Finite Group Databasebeta

← #480575

Subgroups — #480575

Order 512.

848 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35639C211yesmaximal, characteristic
2#17370C211yesmaximal, characteristic
3#17353C211yesmaximal, characteristic
4#11579C214yesmaximal, complemented
5#11577C214yesmaximal, complemented
6#11454C212yesmaximal, complemented
7#11452C212yesmaximal, complemented
8C22×#125C411yescharacteristic
9C2×#520C2211yescharacteristic
10#2982—21no—
11#2975—21no—
12#2579C412yes—
13#2579C2212yes—
14#2487C412yes—
15#2487C2212yes—
16#2454—21no—
17#2454—21no—
18#2453—21no—
19#2453—21no—
20#2442—22no—
21#2441—22no—
22#2317C2211yescharacteristic
23#2316C411yescharacteristic
24#2311C411yescharacteristic
25#2297C411yescharacteristic
26#2296C2211yescharacteristic
27#1465C2214yes—
28#1463—22no—
29#1461—21no—
30#1460—21no—
31#1460C2214yes—
32#1458—22no—
33#1458C2212yes—
34#1458C2212yes—
35#1326C2214yes—
36#1326C2218yes—
37#1326C2214yes—
38C22×#41C2×C411yescharacteristic
39C2×#125—22no—
40C2×#125—21no—
41C2×#125—21no—
42C2×#125—44no—
43#520—44no—
44C2×C4×Q8D411yescharacteristic
45C2×C4×Q8D411yescharacteristic
46C2×C4×D4C2×C412yes—
47C2×C4×D4D411yescharacteristic
48C2×C4×D4D411yescharacteristic
49C2×C4×D4C2312yes—
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117D412yes—
53C2×#117C2311yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4D412yes—
59C22×C42C2×C411yescharacteristic
60C8×Q8—44no—
61#436—44no—
62#431—22no—
63#431—24no—
64#431—22no—
65#407—24no—
66#400—22no—
67#400—22no—
68#399—22no—
69#399—22no—
70#398—22no—
71#398—22no—
72#396—22no—
73#396—22no—
74#391—21no—
75#391—21no—
76#390—21no—
77#390—21no—
78#389—22no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—22no—
84#389—21no—
85#389—21no—
86#389—22no—
87#387—22no—
88#387—22no—
89#386C2×C412yes—
90#386C2×C412yes—
91#386C2×C412yes—
92#386C2×C412yes—
93#385—22no—
94#385—22no—
95#385C2×C412yes—
96#385C2×C411yescharacteristic
97#385C2×C411yescharacteristic
98#385C2312yes—
99#385C2311yescharacteristic
100#385C2311yescharacteristic
101#384—21no—
102#382—21no—
103#380—22no—
104#380—22no—
105#380—22no—
106#380C2×C412yes—
107#380C2×C412yes—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C22⋊C4—22no—
111#324—22no—
112#324—22no—
113#324—24no—
114#323—24no—
115#323C2314yes—
116#323C2314yes—
117C2×#41—24no—
118C2×#41—24no—
119C2×#41—42no—
120C22×Q8C22⋊C411yescharacteristic
121C22×D4C22⋊C411yescharacteristic
122C22×D4C22×C412yes—
123C23×C4C22⋊C412yes—
124C23×C4C22⋊C412yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C412yes—
127C23×C4C22×C412yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D412yes—
131C2×M16—44no—
132#125—48no—
133#125—44no—
134#125—44no—
135C4×Q8—24no—
136C4×Q8—42no—
137C4×D4—24no—
138C4×D4—42no—
139C4×D4—44no—
140C4×D4—44no—
141#117—24no—
142#117—24no—
143#117—42no—
144#117—44no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C22×C411yescharacteristic
179C2×C22⋊C4C22×C411yescharacteristic
180C2×C22⋊C4C22×C412yes—
181C2×C22⋊C4C22×C411yescharacteristic
182C2×C22⋊C4C2×D412yes—
183C2×C22⋊C4C2411yescharacteristic
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—42no—
189C2×C42C22⋊C412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C4⋊C8—42no—
194C4⋊C8—44no—
195C4⋊C8—42no—
196C22⋊C8—24no—
197C22⋊C8—24no—
198C22⋊C8—44no—
199C4×C8—42no—
200C4×C8—44no—
201C4×C8—42no—
202#95—22no—
203#95—22no—
204#95—21no—
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—22no—
214C24C2×C22⋊C412yes—
215C24C23×C411yescharacteristic
216#41—416no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8#10114yes—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—24no—
225C2×D4—42no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—82no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4#13114yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259M16—164no—
260C2×C8—44no—
261C2×C8—44no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—24no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—82no—
277C4⋊C4—81no—
278C4⋊C4—82no—
279C4⋊C4—81no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—81no—
283C4⋊C4—81no—
284C4⋊C4#9914yes—
285C22⋊C4—24no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—24no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—82no—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C42—22no—
297C42—24no—
298C42—22no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—42no—
304C23—21no—
305C23—21no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—41no—
310C23—41no—
311C23—42no—
312C23—42no—
313C23C22×C22⋊C411yescharacteristic
314C23C2×#13112yes—
315Q8—28no—
316D4—48no—
317D4—44no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—28no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—42no—
329C2×C4—48no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—84no—
358C2×C4—162no—
359C2×C4#40812yes—
360C2×C4#40812yes—
361C2×C4C2×#9914yes—
362C2×C4C2×#9912yes—
363C2×C4C2×#10114yes—
364C2×C4C2×#10112yes—
365C8—84no—
366C8—84no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—42no—
371C22—42no—
372C22—44no—
373C22—41no—
374C22—41no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—82no—
379C22C2×#40811yescharacteristic
380C22C22×#9911yescharacteristic
381C22C22×#10111yescharacteristic
382C4—24no—
383C4—22no—
384C4—24no—
385C4—41no—
386C4—42no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—81no—
391C4—81no—
392C4—82no—
393C4—81no—
394C4—81no—
395C4—82no—
396C4#147214yes—
397C2—21no—
398C2—21no—
399C2—42no—
400C2—82no—
401C2C2×#147211yescharacteristic
402C2#3335011yescharacteristic
403C2#3335111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11579C2semidirect
5. #11577C2semidirect
6. #11454C2semidirect
7. #11452C2semidirect

Maximal subgroups

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

subgroupindex
#356392
#173702
#173532
#115792
#115772
#114542
#114522