Finite Group Databasebeta

← #165666

Subgroups — #165666

Order 512.

724 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36403C211yesmaximal, characteristic
2#22736C212yesmaximal, complemented
3#7732C212yesmaximal
4#7697C212yesmaximal, complemented
5#7686C212yesmaximal
6#7635C211yesmaximal, characteristic
7#7615C212yesmaximal, complemented
8#7601C211yesmaximal, characteristic
9#7577C212yesmaximal, complemented
10C2×#446—22no—
11#2864C2211yescharacteristic
12#2860C2212yes—
13#2604—22no—
14#2596—22no—
15#2581C411yescharacteristic
16#2580C411yescharacteristic
17#2575—22no—
18#2509C412yes—
19#2448C412yes—
20#2373—22no—
21#2357C411yescharacteristic
22#2351C411yescharacteristic
23C4×#124C2211yescharacteristic
24#2295C2212yes—
25C42×Q8C2211yescharacteristic
26#1940—22no—
27#1940—22no—
28#1939—22no—
29#1831C2212yes—
30#1831C2212yes—
31#1831C2212yes—
32#1767C2212yes—
33#1767C2212yes—
34C4×C4⋊C8C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2211yescharacteristic
38#1271C2212yes—
39#1271C2212yes—
40#1271C2212yes—
41#1271C2212yes—
42#1270C2211yescharacteristic
43#1270C2212yes—
44#1270C2211yescharacteristic
45C2×#126—22no—
46C2×#124C2×C411yescharacteristic
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8D412yes—
51C2×C4×D4C2×C411yescharacteristic
52#446—48no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×#97—22no—
60C2×#97—22no—
61C2×C4×C8—22no—
62#400—22no—
63#400—21no—
64#400—22no—
65#399—22no—
66#399—21no—
67#397—22no—
68#395—21no—
69#393—21no—
70#391C2×C412yes—
71#391C2311yescharacteristic
72#390C2×C412yes—
73#390C2×C412yes—
74#390C2311yescharacteristic
75#389—22no—
76#389—22no—
77#389C2×C412yes—
78#389C2×C412yes—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2312yes—
82#388—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#388—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#385—22no—
96#382—21no—
97#382C2×C411yescharacteristic
98#382C2×C411yescharacteristic
99#382C2×C411yescharacteristic
100#381—21no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4C2×C411yescharacteristic
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C4×C4⋊C4D412yes—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—21no—
111#376—22no—
112C43C2312yes—
113C43C2311yescharacteristic
114#336—22no—
115#336—22no—
116#336—22no—
117#336C2312yes—
118#336C2312yes—
119#336C2312yes—
120#336C2312yes—
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C22×C411yescharacteristic
123C22×D4C22×C411yescharacteristic
124C23×C4—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129#126—44no—
130#126—44no—
131#124—22no—
132#124—41no—
133C4×Q8—24no—
134C4×Q8—24no—
135C4×Q8—24no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×Q8—42no—
139C4×D4—42no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4C22⋊C412yes—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C22×C412yes—
153C2×C4⋊C4C22×C411yescharacteristic
154C2×C4⋊C4C22×C411yescharacteristic
155C2×C4⋊C4C22×C412yes—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C42—22no—
164C2×C42—21no—
165C2×C42—22no—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C412yes—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C2×C42C2411yescharacteristic
177C4⋊C8—44no—
178C4⋊C8—44no—
179C4⋊C8—44no—
180C4⋊C8—44no—
181C4⋊C8—44no—
182C4⋊C8—44no—
183#97—44no—
184#97—44no—
185C4×C8—44no—
186#95—22no—
187#95—22no—
188#95—21no—
189#95—22no—
190#95—21no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—21no—
195#95—21no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95C22⋊C411yescharacteristic
201#95C22⋊C411yescharacteristic
202#95C22⋊C412yes—
203#95C22⋊C412yes—
204C24—21no—
205C2×Q8—21no—
206C2×Q8—24no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8#13112yes—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—41no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—82no—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C23×C411yescharacteristic
230C22×C4C22×D412yes—
231C2×C8—24no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—41no—
250C4⋊C4—44no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C22⋊C4—44no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—44no—
266C22⋊C4—44no—
267C22⋊C4—82no—
268C22⋊C4—82no—
269C42—22no—
270C42—21no—
271C42—22no—
272C42—21no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—42no—
279C42—42no—
280C42—42no—
281C42—42no—
282C42—41no—
283C42—44no—
284C42—42no—
285C42—82no—
286C42AGL(1,ℤ/8)14yes—
287C42#13714yes—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—42no—
292C23—42no—
293C23C22×C22⋊C411yescharacteristic
294Q8—44no—
295Q8—42no—
296Q8—42no—
297D4—81no—
298D4—81no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—162no—
333C2×C4#41312yes—
334C2×C4#41714yes—
335C2×C4#41714yes—
336C2×C4C2×#13111yescharacteristic
337C2×C4#56811yescharacteristic
338C2×C4C2×AGL(1,ℤ/8)12yes—
339C2×C4C2×#13712yes—
340C8—44no—
341C8—82no—
342C8—82no—
343C8—82no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—82no—
348C22—82no—
349C22#288011yescharacteristic
350C22#288411yescharacteristic
351C22C2×#41311yescharacteristic
352C22C2×#41712yes—
353C22#290112yes—
354C4—21no—
355C4—21no—
356C4—22no—
357C4—24no—
358C4—24no—
359C4—22no—
360C4—22no—
361C4—42no—
362C4—42no—
363C4—41no—
364C4—42no—
365C4—42no—
366C4—81no—
367C4—81no—
368C4—82no—
369C4—82no—
370C2—161no—
371C2#1742311yescharacteristic
372C2#1751512yes—
373C2#1762812yes—
374C2#1768611yescharacteristic
375C2#1775211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22736C2semidirect
4. #7697C2semidirect
7. #7615C2semidirect
9. #7577C2semidirect

Maximal subgroups

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

subgroupindex
#364032
#227362
#77322
#76972
#76862
#76352
#76152
#76012
#75772