Finite Group Databasebeta

← #6369938

Subgroups — #6369938

Order 512.

3206 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2913C211yesmaximal, characteristic
2C2×#2912C211yesmaximal, characteristic
3C2×#2875C211yesmaximal, characteristic
4C4×#520C212yesmaximal
5#20455C213yesmaximal, complemented
6#20454C213yesmaximal, complemented
7#20453C216yesmaximal, complemented
8C4×#428C216yesmaximal, complemented
9#20451C213yesmaximal, complemented
10C2×#1776C213yesmaximal, complemented
11#20348C212yesmaximal
12C22×#131C2211yescharacteristic
13C22×#131C2211yescharacteristic
14C2×#520C2212yes—
15C2×#520C2212yes—
16C2×C4×#41C2211yescharacteristic
17#2913C4116yescomplemented
18#2912C4116yescomplemented
19C2×#428C2216yes—
20C2×#428C2216yes—
21C2×#427C2213yes—
22C2×#427C2213yes—
23C22×#107C2213yes—
24C22×#106C2213yes—
25#2875C4116yescomplemented
26C2×#405C2213yes—
27C2×#405C2213yes—
28C2×C4×M16C2213yes—
29C22×#97C2213yes—
30#2276—22no—
31#2275—26no—
32#2274C2216yes—
33#2274C2216yes—
34C42×Q8—22no—
35C42×D4—26no—
36C4×#117C2216yes—
37C2×C4×C4⋊C4C2216yes—
38#1776C2213yes—
39#1776C2216yes—
40#1776C2216yes—
41#1776C2216yes—
42#1776C2211yescharacteristic
43#1776C2216yes—
44#1776C2216yes—
45#1776C2213yes—
46#1776C2213yes—
47C4×#107C22112yes—
48C4×#106C22112yes—
49#1736C2216yes—
50#1736C2216yes—
51#1735C2216yes—
52C2×#340C2216yes—
53C22×#41C2311yescharacteristic
54C2×#131—24no—
55C2×#131—24no—
56C2×#131C2×C418yes—
57C2×#131C2×C418yes—
58C2×#131C2×C418yes—
59C2×#131C2×C418yes—
60C22×M16C2313yes—
61C22×M16C2313yes—
62C23×C8C2313yes—
63C23×C8C2313yes—
64#520—28no—
65#520—28no—
66#520C2×C4116yes—
67#520C2×C4116yes—
68C4×#41C2×C418yes—
69C4×#41Q818yes—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8D412yes—
73C2×C4×D4—26no—
74C2×C4×D4—26no—
75C2×C4×D4D416yes—
76C2×#117C2316yes—
77C2×#117C2316yes—
78C2×#117C2313yes—
79C22×C4⋊C4C2316yes—
80C22×C4⋊C4C2316yes—
81C22×C42—26no—
82C22×C42C2313yes—
83#428C2×C4148yescomplemented
84#428C2×C4148yescomplemented
85#427C2×C4124yescomplemented
86#427C2×C4124yescomplemented
87C2×#107C2×C4124yescomplemented
88C2×#107C2313yes—
89C2×#107C2316yes—
90C2×#107C2311yescharacteristic
91C2×#107C2316yes—
92C2×#106C2×C4124yescomplemented
93C2×#106C2313yes—
94C2×#106C2316yes—
95C2×#106C2311yescharacteristic
96C2×#106C2316yes—
97#405C2×C4124yescomplemented
98#405C2×C4124yescomplemented
99C4×M16C2×C4124yescomplemented
100C2×#97C2×C4124yescomplemented
101C2×#97C2311yescharacteristic
102C2×#97C2313yes—
103C2×#97C2313yes—
104C2×#97C2313yes—
105C2×C4×C8C2316yes—
106C4×C4⋊C4—26no—
107C4×C4⋊C4—26no—
108C4×C4⋊C4—26no—
109C4×C4⋊C4—26no—
110C4×C4⋊C4—26no—
111C4×C4⋊C4—26no—
112C4×C4⋊C4C2316yes—
113C4×C4⋊C4C23112yes—
114C4×C4⋊C4C2312yes—
115C4×C4⋊C4C2316yes—
116C4×C4⋊C4C2316yes—
117C4×C22⋊C4—26no—
118C4×C22⋊C4—26no—
119C4×C22⋊C4—212no—
120C4×C22⋊C4—212no—
121#376—26no—
122C2×#95—26no—
123C43—26no—
124#340C2316yes—
125#340C23112yes—
126#340C2312yes—
127#340C2316yes—
128#340C2316yes—
129C2×#41C22×C414yes—
130C2×#41C22×C414yes—
131C2×#41C22×C414yes—
132C2×#41C2×Q814yes—
133C2×#41#4114yes—
134C2×#41#4114yes—
135C22×Q8C2×D411yescharacteristic
136C22×D4C2×D413yes—
137C23×C4—26no—
138C23×C4—26no—
139C23×C4C2×D416yes—
140C23×C4C2413yes—
141#131—232no—
142#131C42164yescomplemented
143C2×M16—26no—
144C2×M16—26no—
145C2×M16C22×C4112yes—
146C2×M16C22×C4112yes—
147C2×M16C22×C4112yes—
148C2×M16C22×C4112yes—
149C22×C8—26no—
150C22×C8—26no—
151C22×C8C22×C4112yes—
152C22×C8C22×C4112yes—
153C22×C8C22×C4112yes—
154C22×C8C22×C4112yes—
155C22×C8C2411yescharacteristic
156C22×C8C2413yes—
157C22×C8C2411yescharacteristic
158C22×C8C2413yes—
159C4×Q8—216no—
160C4×Q8—216no—
161C4×Q8C4⋊C4116yes—
162C4×D4—248no—
163C4×D4—248no—
164C4×D4C4⋊C4148yes—
165#117—212no—
166#117—212no—
167#117C22×C4124yes—
168#117C22×C4124yes—
169#117C22×C4112yes—
170#117C2×Q8112yes—
171C2×C4⋊C4—26no—
172C2×C4⋊C4—26no—
173C2×C4⋊C4—212no—
174C2×C4⋊C4—212no—
175C2×C4⋊C4C22×C4124yes—
176C2×C4⋊C4C22×C4124yes—
177C2×C4⋊C4C2×D416yes—
178C2×C4⋊C4C2416yes—
179C2×C4⋊C4C2416yes—
180C2×C4⋊C4C2412yes—
181C2×C4⋊C4C2412yes—
182C2×C22⋊C4—26no—
183C2×C22⋊C4—26no—
184C2×C22⋊C4C2×D416yes—
185C2×C42—26no—
186C2×C42—26no—
187C2×C42—22no—
188C2×C42—26no—
189C2×C42—26no—
190C2×C42—26no—
191C2×C42—212no—
192C2×C42—812no—
193C2×C42C22×C4112yes—
194C2×C42C2×D416yes—
195C2×C42C2×Q8112yes—
196C2×C42C2413yes—
197C2×C42C2411yescharacteristic
198#107C22×C4112yes—
199#107C22×C4124yes—
200#107C22×C414yes—
201#107C22×C4124yes—
202#106C22×C4112yes—
203#106C22×C4124yes—
204#106C22×C414yes—
205#106C22×C4124yes—
206#97C22×C414yes—
207#97C22×C4112yes—
208#97C22×C4112yes—
209#97C22×C4112yes—
210C4×C8C22×C4124yes—
211#95—26no—
212#95—212no—
213#95—22no—
214#95—26no—
215#95—26no—
216C24C22×D413yes—
217#41C2×C42116yes—
218#41#117116yes—
219#41C4×Q8116yes—
220#41C4×Q8116yes—
221C2×Q8C2×C4⋊C414yes—
222C2×Q8C4×D414yes—
223C2×Q8C4×D414yes—
224C2×D4C2×C4⋊C4112yes—
225C2×D4C4×D4112yes—
226C2×D4C4×D4112yes—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—26no—
230C22×C4—26no—
231C22×C4—224no—
232C22×C4—224no—
233C22×C4—46no—
234C22×C4—46no—
235C22×C4—86no—
236C22×C4—86no—
237C22×C4—86no—
238C22×C4—86no—
239C22×C4C2×C4⋊C4124yes—
240C22×C4C23×C416yes—
241C22×C4C23×C416yes—
242C22×C4C23×C416yes—
243C22×C4C22×D412yes—
244C22×C4C22×D416yes—
245C22×C4C22×D413yes—
246C22×C4C22×Q816yes—
247C22×C4C2×#4116yes—
248C22×C4C2×#4116yes—
249C22×C4C2511yescharacteristic
250M16—224no—
251M16C2×C42148yescomplemented
252C2×C8—224no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—23no—
256C2×C8—23no—
257C2×C8C2×C42148yescomplemented
258C2×C8C23×C416yes—
259C2×C8C23×C412yes—
260C2×C8C23×C416yes—
261C2×C8C23×C412yes—
262C2×C8C23×C412yes—
263C2×C8C23×C416yes—
264C2×C8C23×C416yes—
265C2×C8C23×C412yes—
266C4⋊C4—224no—
267C4⋊C4—224no—
268C4⋊C4—26no—
269C4⋊C4—26no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4C2×C4⋊C4124yes—
273C4⋊C4C23×C4112yes—
274C4⋊C4C23×C414yes—
275C4⋊C4C23×C4112yes—
276C4⋊C4C23×C414yes—
277C22⋊C4—224no—
278C22⋊C4—224no—
279C22⋊C4C2×C4⋊C4124yes—
280C42—224no—
281C42—224no—
282C42—44no—
283C42—412no—
284C42—86no—
285C42—86no—
286C42—812no—
287C42C2×C4⋊C4124yes—
288C42C23×C416yes—
289C42C23×C412yes—
290C42C22×Q812yes—
291C42C22×Q816yes—
292C23—26no—
293C23—26no—
294C23C22×C4⋊C416yes—
295C23C2×C4×D416yes—
296C23C2×C4×D416yes—
297C23C23×D411yescharacteristic
298Q8C4×C4⋊C4116yes—
299D4C4×C4⋊C4148yes—
300C2×C4—22no—
301C2×C4—26no—
302C2×C4—22no—
303C2×C4—26no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—212no—
307C2×C4—212no—
308C2×C4—424no—
309C2×C4—46no—
310C2×C4—46no—
311C2×C4—42no—
312C2×C4—46no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—46no—
317C2×C4—824no—
318C2×C4—824no—
319C2×C4C22×C42112yes—
320C2×C4C22×C4⋊C414yes—
321C2×C4C22×C4⋊C4112yes—
322C2×C4C22×C4⋊C416yes—
323C2×C4C2×#117112yes—
324C2×C4C2×C4×D416yes—
325C2×C4C2×C4×D416yes—
326C2×C4C2×C4×Q8112yes—
327C2×C4C2×C4×Q8112yes—
328C2×C4C24×C411yescharacteristic
329C2×C4C24×C411yescharacteristic
330C2×C4C24×C411yescharacteristic
331C2×C4C23×Q811yescharacteristic
332C2×C4C22×#4111yescharacteristic
333C2×C4C22×#4111yescharacteristic
334C8—22no—
335C8—26no—
336C8C22×C4214yes—
337C8C22×C42112yes—
338C22—212no—
339C22—26no—
340C22—26no—
341C22—212no—
342C22C2×C4×C4⋊C4112yes—
343C22C23×C4⋊C411yescharacteristic
344C22C22×C4×D411yescharacteristic
345C22C22×C4×D411yescharacteristic
346C22#358512yes—
347C22#358612yes—
348C4—24no—
349C4—212no—
350C4—44no—
351C4—412no—
352C4—44no—
353C4—412no—
354C4C2×C4×C4⋊C4112yes—
355C4C23×C4211yescharacteristic
356C4C22×#11711yescharacteristic
357C4C22×C4×Q811yescharacteristic
358C4C22×C4×Q811yescharacteristic
359C2—212no—
360C2C22×C4×C4⋊C411yescharacteristic
361C2#6010912yes—
362C2#6017412yes—
363C2#6017511yescharacteristic
364C2#6017611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC4C23×C42
Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #20455C2semidirect
6. #20454C2semidirect
7. #20453C2semidirect
8. C4×#428C2semidirect
9. #20451C2semidirect
10. C2×#1776C2semidirect
17. #2913C4semidirect
18. #2912C4semidirect
25. #2875C4semidirect
83. #428C2×C4semidirect
84. #428C2×C4semidirect
85. #427C2×C4semidirect
86. #427C2×C4semidirect
87. C2×#107C2×C4semidirect
92. C2×#106C2×C4semidirect
97. #405C2×C4semidirect
98. #405C2×C4semidirect
99. C4×M16C2×C4semidirect
100. C2×#97C2×C4semidirect
142. #131C42semidirect
251. M16C2×C42semidirect
257. C2×C8C2×C42semidirect

Maximal subgroups

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

subgroupindex
C2×#29132
C2×#29122
C2×#28752
C4×#5202
#204552
#204542
#204532
C4×#4282
#204512
C2×#17762
#203482