Finite Group Databasebeta

← #6389809

Subgroups — #6389809

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60660C211yesmaximal, characteristic
2#60252C212yesmaximal
3#23172C214yesmaximal, complemented
4#23171C214yesmaximal, complemented
5#22608C214yesmaximal
6#22607C214yesmaximal
7#22259C212yesmaximal, complemented
8#22257C212yesmaximal, complemented
9#21846C212yesmaximal
10#21734C212yesmaximal, complemented
11#21733C212yesmaximal, complemented
12#20828C211yesmaximal, characteristic
13#20475C211yesmaximal, characteristic
14#3458C2212yes—
15#3164C414yescomplemented
16#3164C414yescomplemented
17#3163C418yescomplemented
18#3162C2212yes—
19#3162C2212yes—
20#3160C2212yes—
21#3160C2212yes—
22#2987—24no—
23#2986—24no—
24#2985—28no—
25#2962C2218yes—
26#2961C2214yes—
27#2960C2214yes—
28#2924C2218yes—
29#2921C2212yes—
30C2×#433C2211yescharacteristic
31#2911C2212yes—
32C2×#427C2212yes—
33C2×#405C2211yescharacteristic
34#2298C2211yescharacteristic
35#1995C2214yes—
36#1994C2214yes—
37#1986C2218yes—
38#1946C2214yes—
39#1946C2214yes—
40#1946C2214yes—
41#1945C2214yes—
42#1922C2218yes—
43#1922C2218yes—
44#1922C2218yes—
45#1922C2218yes—
46#1848C2212yes—
47#1838C2212yes—
48#1826C2218yes—
49#1811C2212yes—
50#1810C2212yes—
51#1778C2212yes—
52C4×#108C2212yes—
53#1773C2214yes—
54#1773C2214yes—
55#1773C2214yes—
56#1772C2214yes—
57#1771C2212yes—
58#1770C2212yes—
59#1753C2212yes—
60#1752C2212yes—
61#1746C2211yescharacteristic
62#1745C2211yescharacteristic
63#1726C2211yescharacteristic
64C4×#97C2211yescharacteristic
65C22×M16C2311yescharacteristic
66C23×C8C2311yescharacteristic
67#544—24no—
68#544C2×C418yes—
69#544C2×C418yes—
70#543—24no—
71#533—24no—
72#533—24no—
73#533C2314yes—
74#533C2314yes—
75C2×#117C2314yes—
76C2×#117C2311yescharacteristic
77#501C2×C418yes—
78#501C2×C418yes—
79#501C2×C418yes—
80#501C2×C418yes—
81#500C2×C414yes—
82#500C2×C412yes—
83#500C2×C412yes—
84#499C2×C414yes—
85#499C2×C414yes—
86#499C2×C412yes—
87#499C2×C412yes—
88#499C2×C414yes—
89#498C2×C414yes—
90#498C2×C414yes—
91#446—28no—
92#446—28no—
93#446—28no—
94#446—28no—
95#446—28no—
96#446—24no—
97#446—24no—
98C8×Q8—28no—
99C8×Q8—24no—
100C8×Q8—24no—
101#433—24no—
102#433—24no—
103#433—28no—
104#433—28no—
105#433D414yes—
106#433D414yes—
107#432—28no—
108#431—28no—
109C2×#108C2314yes—
110C2×#108C2314yes—
111#428C2×C418yes—
112#428C2×C418yes—
113#427C2×C414yes—
114#427C2×C412yes—
115#427C2×C412yes—
116#427C2314yes—
117#427C2314yes—
118#427C2314yes—
119#427C2314yes—
120C2×#107C2×C414yes—
121C2×#106C2×C414yes—
122#424C2318yes—
123#424C2318yes—
124#423C2314yes—
125C2×C4⋊C8C2312yes—
126#407C2314yes—
127C2×C22⋊C8C2312yes—
128#405C2×C414yes—
129#405C2×C414yes—
130#405C2318yes—
131#405C2318yes—
132C4×M16C2314yes—
133C2×#97C2312yes—
134C2×#97C2312yes—
135C2×#97C2312yes—
136C2×#97C2311yescharacteristic
137C2×#97C2311yescharacteristic
138C2×#97C2311yescharacteristic
139C2×C4×C8C2312yes—
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143#388—22no—
144#387—22no—
145C4×C4⋊C4C2312yes—
146C4×C22⋊C4C2312yes—
147#376C2311yescharacteristic
148C43C2311yescharacteristic
149#341C2314yes—
150#341C2314yes—
151#339C2318yes—
152#339C2318yes—
153#339C2318yes—
154#339C2318yes—
155#336C2312yes—
156#336C2312yes—
157#336C2312yes—
158#336C2312yes—
159C23×C4C2411yescharacteristic
160C2×M16—24no—
161C2×M16C22×C414yes—
162C2×M16C2418yes—
163C22×C8—24no—
164C22×C8C22×C412yes—
165C22×C8C22×C412yes—
166C22×C8C2411yescharacteristic
167C22×C8C2411yescharacteristic
168C22×C8C2412yes—
169C22×C8C2412yes—
170C22×C8C2411yescharacteristic
171C22×C8C2411yescharacteristic
172C4⋊Q8—28no—
173C4⋊Q8—28no—
174C4⋊Q8C22×C414yes—
175C4⋊Q8C22×C414yes—
176C4⋊Q8C22×C418yes—
177#125—28no—
178#125—28no—
179#125C22×C418yes—
180#125C22×C414yes—
181#125C22×C414yes—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C22⋊Q8—28no—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C4×Q8—28no—
190C4×Q8—24no—
191C4×Q8—24no—
192#117—24no—
193#117—24no—
194#117C22×C414yes—
195#117C22×C414yes—
196#117C22×C414yes—
197#117C2×D412yes—
198#117C2×D412yes—
199#117C2418yes—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C22×C418yes—
202C2×C4⋊C4#4112yes—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4#4112yes—
206C2×C42—24no—
207C2×C42—22no—
208C2×C42C2412yes—
209C2×C42C2412yes—
210C2×C42C2411yescharacteristic
211C2×C42C2411yescharacteristic
212#108—28no—
213#108—28no—
214#107C22×C414yes—
215#107C22×C414yes—
216#107C22×C418yes—
217#107C22×C418yes—
218#107C22×C414yes—
219#107C22×C414yes—
220#106C22×C418yes—
221#106C22×C414yes—
222#106C22×C414yes—
223#106C22×C414yes—
224#106C22×C414yes—
225#106C22×C418yes—
226C4⋊C8—28no—
227C4⋊C8—28no—
228C4⋊C8—28no—
229C4⋊C8—28no—
230C4⋊C8—24no—
231C4⋊C8—24no—
232C4⋊C8C2×D418yes—
233C22⋊C8—28no—
234C22⋊C8C2×D418yes—
235#97—28no—
236#97—28no—
237#97—24no—
238#97—24no—
239#97—24no—
240#97—24no—
241#97C22×C414yes—
242#97C22×C414yes—
243#97C2×D412yes—
244#97C2×D412yes—
245C4×C8—28no—
246C4×C8—28no—
247C4×C8—24no—
248C4×C8—24no—
249C4×C8C22×C412yes—
250C4×C8C22×C412yes—
251C4×C8C22×C412yes—
252C4×C8C22×C412yes—
253C4×C8C2×D412yes—
254C4×C8C2×D412yes—
255#95—22no—
256#95—22no—
257#95—22no—
258C24C2×#4111yescharacteristic
259C2×Q8—28no—
260C2×Q8—24no—
261C2×Q8—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—28no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4C23×C412yes—
268C22×C4C22×D412yes—
269C22×C4C2×#4111yescharacteristic
270C22×C4C2×#4112yes—
271C22×C4C2×#4112yes—
272C22×C4C2511yescharacteristic
273M16—28no—
274M16—28no—
275M16—28no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—28no—
285C2×C8C23×C412yes—
286C2×C8C23×C412yes—
287C2×C8C23×C411yescharacteristic
288C2×C8C23×C411yescharacteristic
289C2×C8C23×C412yes—
290C2×C8C22×D414yes—
291C2×C8C22×D414yes—
292C2×C82−1+414yes—
293C2×C82−1+414yes—
294C4⋊C4—28no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—28no—
299C4⋊C4—28no—
300C4⋊C4—28no—
301C4⋊C4—28no—
302C4⋊C4—82no—
303C4⋊C4C4×D414yes—
304C4⋊C4C4×D414yes—
305C4⋊C4C23×C418yes—
306C4⋊C4C23×C418yes—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—28no—
310C22⋊C4—82no—
311C22⋊C4C4×D414yes—
312C22⋊C4C4×D414yes—
313C42—24no—
314C42—24no—
315C42—28no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—82no—
321C42C23×C412yes—
322C42C23×C412yes—
323C42C22×D412yes—
324C42C22×D411yescharacteristic
325C42C22×D411yescharacteristic
326C23—22no—
327C23—42no—
328C23C2×C4×D412yes—
329C23C22×#4111yescharacteristic
330Q8—84no—
331Q8—84no—
332Q8—88no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—28no—
338C2×C4—28no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—88no—
348C2×C4C2×C4×D412yes—
349C2×C4C2×C4×D414yes—
350C2×C4C2×C4×D412yes—
351C2×C4C2×C4×D412yes—
352C2×C4C24×C411yescharacteristic
353C2×C4C23×D411yescharacteristic
354C2×C4C2×2−1+412yes—
355C2×C4C2×2−1+412yes—
356C8—22no—
357C8—22no—
358C8—24no—
359C8—24no—
360C8—48no—
361C8#51912yes—
362C8#51912yes—
363C8#51914yes—
364C22—22no—
365C22—24no—
366C22—22no—
367C22—82no—
368C22C22×C4×D411yescharacteristic
369C22#350912yes—
370C4—22no—
371C4—22no—
372C4—24no—
373C4—22no—
374C4—48no—
375C4—48no—
376C4—82no—
377C4—82no—
378C4C2×#51911yescharacteristic
379C4C2×#51911yescharacteristic
380C4#349912yes—
381C2—22no—
382C2—42no—
383C2#3420911yescharacteristic
384C2#6018412yes—
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C22×C4×D4
prime radicalC24C2×#41
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#433 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #23172C2semidirect
4. #23171C2semidirect
7. #22259C2semidirect
8. #22257C2semidirect
10. #21734C2semidirect
11. #21733C2semidirect
15. #3164C4semidirect
16. #3164C4semidirect
17. #3163C4semidirect

Maximal subgroups

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

subgroupindex
#606602
#602522
#231722
#231712
#226082
#226072
#222592
#222572
#218462
#217342
#217332
#208282
#204752