Finite Group Databasebeta

← #481550

Subgroups — #481550

Order 512.

887 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#35749C211yesmaximal, characteristic
2#20399C211yesmaximal, characteristic
3#17352C211yesmaximal, characteristic
4#11740C214yesmaximal, complemented
5#11736C214yesmaximal, complemented
6#11721C212yesmaximal
7#11700C212yesmaximal
8C2×#557C411yescharacteristic
9C2×#519C2211yescharacteristic
10#2982—21no—
11C22×#101—21no—
12#2873—21no—
13#2634C2212yes—
14#2621C2211yescharacteristic
15#2611C412yes—
16#2599C412yes—
17#2581C411yescharacteristic
18#2546C411yescharacteristic
19#2475—22no—
20#2462—22no—
21#2446—21no—
22#2434—21no—
23#2399C2212yes—
24#2396C2211yescharacteristic
25#2365—21no—
26#2362—21no—
27#2356C411yescharacteristic
28C4×#101—22no—
29C4×#101C2214yes—
30#1742C2212yes—
31#1461—21no—
32#1461C2214yes—
33#1458—22no—
34#1458C2212yes—
35#1354C2214yes—
36#1353C2214yes—
37#1346C2218yes—
38C23×Q8D411yescharacteristic
39C22×M16—22no—
40#557—44no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—22no—
45C2×#125C2×C412yes—
46C2×C22⋊Q8—21no—
47#519—44no—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8D412yes—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8C2312yes—
57C2×#117C2×C412yes—
58C2×#117D412yes—
59C2×#117D411yescharacteristic
60C2×#117C2311yescharacteristic
61C8×Q8—44no—
62#431—24no—
63#431—22no—
64C2×#101—28no—
65C2×#101—24no—
66#405—44no—
67C4×M16—24no—
68#400—22no—
69#400C2×C412yes—
70#400C2311yescharacteristic
71#399C2×C411yescharacteristic
72#398—21no—
73#398—21no—
74#398—22no—
75#398C2×C412yes—
76#398C2×C412yes—
77#398C2312yes—
78#397—21no—
79#397C2×C412yes—
80#396—21no—
81#395—22no—
82#395—22no—
83#395C2×C412yes—
84#395C2311yescharacteristic
85#393C2×C411yescharacteristic
86#391—21no—
87#391—21no—
88#391—21no—
89#391—21no—
90#389—21no—
91#389—22no—
92#389—22no—
93#388—22no—
94#388—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—22no—
103#385—22no—
104#383C2×C412yes—
105#382C2×C411yescharacteristic
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C411yescharacteristic
111C4×C22⋊C4—21no—
112#376C2×C412yes—
113#343—22no—
114#343C2314yes—
115#324—24no—
116#324C2314yes—
117#323—22no—
118C22×Q8—24no—
119C22×Q8—21no—
120C22×Q8—24no—
121C22×Q8—42no—
122C22×Q8C22⋊C411yescharacteristic
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8C22×C411yescharacteristic
125C22×Q8C2×D412yes—
126C23×C4C22⋊C412yes—
127C23×C4C2×D411yescharacteristic
128C2×M16—28no—
129C2×M16—24no—
130C2×M16—42no—
131C2×M16—44no—
132C22×C8—42no—
133C4⋊Q8—42no—
134C4⋊Q8—44no—
135C4⋊Q8—44no—
136#125—44no—
137#125—48no—
138C22⋊Q8—44no—
139C4×Q8—24no—
140C4×Q8—24no—
141C4×Q8—44no—
142C4×Q8—44no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×Q8—44no—
146C4×Q8—42no—
147C4×Q8—44no—
148#117—24no—
149#117—42no—
150#117—44no—
151#117—44no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2411yescharacteristic
175C2×C22⋊C4—22no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4C22⋊C412yes—
179C2×C22⋊C4C22×C411yescharacteristic
180C2×C22⋊C4C2×D411yescharacteristic
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C412yes—
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D411yescharacteristic
192C4⋊C8—44no—
193C4⋊C8—42no—
194#101—216no—
195C22⋊C8—24no—
196#97—44no—
197C4×C8—44no—
198C4×C8—44no—
199C4×C8—42no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—21no—
204#95—22no—
205#95—21no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—22no—
215#95C22×C411yescharacteristic
216#95C22×C412yes—
217#95C22×C411yescharacteristic
218C24C2×C22⋊C411yescharacteristic
219C2×Q8—24no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—22no—
226C2×Q8—42no—
227C2×Q8—48no—
228C2×Q8—48no—
229C2×Q8—42no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8#10414yes—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—42no—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4#13514yes—
254C22×C4C23×C411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257M16—28no—
258M16—48no—
259M16—44no—
260C2×C8—44no—
261C2×C8—48no—
262C2×C8—44no—
263C4⋊C4—22no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4#9914yes—
286C22⋊C4—24no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—44no—
292C42—24no—
293C42—24no—
294C42—22no—
295C42—24no—
296C42—22no—
297C42—42no—
298C42—42no—
299C42—44no—
300C42—44no—
301C42—42no—
302C42—42no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23#41612yes—
308C23C22×C22⋊C411yescharacteristic
309Q8—28no—
310Q8—48no—
311Q8—42no—
312Q8—42no—
313Q8—44no—
314Q8—84no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—48no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4C2×#9914yes—
355C2×C4C2×#9912yes—
356C2×C4#41612yes—
357C2×C4C2×#10414yes—
358C2×C4C2×#10412yes—
359C2×C4C2×#13512yes—
360C8—48no—
361C8—84no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—81no—
369C22C22×#9911yescharacteristic
370C22C2×#41611yescharacteristic
371C22C22×#10411yescharacteristic
372C4—24no—
373C4—22no—
374C4—24no—
375C4—42no—
376C4—41no—
377C4—42no—
378C4—42no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—42no—
383C4—81no—
384C4—82no—
385C4—81no—
386C4—82no—
387C4#150114yes—
388C2—21no—
389C2—21no—
390C2—42no—
391C2C2×#150111yescharacteristic
392C2#3334911yescharacteristic
393C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11740C2semidirect
5. #11736C2semidirect

Maximal subgroups

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

subgroupindex
#357492
#203992
#173522
#117402
#117362
#117212
#117002