Finite Group Databasebeta

← #7500253

Subgroups — #7500253

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59869C211yesmaximal, characteristic, complemented
2#39506C211yesmaximal, characteristic
3#32621C214yesmaximal
4#32182C214yesmaximal
5#32100C214yesmaximal, complemented
6#30079C212yesmaximal, complemented
7#29368C212yesmaximal
8#27508C214yesmaximal
9#27378C214yesmaximal, complemented
10#26333C214yesmaximal, complemented
11#23366C211yesmaximal, characteristic, complemented
12#2859C2211yescharacteristic
13#2809—22no—
14#2783—24no—
15#2621—21no—
16#2571—21no—
17#2568C2214yes—
18#2546C2214yes—
19#2538C2212yes—
20#2513C2214yes—
21#2508—24no—
22#2508C2214yes—
23#2489—22no—
24#2369—21no—
25#2355C2214yes—
26#2355C2212yes—
27#2314—21no—
28C4×C4⋊Q8C2212yes—
29C4×C4⋊Q8C2211yescharacteristic
30#2293C2211yescharacteristic
31#2100C2212yes—
32#2097C2214yes—
33#2095C2214yes—
34#2094C2212yes—
35#2083C2214yes—
36#2074C2214yes—
37#2072C2214yes—
38#2068C2214yes—
39#2055C2214yes—
40#2043C2214yes—
41#2043C2212yes—
42#2043C2214yes—
43#2013C2214yes—
44#2013C2212yes—
45#1985C2214yes—
46#1981C2214yes—
47#1973C2214yes—
48#1960C2214yes—
49#1949C2214yes—
50#1935C2214yes—
51#1928C2214yes—
52#1927C2214yes—
53#1927C2214yes—
54#1925C2214yes—
55#1918C2214yes—
56#1917C2214yes—
57#1917C2214yes—
58#1880C2214yes—
59#1877C2214yes—
60#1865C2214yes—
61#1863C2214yes—
62#1863C2214yes—
63#1839C2212yes—
64#1776C2214yes—
65C22×SD16C2314yes—
66C2×C4⋊Q8D411yescharacteristic
67C2×C4⋊Q8C2312yes—
68C2×C4⋊Q8C2314yes—
69C2×#125—21no—
70C2×C22⋊Q8—22no—
71C2×C4×Q8C2314yes—
72C2×C4×Q8C2312yes—
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75C2×#107C2314yes—
76C2×#107C2314yes—
77C2×#106C2314yes—
78C2×C4⋊C8C2314yes—
79C2×#103C2314yes—
80C2×#103C2314yes—
81C2×#103C2314yes—
82C2×#103C2314yes—
83C2×#103C2314yes—
84C2×#103C2314yes—
85C2×#103C2314yes—
86C2×#102C2314yes—
87C2×#97C2314yes—
88#400—24no—
89#400—24no—
90#400—24no—
91#398—24no—
92#398—24no—
93#398—24no—
94#397—24no—
95#397—24no—
96#396—24no—
97#391—24no—
98#391—22no—
99#391D412yes—
100#391C2314yes—
101#391C2314yes—
102#391C2314yes—
103#390C2314yes—
104#390C2312yes—
105#389—24no—
106#389—24no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389D414yes—
111#389C2314yes—
112#389C2314yes—
113#389C2312yes—
114#388—24no—
115#388—22no—
116#388—24no—
117#387—24no—
118#387—24no—
119#387—24no—
120#387—24no—
121#384D411yescharacteristic
122#384C2312yes—
123#384C2312yes—
124#382—24no—
125#382—24no—
126#382—21no—
127#382—24no—
128#382C2314yes—
129#380—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4C2314yes—
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2314yes—
137C4×C4⋊C4C2314yes—
138C4×C22⋊C4—21no—
139C43C2311yescharacteristic
140#340C2314yes—
141#340C2314yes—
142#340C2314yes—
143#340C2314yes—
144#336C2314yes—
145#336C2314yes—
146#336C2314yes—
147#336C2312yes—
148#336C2312yes—
149#336C2314yes—
150C22×Q8C2×D412yes—
151C22×Q8C2414yes—
152C22×D4C2411yescharacteristic
153C23×C4—21no—
154C2×SD16—44no—
155C2×SD16—44no—
156C2×SD16—44no—
157C22×C8C2414yes—
158C22×C8C2414yes—
159C4⋊Q8—24no—
160C4⋊Q8—44no—
161C4⋊Q8—44no—
162C4⋊Q8—44no—
163#125—82no—
164C22⋊Q8—84no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×Q8—44no—
168C4×Q8—44no—
169C4×D4—42no—
170#117—44no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4C2×D414yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D414yes—
186C2×C4⋊C4C2×D414yes—
187C2×C4⋊C4C2411yescharacteristic
188C2×C4⋊C4C2414yes—
189C2×C4⋊C4C2412yes—
190C2×C4⋊C4C2414yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C42—21no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2411yescharacteristic
204C2×C42C2412yes—
205C2×C42C2414yes—
206#107—44no—
207#107—44no—
208#106—44no—
209C4⋊C8—44no—
210#103—44no—
211#103—44no—
212#103—44no—
213#103—44no—
214#103—44no—
215#103—44no—
216#103—44no—
217#102—44no—
218#97—44no—
219#95—22no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—21no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95C2×D414yes—
234#95C2×D414yes—
235C24—21no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×D4—24no—
246C2×D4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—21no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4C22×D414yes—
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D414yes—
258C22×C4C22×D412yes—
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D412yes—
261C22×C4C2511yescharacteristic
262SD16—88no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C4⋊C4—28no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—48no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—84no—
288C4⋊C4—84no—
289C4⋊C4—84no—
290C4⋊C4—84no—
291C22⋊C4—42no—
292C22⋊C4—44no—
293C22⋊C4—82no—
294C22⋊C4—84no—
295C22⋊C4—84no—
296C42—24no—
297C42—24no—
298C42—22no—
299C42—24no—
300C42—42no—
301C42—41no—
302C42—48no—
303C42—42no—
304C42—84no—
305C42—84no—
306C42#13714yes—
307C23—41no—
308C23—42no—
309C23—41no—
310C23—42no—
311C23—41no—
312C23C23×D411yescharacteristic
313Q8—44no—
314Q8—44no—
315Q8—44no—
316Q8—44no—
317Q8—84no—
318D4—44no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—21no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—41no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—84no—
350C2×C4C2×#13712yes—
351C2×C4C2×#13714yes—
352C2×C4#58512yes—
353C2×C4#58514yes—
354C8—44no—
355C8—44no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—82no—
360C22—82no—
361C22#348511yescharacteristic
362C22#348712yes—
363C22C22×#13711yescharacteristic
364C22#358511yescharacteristic
365C22#358612yes—
366C4—22no—
367C4—24no—
368C4—44no—
369C4—41no—
370C4—44no—
371C4—42no—
372C4—81no—
373C4—84no—
374C4—82no—
375C4—84no—
376C4—84no—
377C4#356614yes—
378C2—161no—
379C2#5208411yescharacteristic
380C2#6172111yescharacteristic
381C2#6179811yescharacteristic
382C2#6189312yes—
383C2#6194012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59869C2semidirect
5. #32100C2semidirect
6. #30079C2semidirect
9. #27378C2semidirect
10. #26333C2semidirect
11. #23366C2semidirect

Maximal subgroups

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

subgroupindex
#598692
#395062
#326212
#321822
#321002
#300792
#293682
#275082
#273782
#263332
#233662