Finite Group Databasebeta

← #6343460

Subgroups — #6343460

Order 512.

3099 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62813C211yesmaximal, characteristic
2#60276C211yesmaximal, characteristic
3C4⋊C4×#41C211yesmaximal, characteristic
4#17619C214yesmaximal, complemented
5#17614C2112yesmaximal, complemented
6#17575C216yesmaximal, complemented
7C2×#1503C216yesmaximal, complemented
8#3504—24no—
9#3503C414yescomplemented
10#3503C414yescomplemented
11#3502—21no—
12#3502C2212yes—
13#3501—23no—
14#3501C2216yes—
15Q8×#41—24no—
16Q8×#41C418yescomplemented
17#3458—23no—
18C2×#533C2216yes—
19#3428—21no—
20C2×C4×#41—21no—
21C2×C4×#41C2211yescharacteristic
22#2988—24no—
23#2982C2212yes—
24#2972C2216yes—
25C8×#41—24no—
26C2×#431C2216yes—
27#2351C2212yes—
28Q8×C4⋊C4—21no—
29#2349C2216yes—
30D4×C4⋊C4—23no—
31#2296—23no—
32#2285C2213yes—
33C2×C4×C4⋊C4C2213yes—
34#1520C22124yes—
35#1518C22124yes—
36#1503C2214yes—
37#1503C22112yes—
38#1503C22124yes—
39C2×#328C22124yes—
40C22×#41D411yescharacteristic
41Q82—24no—
42Q82C2×C418yescomplemented
43#557—24no—
44#557—28no—
45#557—28no—
46#557C2×C418yescomplemented
47#554—212no—
48#554—224no—
49#554—224no—
50#554C2×C4124yescomplemented
51D4×Q8—212no—
52D4×Q8C2×C4124yescomplemented
53#544—212no—
54#544—212no—
55#544—26no—
56#544—26no—
57#543—212no—
58#533—224no—
59#533—26no—
60#533—26no—
61#533—26no—
62#533—26no—
63#533C2×C4112yescomplemented
64#533C2×C4112yescomplemented
65#533C2314yes—
66#533C23112yes—
67C2×C4⋊Q8C2×C416yescomplemented
68C2×C4⋊Q8C2×C416yescomplemented
69C2×#125—26no—
70C2×C22⋊Q8—26no—
71C2×C22⋊Q8C23112yes—
72#520—24no—
73#520—24no—
74#519—22no—
75#518—26no—
76C4×#41—28no—
77C4×#41—48no—
78C4×#41C2×C414yes—
79C4×#41C2×C414yes—
80C4×#41D414yes—
81C4×#41D414yes—
82C2×C4×Q8—22no—
83C2×C4×Q8—26no—
84C2×C4×Q8C2×C4112yescomplemented
85C2×C4×Q8D411yescharacteristic
86C2×C4×Q8C2311yescharacteristic
87C2×C4×D4—26no—
88C2×C4×D4D413yes—
89C2×C4×D4C2313yes—
90C2×#117—26no—
91C2×#117—23no—
92C2×#117D413yes—
93C22×C4⋊C4—23no—
94C22×C42—23no—
95C22×C42C2313yes—
96C8×Q8—28no—
97C8×D4—224no—
98#431—26no—
99#431—26no—
100#431C2314yes—
101#431C23112yes—
102C2×C4⋊C8—26no—
103C2×C22⋊C8C23112yes—
104C2×C4×C8—26no—
105#389—26no—
106#389—26no—
107#385—212no—
108#384—23no—
109#382C2316yes—
110#380—23no—
111#380C2316yes—
112C4×C4⋊C4—23no—
113C4×C4⋊C4—23no—
114C4×C4⋊C4—23no—
115C4×C4⋊C4C2316yes—
116C4×C4⋊C4C2311yescharacteristic
117C4×C4⋊C4C2313yes—
118C2×#95C2316yes—
119#328C23116yes—
120#328C23148yes—
121C2×#41C22⋊C414yes—
122C2×#41C22⋊C414yes—
123C2×#41SD1618yes—
124C2×#41Q1618yes—
125C22×Q8—26no—
126C22×Q8C22×C4112yes—
127C22×Q8C2×D411yescharacteristic
128C22×D4C2×D413yes—
129C23×C4—26no—
130C23×C4C2×D413yes—
131C23×C4C2×D413yes—
132C23×C4C2413yes—
133C22×C8—212no—
134C4⋊Q8—224no—
135C4⋊Q8—224no—
136C4⋊Q8—26no—
137C4⋊Q8—26no—
138C4⋊Q8—26no—
139C4⋊Q8—26no—
140C4⋊Q8C22×C414yes—
141C4⋊Q8C22×C4112yes—
142C4⋊Q8C22×C414yes—
143C4⋊Q8C22×C4112yes—
144#125—224no—
145#125—224no—
146#125—212no—
147#125—212no—
148#125—24no—
149#125—212no—
150C22⋊Q8—212no—
151C22⋊Q8—212no—
152C22⋊Q8—24no—
153C22⋊Q8—212no—
154C22⋊Q8—212no—
155C22⋊Q8—212no—
156C22⋊Q8—248no—
157C22⋊Q8—248no—
158C22⋊Q8C22×C4124yes—
159C22⋊Q8C2418yes—
160C4×Q8—28no—
161C4×Q8—28no—
162C4×Q8—212no—
163C4×Q8—212no—
164C4×Q8—212no—
165C4×Q8—212no—
166C4×Q8—24no—
167C4×Q8—212no—
168C4×Q8—48no—
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×D4—424no—
178C4×D4C22⋊C4112yes—
179C4×D4C22⋊C4112yes—
180C4×D4C22×C4112yes—
181C4×D4C2×D4112yes—
182#117—212no—
183#117—212no—
184#117—24no—
185#117—212no—
186#117—412no—
187#117C22⋊C4112yes—
188#117C22⋊C4112yes—
189C2×C4⋊C4—26no—
190C2×C4⋊C4—26no—
191C2×C4⋊C4—26no—
192C2×C4⋊C4—212no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—212no—
195C2×C4⋊C4—23no—
196C2×C4⋊C4—26no—
197C2×C4⋊C4—26no—
198C2×C4⋊C4—26no—
199C2×C4⋊C4—412no—
200C2×C4⋊C4C22×C4112yes—
201C2×C4⋊C4C2×D413yes—
202C2×C4⋊C4C2×D413yes—
203C2×C22⋊C4—212no—
204C2×C22⋊C4—26no—
205C2×C22⋊C4C2×D416yes—
206C2×C42—212no—
207C2×C42—23no—
208C2×C42—26no—
209C2×C42—21no—
210C2×C42—412no—
211C2×C42C22×C416yes—
212C2×C42C22×C416yes—
213C2×C42C2×D413yes—
214C2×C42C2×D413yes—
215C2×C42C2×D416yes—
216C2×C42C2×D416yes—
217C2×C42C2413yes—
218C2×C42C2411yescharacteristic
219C4⋊C8—24no—
220C4⋊C8—212no—
221C22⋊C8—212no—
222C22⋊C8C2418yes—
223C4×C8—24no—
224C4×C8—212no—
225#95—26no—
226#95—26no—
227#95C2412yes—
228#95C2416yes—
229C24C22×D413yes—
230#41#103132yes—
231#41#103132yes—
232C2×Q8—24no—
233C2×Q8—212no—
234C2×Q8—212no—
235C2×Q8—812no—
236C2×Q8—812no—
237C2×Q8C2×C22⋊C414yes—
238C2×Q8C2×SD1614yes—
239C2×Q8C2×Q1614yes—
240C2×Q8C23×C418yes—
241C2×D4C2×C22⋊C4112yes—
242C2×D4C2×SD16112yes—
243C2×D4C2×Q16112yes—
244C22×C4—22no—
245C22×C4—212no—
246C22×C4—212no—
247C22×C4—26no—
248C22×C4—46no—
249C22×C4—412no—
250C22×C4—412no—
251C22×C4—83no—
252C22×C4—86no—
253C22×C4C2×C22⋊C416yes—
254C22×C4C2×C22⋊C416yes—
255C22×C4C2×C22⋊C416yes—
256C22×C4C2×C22⋊C416yes—
257C22×C4C2×SD16112yes—
258C22×C4C2×Q16112yes—
259C22×C4C23×C416yes—
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D413yes—
262C22×C4C22×D413yes—
263C22×C4C22×D416yes—
264C22×C4C22×D413yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C2511yescharacteristic
267C2×C8—28no—
268C2×C8—812no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—28no—
274C4⋊C4—212no—
275C4⋊C4—212no—
276C4⋊C4—212no—
277C4⋊C4—224no—
278C4⋊C4—224no—
279C4⋊C4—212no—
280C4⋊C4—28no—
281C4⋊C4—46no—
282C4⋊C4—412no—
283C4⋊C4—412no—
284C4⋊C4—86no—
285C4⋊C4C2×C22⋊C4112yes—
286C4⋊C4C2×C22⋊C4112yes—
287C4⋊C4C23×C418yes—
288C22⋊C4—224no—
289C22⋊C4—28no—
290C22⋊C4—412no—
291C22⋊C4C2×C22⋊C4124yes—
292C42—224no—
293C42—28no—
294C42—42no—
295C42—46no—
296C42—412no—
297C42C2×C22⋊C416yes—
298C42C2×C22⋊C416yes—
299C42C2×C22⋊C416yes—
300C42C2×C22⋊C416yes—
301C42C23×C411yescharacteristic
302C42C23×C416yes—
303C42C23×C411yescharacteristic
304C42C22×D416yes—
305C42C22×D411yescharacteristic
306C42C22×D411yescharacteristic
307C23—43no—
308C23—43no—
309C23C22×C22⋊C416yes—
310C23C22×SD1616yes—
311C23C22×Q1616yes—
312C23C23×D411yescharacteristic
313Q8—48no—
314Q8—48no—
315Q8—812no—
316Q8—812no—
317Q8C2×#103116yes—
318D4C2×#103148yes—
319C2×C4—22no—
320C2×C4—212no—
321C2×C4—24no—
322C2×C4—28no—
323C2×C4—28no—
324C2×C4—43no—
325C2×C4—43no—
326C2×C4—46no—
327C2×C4—46no—
328C2×C4—41no—
329C2×C4—46no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—46no—
333C2×C4—46no—
334C2×C4—46no—
335C2×C4—812no—
336C2×C4—812no—
337C2×C4—824no—
338C2×C4C2×#103124yes—
339C2×C4C2×#103124yes—
340C2×C4C22×C22⋊C411yescharacteristic
341C2×C4C22×C22⋊C416yes—
342C2×C4C22×C22⋊C416yes—
343C2×C4C22×C22⋊C416yes—
344C2×C4C22×C22⋊C411yescharacteristic
345C2×C4C22×C22⋊C411yescharacteristic
346C2×C4C22×C22⋊C411yescharacteristic
347C2×C4C22×SD1612yes—
348C2×C4C22×SD1616yes—
349C2×C4C22×Q1612yes—
350C2×C4C22×Q1616yes—
351C2×C4C24×C411yescharacteristic
352C2×C4C23×D411yescharacteristic
353C8—48no—
354C22—21no—
355C22—26no—
356C22—21no—
357C22—26no—
358C22—46no—
359C22—46no—
360C22C22×#103112yes—
361C22C23×C22⋊C411yescharacteristic
362C22C23×SD1611yescharacteristic
363C22C23×Q1611yescharacteristic
364C4—21no—
365C4—21no—
366C4—26no—
367C4—21no—
368C4—26no—
369C4—44no—
370C4—48no—
371C4—48no—
372C4—86no—
373C4C22×#10312yes—
374C4C22×#103112yes—
375C4C22×#10312yes—
376C4#290112yes—
377C2—21no—
378C2—21no—
379C2—212no—
380C2#6007011yescharacteristic
381C2C23×#10311yescharacteristic
382C2C2×#290111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C23×C22⋊C4
prime radicalC22×#41D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17619C2semidirect
5. #17614C2semidirect
6. #17575C2semidirect
7. C2×#1503C2semidirect
9. #3503C4semidirect
10. #3503C4semidirect
16. Q8×#41C4semidirect
42. Q82C2×C4semidirect
46. #557C2×C4semidirect
50. #554C2×C4semidirect
52. D4×Q8C2×C4semidirect
63. #533C2×C4semidirect
64. #533C2×C4semidirect
67. C2×C4⋊Q8C2×C4semidirect
68. C2×C4⋊Q8C2×C4semidirect
84. C2×C4×Q8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#628132
#602762
C4⋊C4×#412
#176192
#176142
#175752
C2×#15032