Finite Group Databasebeta

← #6389808

Subgroups — #6389808

Order 512.

1524 conjugacy classes of subgroups.

The subgroup lattice has 392 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#23170C214yesmaximal, complemented
5#22608C214yesmaximal
6#22606C214yesmaximal
7#22259C212yesmaximal, complemented
8#22257C212yesmaximal, complemented
9#21846C212yesmaximal
10#21733C212yesmaximal, complemented
11#21732C212yesmaximal, complemented
12#20828C211yesmaximal, characteristic
13#20474C211yesmaximal, 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#1945C2214yes—
40#1945C2214yes—
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#1772C2214yes—
55#1772C2214yes—
56#1772C2214yes—
57#1771C2212yes—
58C8×C4⋊C4C2212yes—
59#1753C2212yes—
60C8×C22⋊C4C2212yes—
61#1746C2211yescharacteristic
62#1745C2211yescharacteristic
63#1726C2211yescharacteristic
64C42×C8C2211yescharacteristic
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#433D412yes—
107#433D412yes—
108#432—28no—
109#431—28no—
110C2×#108C2314yes—
111C2×#108C2314yes—
112#428C2×C418yes—
113#428C2×C418yes—
114#427C2×C414yes—
115#427C2×C412yes—
116#427C2×C412yes—
117#427C2314yes—
118#427C2314yes—
119#427C2314yes—
120#427C2314yes—
121C2×#107C2×C414yes—
122C2×#106C2×C414yes—
123#424C2318yes—
124#424C2318yes—
125#423C2314yes—
126C2×C4⋊C8C2312yes—
127#407C2314yes—
128C2×C22⋊C8C2312yes—
129#405C2×C414yes—
130#405C2×C414yes—
131#405C2318yes—
132#405C2318yes—
133C4×M16C2314yes—
134C2×#97C2312yes—
135C2×#97C2311yescharacteristic
136C2×#97C2311yescharacteristic
137C2×#97C2311yescharacteristic
138C2×C4×C8C2312yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144#388—22no—
145#387—22no—
146C4×C4⋊C4C2312yes—
147C4×C22⋊C4C2312yes—
148#376C2311yescharacteristic
149C43C2311yescharacteristic
150#341C2314yes—
151#341C2314yes—
152#339C2318yes—
153#339C2318yes—
154#339C2318yes—
155#339C2318yes—
156#336C2312yes—
157#336C2312yes—
158#336C2312yes—
159#336C2312yes—
160C23×C4C2411yescharacteristic
161C2×M16—24no—
162C2×M16C22×C414yes—
163C2×M16C2418yes—
164C22×C8—24no—
165C22×C8C22×C412yes—
166C22×C8C22×C412yes—
167C22×C8C2411yescharacteristic
168C22×C8C2411yescharacteristic
169C22×C8C2412yes—
170C22×C8C2412yes—
171C22×C8C2411yescharacteristic
172C22×C8C2411yescharacteristic
173C4⋊Q8—28no—
174C4⋊Q8—28no—
175C4⋊Q8C22×C414yes—
176C4⋊Q8C22×C414yes—
177C4⋊Q8C22×C418yes—
178#125—28no—
179#125—28no—
180#125C22×C418yes—
181#125C22×C414yes—
182#125C22×C414yes—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8—28no—
190C4×Q8—28no—
191C4×Q8—24no—
192C4×Q8—24no—
193#117—24no—
194#117—24no—
195#117C22×C414yes—
196#117C22×C414yes—
197#117C22×C414yes—
198#117C2×D412yes—
199#117C2×D412yes—
200#117C2418yes—
201C2×C4⋊C4—22no—
202C2×C4⋊C4C22×C418yes—
203C2×C4⋊C4#4112yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4#4112yes—
207C2×C42—24no—
208C2×C42—22no—
209C2×C42C2412yes—
210C2×C42C2412yes—
211C2×C42C2411yescharacteristic
212C2×C42C2411yescharacteristic
213#108—28no—
214#108—28no—
215#107C22×C414yes—
216#107C22×C414yes—
217#107C22×C418yes—
218#107C22×C418yes—
219#107C22×C414yes—
220#107C22×C414yes—
221#106C22×C418yes—
222#106C22×C414yes—
223#106C22×C414yes—
224#106C22×C414yes—
225#106C22×C414yes—
226#106C22×C418yes—
227C4⋊C8—28no—
228C4⋊C8—28no—
229C4⋊C8—28no—
230C4⋊C8—28no—
231C4⋊C8—24no—
232C4⋊C8—24no—
233C4⋊C8C2×D414yes—
234C4⋊C8C2×D414yes—
235C22⋊C8—28no—
236C22⋊C8C2×D414yes—
237C22⋊C8C2×D414yes—
238#97—28no—
239#97—28no—
240#97—24no—
241#97—24no—
242#97C22×C414yes—
243#97C22×C414yes—
244#97C2×D412yes—
245#97C2×D412yes—
246C4×C8—28no—
247C4×C8—28no—
248C4×C8—24no—
249C4×C8—24no—
250C4×C8—24no—
251C4×C8—24no—
252C4×C8C22×C412yes—
253C4×C8C22×C412yes—
254C4×C8C22×C412yes—
255C4×C8C22×C412yes—
256C4×C8C2×D411yescharacteristic
257C4×C8C2×D412yes—
258C4×C8C2×D411yescharacteristic
259#95—22no—
260#95—22no—
261#95—22no—
262C24C2×#4111yescharacteristic
263C2×Q8—28no—
264C2×Q8—24no—
265C2×Q8—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—28no—
270C22×C4—22no—
271C22×C4C23×C412yes—
272C22×C4C22×D412yes—
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4112yes—
275C22×C4C2×#4112yes—
276C22×C4C2511yescharacteristic
277M16—28no—
278M16—28no—
279M16—28no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—28no—
289C2×C8C23×C412yes—
290C2×C8C23×C412yes—
291C2×C8C23×C411yescharacteristic
292C2×C8C23×C411yescharacteristic
293C2×C8C23×C412yes—
294C2×C8C22×D412yes—
295C2×C8C22×D414yes—
296C2×C8C22×D412yes—
297C2×C82−1+414yes—
298C2×C82−1+414yes—
299C4⋊C4—28no—
300C4⋊C4—28no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C4⋊C4—28no—
307C4⋊C4—44no—
308C4⋊C4C4×D414yes—
309C4⋊C4C4×D414yes—
310C4⋊C4C23×C418yes—
311C4⋊C4C23×C418yes—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—28no—
315C22⋊C4—44no—
316C22⋊C4C4×D414yes—
317C22⋊C4C4×D414yes—
318C42—24no—
319C42—24no—
320C42—28no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—44no—
326C42C23×C412yes—
327C42C23×C412yes—
328C42C22×D412yes—
329C42C22×D411yescharacteristic
330C42C22×D411yescharacteristic
331C23—22no—
332C23—42no—
333C23C2×C4×D412yes—
334C23C22×#4111yescharacteristic
335Q8—84no—
336Q8—84no—
337Q8—88no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—88no—
353C2×C4C2×C4×D412yes—
354C2×C4C2×C4×D414yes—
355C2×C4C2×C4×D412yes—
356C2×C4C2×C4×D412yes—
357C2×C4C24×C411yescharacteristic
358C2×C4C23×D411yescharacteristic
359C2×C4C2×2−1+412yes—
360C2×C4C2×2−1+412yes—
361C8—22no—
362C8—22no—
363C8—22no—
364C8—22no—
365C8—24no—
366C8—48no—
367C8#51912yes—
368C8#51912yes—
369C8#51914yes—
370C22—22no—
371C22—24no—
372C22—22no—
373C22—82no—
374C22C22×C4×D411yescharacteristic
375C22#350912yes—
376C4—22no—
377C4—22no—
378C4—24no—
379C4—22no—
380C4—44no—
381C4—48no—
382C4—48no—
383C4—82no—
384C4C2×#51911yescharacteristic
385C4C2×#51911yescharacteristic
386C4#349912yes—
387C2—22no—
388C2—42no—
389C2#3420911yescharacteristic
390C2#6018312yes—
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. #23170C2semidirect
7. #22259C2semidirect
8. #22257C2semidirect
10. #21733C2semidirect
11. #21732C2semidirect
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
#231702
#226082
#226062
#222592
#222572
#218462
#217332
#217322
#208282
#204742