Finite Group Databasebeta

← #6343992

Subgroups — #6343992

Order 512.

2989 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62813C211yesmaximal, characteristic
2#60276C211yesmaximal, characteristic
3C2×#2906C211yesmaximal, characteristic
4#17723C218yesmaximal, complemented
5#17722C218yesmaximal, complemented
6#17708C216yesmaximal, complemented
7C2×#1542C216yesmaximal, complemented
8C22×#131—21no—
9#3504—24no—
10#3503C2214yes—
11#3503C2214yes—
12#3502—21no—
13#3502C412yes—
14#3501—23no—
15#3501C416yes—
16Q8×#41—24no—
17Q8×#41C418yes—
18#3458—23no—
19C2×#533C2216yes—
20#3428—21no—
21C2×C4×#41C2211yescharacteristic
22#2988C2218yes—
23#2982—21no—
24#2972—23no—
25C8×#41—24no—
26C2×#431C2216yes—
27#2906—24no—
28#2906C2218yes—
29C2×#424—23no—
30C2×#423C2213yes—
31C22×C4⋊C8C2213yes—
32#1550C22124yescomplemented
33#1549C22112yescomplemented
34#1548C22112yescomplemented
35#1542C2214yes—
36#1542C22112yes—
37#1542C22124yescomplemented
38C2×#333C22112yescomplemented
39C2×#332C22112yescomplemented
40C22×#41D411yescharacteristic
41C2×#131—28no—
42C2×#131—48no—
43C22×M16—23no—
44C23×C8—23no—
45Q82—24no—
46Q82C2×C418yes—
47#557—28no—
48#557—28no—
49#557—24no—
50#557C2×C418yes—
51#554—224no—
52#554—224no—
53#554—212no—
54#554C2×C4124yes—
55D4×Q8—212no—
56D4×Q8C2×C4124yes—
57#544—212no—
58#544—212no—
59#544—26no—
60#544—26no—
61#543—212no—
62#533—224no—
63#533—26no—
64#533—26no—
65#533—26no—
66#533—26no—
67#533C2×C4112yes—
68#533C2×C4112yes—
69#533C2314yes—
70#533C23112yes—
71C2×C4⋊Q8C2316yes—
72C2×C4⋊Q8C2316yes—
73C2×#125—26no—
74C2×C22⋊Q8—26no—
75C2×C22⋊Q8C2×C4112yes—
76#520—24no—
77#520—24no—
78#519—22no—
79#518—26no—
80C4×#41—28no—
81C4×#41C2×C414yes—
82C4×#41D414yes—
83C4×#41D414yes—
84C4×#41C2314yes—
85C2×C4×Q8—26no—
86C2×C4×Q8C2×C4112yes—
87C2×C4×Q8C2×C411yescharacteristic
88C2×C4×Q8D411yescharacteristic
89C2×C4×D4C2×C413yes—
90C2×C4×D4D413yes—
91C2×#117—26no—
92C2×#117D413yes—
93C22×C42C2313yes—
94C8×Q8—28no—
95C8×D4—224no—
96#431—26no—
97#431—26no—
98#431C2314yes—
99#431C23112yes—
100#424—224no—
101#423—26no—
102#423C23112yes—
103C2×C4⋊C8—23no—
104C2×C4⋊C8—26no—
105C2×C4⋊C8—23no—
106C2×C4⋊C8C23112yes—
107C2×C4⋊C8C23112yes—
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2313yes—
110C2×C22⋊C8—26no—
111C2×C4×C8—26no—
112#333C2318yes—
113#333C23124yes—
114#332C2318yes—
115#332C23124yes—
116C2×#41C22⋊C414yes—
117C2×#41SD1618yes—
118C2×#41Q1618yes—
119C2×#41C2×D414yes—
120C22×Q8—26no—
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C22×C4112yes—
123C22×D4C22⋊C413yes—
124C23×C4C22×C413yes—
125C23×C4C2×D413yes—
126C23×C4C2×D413yes—
127#131—432no—
128C2×M16—212no—
129C2×M16—412no—
130C22×C8—21no—
131C22×C8—23no—
132C22×C8—212no—
133C22×C8—212no—
134C22×C8—412no—
135C4⋊Q8—224no—
136C4⋊Q8—224no—
137C4⋊Q8—26no—
138C4⋊Q8—26no—
139C4⋊Q8—26no—
140C4⋊Q8—26no—
141C4⋊Q8C22×C4112yes—
142C4⋊Q8C22×C4112yes—
143C4⋊Q8C2414yes—
144C4⋊Q8C2414yes—
145#125—224no—
146#125—224no—
147#125—212no—
148#125—212no—
149#125—24no—
150#125—212no—
151C22⋊Q8—212no—
152C22⋊Q8—212no—
153C22⋊Q8—212no—
154C22⋊Q8—212no—
155C22⋊Q8—24no—
156C22⋊Q8—212no—
157C22⋊Q8—248no—
158C22⋊Q8—248no—
159C22⋊Q8C22×C418yes—
160C22⋊Q8C22×C4124yes—
161C4×Q8—28no—
162C4×Q8—28no—
163C4×Q8—212no—
164C4×Q8—212no—
165C4×Q8—212no—
166C4×Q8—212no—
167C4×Q8—24no—
168C4×Q8—212no—
169C4×Q8C22⋊C414yes—
170C4×Q8C22⋊C414yes—
171C4×Q8C22×C418yes—
172C4×Q8C22×C4124yes—
173C4×Q8C22×C414yes—
174C4×Q8C2×D414yes—
175C4×D4—224no—
176C4×D4—224no—
177C4×D4C22⋊C4112yes—
178C4×D4C22⋊C4112yes—
179C4×D4C22×C4112yes—
180C4×D4C2×D4112yes—
181#117—212no—
182#117—212no—
183#117—24no—
184#117—212no—
185#117C22⋊C4112yes—
186#117C2×D4112yes—
187C2×C4⋊C4—212no—
188C2×C4⋊C4—212no—
189C2×C4⋊C4—26no—
190C2×C4⋊C4—26no—
191C2×C4⋊C4—26no—
192C2×C4⋊C4C22⋊C413yes—
193C2×C4⋊C4C22⋊C413yes—
194C2×C4⋊C4C22×C4112yes—
195C2×C22⋊C4—212no—
196C2×C22⋊C4C22⋊C416yes—
197C2×C42—212no—
198C2×C42C22×C413yes—
199C2×C42C22×C416yes—
200C2×C42C2×D413yes—
201C2×C42C2×D413yes—
202C2×C42C2×D416yes—
203C2×C42C2×D416yes—
204C2×C42C2416yes—
205C2×C42C2411yescharacteristic
206C4⋊C8—21no—
207C4⋊C8—23no—
208C4⋊C8—212no—
209C4⋊C8—212no—
210C4⋊C8—212no—
211C4⋊C8C2412yes—
212C4⋊C8C2416yes—
213C4⋊C8C2418yes—
214C22⋊C8—24no—
215C22⋊C8—212no—
216C4×C8—24no—
217C4×C8—212no—
218C24C2×C22⋊C413yes—
219#41#103132yes—
220#41C2×SD16116yes—
221#41C2×Q16116yes—
222C2×Q8—212no—
223C2×Q8—212no—
224C2×Q8—24no—
225C2×Q8—812no—
226C2×Q8—812no—
227C2×Q8#10318yes—
228C2×Q8C2×C22⋊C414yes—
229C2×Q8C23×C418yes—
230C2×D4#103124yes—
231C2×D4C2×C22⋊C4112yes—
232C22×C4—212no—
233C22×C4—212no—
234C22×C4—46no—
235C22×C4—83no—
236C22×C4C2×C22⋊C413yes—
237C22×C4C2×C22⋊C416yes—
238C22×C4C2×C22⋊C413yes—
239C22×C4C2×C22⋊C416yes—
240C22×C4C2×C22⋊C413yes—
241C22×C4C2×C22⋊C416yes—
242C22×C4C2×SD16112yes—
243C22×C4C2×Q16112yes—
244C22×C4C23×C411yescharacteristic
245C22×C4C23×C416yes—
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D416yes—
248C22×C4C22×D416yes—
249C22×C4C22×D411yescharacteristic
250M16—424no—
251C2×C8—28no—
252C2×C8—22no—
253C2×C8—26no—
254C2×C8—42no—
255C2×C8—46no—
256C2×C8—424no—
257C2×C8—812no—
258C4⋊C4—28no—
259C4⋊C4—28no—
260C4⋊C4—212no—
261C4⋊C4—212no—
262C4⋊C4—212no—
263C4⋊C4—224no—
264C4⋊C4—224no—
265C4⋊C4—212no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4C2×C22⋊C4112yes—
270C4⋊C4C2×C22⋊C4112yes—
271C4⋊C4C23×C418yes—
272C22⋊C4—28no—
273C22⋊C4—224no—
274C22⋊C4C2×C22⋊C4124yes—
275C42—224no—
276C42—28no—
277C42C2×C22⋊C416yes—
278C42C2×C22⋊C416yes—
279C42C2×C22⋊C416yes—
280C42C23×C416yes—
281C42C23×C411yescharacteristic
282C42C22×D416yes—
283C42C22×D411yescharacteristic
284C42C22×D416yes—
285C42C22×D411yescharacteristic
286C42C2511yescharacteristic
287C23—43no—
288C23—43no—
289C23C2×#103112yes—
290C23C22×C22⋊C411yescharacteristic
291C23C22×C22⋊C416yes—
292Q8—48no—
293Q8—48no—
294Q8—812no—
295Q8—812no—
296Q8C2×#103116yes—
297D4C2×#103148yes—
298C2×C4—212no—
299C2×C4—22no—
300C2×C4—28no—
301C2×C4—28no—
302C2×C4—46no—
303C2×C4—43no—
304C2×C4—41no—
305C2×C4—46no—
306C2×C4—43no—
307C2×C4—46no—
308C2×C4—812no—
309C2×C4—812no—
310C2×C4C2×#103112yes—
311C2×C4C2×#103124yes—
312C2×C4C22×C22⋊C411yescharacteristic
313C2×C4C22×C22⋊C411yescharacteristic
314C2×C4C22×C22⋊C416yes—
315C2×C4C22×C22⋊C416yes—
316C2×C4C22×C22⋊C416yes—
317C2×C4C22×C22⋊C411yescharacteristic
318C2×C4C22×SD1612yes—
319C2×C4C22×SD16112yes—
320C2×C4C22×Q1612yes—
321C2×C4C22×Q16112yes—
322C2×C4C24×C411yescharacteristic
323C2×C4C23×D411yescharacteristic
324C2×C4C23×D411yescharacteristic
325C8—48no—
326C8—42no—
327C8—46no—
328C22—21no—
329C22—26no—
330C22—21no—
331C22—26no—
332C22—412no—
333C22C22×#10312yes—
334C22C22×#103112yes—
335C22C23×C22⋊C411yescharacteristic
336C4—21no—
337C4—21no—
338C4—26no—
339C4—21no—
340C4—26no—
341C4—48no—
342C4—48no—
343C4C22×#103112yes—
344C4C22×#10312yes—
345C4C23×SD1611yescharacteristic
346C4C23×Q1611yescharacteristic
347C4#358412yes—
348C2—22no—
349C2—212no—
350C2#6007611yescharacteristic
351C2C23×#10311yescharacteristic
352C2#6008611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C23×C22⋊C4
prime radicalC22×#41D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17723C2semidirect
5. #17722C2semidirect
6. #17708C2semidirect
7. C2×#1542C2semidirect
32. #1550C22semidirect
33. #1549C22semidirect
34. #1548C22semidirect
37. #1542C22semidirect
38. C2×#333C22semidirect
39. C2×#332C22semidirect

Maximal subgroups

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

subgroupindex
#628132
#602762
C2×#29062
#177232
#177222
#177082
C2×#15422