Finite Group Databasebeta

← #224650

Subgroups — #224650

Order 512.

763 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36602C211yesmaximal, characteristic
2#21346C212yesmaximal, complemented
3#9689C212yesmaximal
4#9688C212yesmaximal, complemented
5#9687C212yesmaximal, complemented
6#8580C211yesmaximal, characteristic, complemented
7#8579C212yesmaximal, complemented
8#8226C212yesmaximal, complemented
9#8225C211yesmaximal, characteristic, complemented
10#2861C2212yes—
11#2858C2211yescharacteristic
12#2597—22no—
13#2593—22no—
14#2592C412yes—
15#2458—22no—
16#2446C412yes—
17#2445C412yes—
18#2374—22no—
19#2367C412yes—
20C4×#125C2211yescharacteristic
21C4×#124C2212yes—
22C42×Q8C2211yescharacteristic
23#1985—22no—
24#1940—22no—
25#1916—22no—
26#1916—22no—
27#1767C2212yes—
28#1767C2212yes—
29#1767C2212yes—
30#1745C2212yes—
31#1745C2212yes—
32C4×#97C2212yes—
33C2×#330C2212yes—
34C2×#330C2211yescharacteristic
35C2×#330C2211yescharacteristic
36C2×#329C2212yes—
37C2×#329C2212yes—
38C2×#329C2212yes—
39C2×#329C2212yes—
40#1295C2211yescharacteristic
41#1295C2211yescharacteristic
42#1295C2212yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45C2×#125C2311yescharacteristic
46C2×#124C2×C412yes—
47C2×#124C2312yes—
48C2×#123—22no—
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8D412yes—
52C2×C4×D4C2×C412yes—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—22no—
56C2×#97—22no—
57C2×#97C2312yes—
58C2×#97C2312yes—
59C2×#97C2312yes—
60C2×#97C2312yes—
61C2×C4×C8—22no—
62#400—22no—
63#398—22no—
64#397—22no—
65#394—22no—
66#394—22no—
67#393—22no—
68#391C2×C412yes—
69#391C2×C412yes—
70#391C2×C412yes—
71#390—22no—
72#390—22no—
73#390C2×C412yes—
74#390C2×C412yes—
75#390C2×C412yes—
76#389C2×C412yes—
77#388—22no—
78#388—22no—
79#388—22no—
80#388—22no—
81#388—22no—
82#388—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#386—22no—
91#385—22no—
92#382C2311yescharacteristic
93C4×C4⋊C4—22no—
94C4×C4⋊C4C2×C412yes—
95C4×C4⋊C4C2×C411yescharacteristic
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4D412yes—
98C4×C4⋊C4D412yes—
99C4×C4⋊C4D412yes—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C43C2311yescharacteristic
104C43C2312yes—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#330—24no—
113#330—41no—
114#330—41no—
115#329—24no—
116#329—24no—
117#329—42no—
118#329—42no—
119C22×Q8C22⋊C411yescharacteristic
120C22×Q8C22×C412yes—
121C22×D4—22no—
122C22×D4C22×C412yes—
123C23×C4—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C4⋊Q8—81no—
129#127—81no—
130#125—22no—
131#125—21no—
132#125—21no—
133#124—22no—
134#124—22no—
135#124—22no—
136#124—22no—
137#124—44no—
138#123—48no—
139C4×Q8—42no—
140C4×Q8—44no—
141C4×Q8—82no—
142C4×D4—82no—
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⋊C411yescharacteristic
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22×C412yes—
153C2×C4⋊C4C22×C412yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
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×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C412yes—
169C2×C42C22×C412yes—
170C2×C42C22×C411yescharacteristic
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2411yescharacteristic
178C4⋊C8—44no—
179C4⋊C8—44no—
180C4⋊C8—44no—
181#97—22no—
182#97—22no—
183#97—22no—
184#97—22no—
185#97—22no—
186#97—22no—
187#97—22no—
188#97—44no—
189#97#4114yes—
190C4×C8—44no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95C22⋊C411yescharacteristic
201#95C22⋊C412yes—
202#95C22⋊C411yescharacteristic
203#95C22⋊C412yes—
204C24—22no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—41no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—44no—
216C2×D4—82no—
217C2×D4—82no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—44no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C412yes—
231C22×C4C23×C411yescharacteristic
232C22×C4C22×D412yes—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—44no—
266C22⋊C4—44no—
267C22⋊C4—82no—
268C22⋊C4—82no—
269C22⋊C4—82no—
270C22⋊C4—82no—
271C42—22no—
272C42—24no—
273C42—42no—
274C42—42no—
275C42—42no—
276C42—42no—
277C42—42no—
278C42—41no—
279C42—42no—
280C42—42no—
281C42—82no—
282C42—82no—
283C42C2×#4112yes—
284C42H(5,𝔽2)11yescharacteristic
285C422−1+411yescharacteristic
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23C22×C22⋊C411yescharacteristic
294Q8—44no—
295Q8—82no—
296Q8—82no—
297D4—82no—
298D4—82no—
299D4—164no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
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—22no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—164no—
331C2×C4#41312yes—
332C2×C4#42114yes—
333C2×C4C4×#4111yescharacteristic
334C2×C4#52011yescharacteristic
335C2×C4#54812yes—
336C2×C4#55012yes—
337C8—44no—
338C8—44no—
339C8—82no—
340C8—82no—
341C22—82no—
342C22—82no—
343C22—82no—
344C22—82no—
345C22—82no—
346C22—82no—
347C22—82no—
348C22#234111yescharacteristic
349C22C2×#41311yescharacteristic
350C22#288811yescharacteristic
351C22C2×#42112yes—
352C22#290112yes—
353C4—21no—
354C4—21no—
355C4—24no—
356C4—22no—
357C4—22no—
358C4—22no—
359C4—41no—
360C4—42no—
361C4—42no—
362C4—42no—
363C4—42no—
364C4—82no—
365C4—82no—
366C4—82no—
367C2—162no—
368C2#1767311yescharacteristic
369C2#1767912yes—
370C2#1769011yescharacteristic
371C2#2253911yescharacteristic
372C2#2255112yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21346C2semidirect
4. #9688C2semidirect
5. #9687C2semidirect
6. #8580C2semidirect
7. #8579C2semidirect
8. #8226C2semidirect
9. #8225C2semidirect

Maximal subgroups

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

subgroupindex
#366022
#213462
#96892
#96882
#96872
#85802
#85792
#82262
#82252