8—42no—
320C8—41no—
321C8—44no—
322C8—42no—
323C8—81no—
324C8C4×SD1614yes—
325C22#236111yescharacteristic
326C22D4×SD1611yescharacteristic
327C22#328311yescharacteristic
328C22#328511yescharacteristic
329C22#328711yescharacteristic
330C22#329011yescharacteristic
331C22#329311yescharacteristic
332C4—21no—
333C4—21no—
334C4—42no—
335C4—44no—
336C4—42no—
337C4—42no—
338C4—81no—
339C4—82no—
340C4—84no—
341C4—162no—
342C4C2×C4×SD1611yescharacteristic
343C4C4×#13511yescharacteristic
344C4#294611yescharacteristic
345C4#295011yescharacteristic
346C4#333611yescharacteristic
347C4#334211yescharacteristic
348C4#336211yescharacteristic
349C4#336511yescharacteristic
350C2#1998011yescharacteristic
351C2#1998311yescharacteristic
352C2#2001511yescharacteristic
353C2#2002011yescharacteristic
354C2#2299711yescharacteristic
355C2#2302611yescharacteristic
356C2#2304511yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
37. #1670C4semidirect

Maximal subgroups

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

subgroupindex
#231842
#230622
#227272
C2×#16702
#98252
#96152
#95652
#89822
#89252
C8×#1072
C8×#1032