Finite Group Databasebeta

← #220111

Subgroups — #220111

Order 512.

945 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36151C211yesmaximal, characteristic
2#22751C212yesmaximal
3C2×#1559C212yesmaximal, complemented
4#9700C212yesmaximal, complemented
5#9140C212yesmaximal, complemented
6#9137C212yesmaximal, complemented
7#8594C211yesmaximal, characteristic, complemented
8#8343C212yesmaximal, complemented
9C4×#333C211yesmaximal, characteristic, complemented
10C2×#533C2212yes—
11C2×#432C2212yes—
12C2×#423C2212yes—
13#2862C2211yescharacteristic
14#2857C2212yes—
15#2536C412yes—
16#2489—22no—
17#2453—22no—
18#2421C414yes—
19#2415—22no—
20#2362—22no—
21#2351C412yes—
22C4×C4⋊Q8C2211yescharacteristic
23C42×D4C2211yescharacteristic
24#1923—22no—
25#1917—22no—
26#1870—22no—
27#1831C2212yes—
28#1797C2212yes—
29#1797C2212yes—
30C4×C4⋊C8C2212yes—
31#1753—22no—
32#1559—28no—
33C2×#333C2211yescharacteristic
34C2×#333C2212yes—
35C2×#333C2211yescharacteristic
36C2×#332C2212yes—
37C2×#332C2212yes—
38C2×#332C2212yes—
39C2×#332C2212yes—
40#1296C2212yes—
41#1296C2211yescharacteristic
42#1296C2211yescharacteristic
43C22×M16—22no—
44#533—24no—
45#533—24no—
46#533—42no—
47#533—42no—
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2312yes—
50C2×#125—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8C2×C414yes—
54C2×C4×Q8—22no—
55C2×C4×Q8C2×C414yes—
56C2×C4×Q8C2×C412yes—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4D412yes—
59C2×#117—22no—
60C22×C42C2312yes—
61#432—24no—
62#432—42no—
63#423—24no—
64#423—42no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×#97—22no—
74C2×C4×C8—22no—
75#400—24no—
76#398—22no—
77#397—24no—
78#394—22no—
79#391—22no—
80#391—22no—
81#391—22no—
82#391C2×C412yes—
83#389—24no—
84#389—22no—
85#389—22no—
86#389C2×C414yes—
87#388—24no—
88#388—24no—
89#387—24no—
90#384C2311yescharacteristic
91#382—22no—
92#382C2×C412yes—
93#382C2×C412yes—
94#380—22no—
95#380C2×C414yes—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4C2×C411yescharacteristic
99C4×C4⋊C4D412yes—
100C4×C22⋊C4C2×C412yes—
101C4×C22⋊C4D412yes—
102C4×C22⋊C4D412yes—
103C43C2311yescharacteristic
104#336—22no—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#333—24no—
110#333—22no—
111#333D414yescomplemented
112#332—24no—
113#332—24no—
114#332—42no—
115#332—42no—
116C22×Q8—22no—
117C22×Q8C22×C412yes—
118C22×D4C22⋊C411yescharacteristic
119C23×C4C22×C412yes—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C2×M16—44no—
123C2×M16—44no—
124C2×M16—82no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8C2×D412yes—
135C4⋊Q8#4112yes—
136#125—44no—
137#125—44no—
138C22⋊Q8—28no—
139C22⋊Q8—44no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142C22⋊Q8—44no—
143C22⋊Q8—44no—
144C4×Q8—28no—
145C4×Q8—44no—
146C4×Q8—44no—
147C4×Q8—44no—
148C4×Q8—44no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—42no—
152#117—44no—
153#117—44no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22×C414yes—
164C2×C4⋊C4C22×C412yes—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4C22⋊C412yes—
168C2×C22⋊C4C22⋊C412yes—
169C2×C22⋊C4C22⋊C412yes—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—24no—
175C2×C42—22no—
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C412yes—
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2411yescharacteristic
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—44no—
192C4⋊C8C2×D414yes—
193C22⋊C8—44no—
194C22⋊C8—44no—
195C22⋊C8—82no—
196#97—44no—
197C4×C8—82no—
198#95—22no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—22no—
204#95C22⋊C412yes—
205#95C22⋊C412yes—
206#95C22⋊C411yescharacteristic
207#95C22⋊C411yescharacteristic
208C24C2×C22⋊C412yes—
209C2×Q8—22no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8C4×D414yes—
219C2×D4—41no—
220C2×D4—42no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C23×C411yescharacteristic
247C22×C4C22×D412yes—
248M16—88no—
249C2×C8—24no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—48no—
268C4⋊C4—48no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—48no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C22⋊C4—24no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—48no—
280C22⋊C4—48no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—41no—
293C42—48no—
294C42—44no—
295C42C22×D411yescharacteristic
296C42C2×#4111yescharacteristic
297C42H(5,𝔽2)12yes—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23C22×C22⋊C411yescharacteristic
306Q8—44no—
307Q8—44no—
308Q8—82no—
309Q8—82no—
310Q8—82no—
311Q8—82no—
312D4—44no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4#41312yes—
349C2×C4C2×C4×D411yescharacteristic
350C2×C4#51811yescharacteristic
351C2×C4D4212yes—
352C2×C4#54612yes—
353C2×C4#57512yes—
354C2×C4#57512yes—
355C2×C4#57712yes—
356C2×C4#57812yes—
357C8—44no—
358C8—82no—
359C8—82no—
360C8—82no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22D4×C22⋊C411yescharacteristic
369C22C2×#41311yescharacteristic
370C22#288911yescharacteristic
371C22#289612yes—
372C22#289812yes—
373C4—22no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—42no—
378C4—42no—
379C4—44no—
380C4—42no—
381C4—42no—
382C4—82no—
383C4—84no—
384C4#327512yes—
385C4#327712yes—
386C4#332612yes—
387C4#332712yes—
388C2—82no—
389C2#1774011yescharacteristic
390C2#1776712yes—
391C2#1778011yescharacteristic
392C2#2241111yescharacteristic
393C2#2242912yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1559C2semidirect
4. #9700C2semidirect
5. #9140C2semidirect
6. #9137C2semidirect
7. #8594C2semidirect
8. #8343C2semidirect
9. C4×#333C2semidirect
111. #333D4semidirect

Maximal subgroups

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

subgroupindex
#361512
#227512
C2×#15592
#97002
#91402
#91372
#85942
#83432
C4×#3332