Finite Group Databasebeta

← #480566

Subgroups — #480566

Order 512.

817 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35668C211yesmaximal, characteristic
2#17371C211yesmaximal, characteristic
3#17352C211yesmaximal, characteristic
4#11574C214yesmaximal, complemented
5#11571C214yesmaximal, complemented
6#11450C212yesmaximal
7#11447C212yesmaximal
8C2×#544C411yescharacteristic
9C2×#519C2211yescharacteristic
10#2983—21no—
11#2982—21no—
12#2568C412yes—
13#2567C2212yes—
14#2565C412yes—
15#2565C2212yes—
16#2453—21no—
17#2453—21no—
18#2451—21no—
19#2451—21no—
20#2438—22no—
21#2438—22no—
22#2317C2211yescharacteristic
23#2310C411yescharacteristic
24#2296C411yescharacteristic
25C2×#391C411yescharacteristic
26#2285C2211yescharacteristic
27#1467C2214yes—
28#1464—22no—
29#1461—21no—
30#1461—21no—
31#1461C2214yes—
32#1458—22no—
33#1458C2212yes—
34#1458C2212yes—
35#1325C2218yes—
36#1325C2214yes—
37#1324C2214yes—
38C23×Q8C2×C411yescharacteristic
39#544—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8C2×C412yes—
42C2×#125—21no—
43C2×#125C2×C412yes—
44C2×C22⋊Q8—22no—
45#519—44no—
46C2×C4×Q8C2×C412yes—
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8D411yescharacteristic
51C2×C4×Q8C2312yes—
52C2×#117C2×C411yescharacteristic
53C2×#117C2×C411yescharacteristic
54C2×#117C2×C411yescharacteristic
55C2×#117D412yes—
56C2×#117D412yes—
57C2×#117C2311yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60#446—44no—
61C8×Q8—44no—
62#432—24no—
63#431—24no—
64#431—22no—
65#431—22no—
66#400—22no—
67#400—22no—
68#398—22no—
69#398—22no—
70#398—22no—
71#398—22no—
72#395—22no—
73#395—22no—
74#391—21no—
75#391—21no—
76#391—21no—
77#391—21no—
78#391—22no—
79#391—22no—
80#391C2×C412yes—
81#391C2×C412yes—
82#391C2×C412yes—
83#391C2×C412yes—
84#389—21no—
85#389—21no—
86#389—22no—
87#389—22no—
88#389—21no—
89#389—21no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#385—22no—
95#385—21no—
96#385—21no—
97#384C2×C411yescharacteristic
98#384C2312yes—
99#384C2311yescharacteristic
100#382C2×C412yes—
101#382C2×C411yescharacteristic
102#382C2311yescharacteristic
103#381—22no—
104#380—21no—
105#380—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C412yes—
110C2×#95—22no—
111#324—24no—
112#324—24no—
113#324C2314yes—
114#324C2314yes—
115#323—22no—
116#323—22no—
117C22×Q8—24no—
118C22×Q8—22no—
119C22×Q8—24no—
120C22×Q8—42no—
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C22⋊C411yescharacteristic
123C22×Q8C22×C412yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4C22⋊C412yes—
127C23×C4C22×C411yescharacteristic
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130#125—44no—
131#125—44no—
132C22⋊Q8—48no—
133C4×Q8—24no—
134C4×Q8—24no—
135C4×Q8—44no—
136C4×Q8—44no—
137C4×Q8—42no—
138C4×Q8—42no—
139#117—24no—
140#117—24no—
141#117—42no—
142#117—44no—
143#117—44no—
144#117—42no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2411yescharacteristic
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4C22⋊C412yes—
173C2×C22⋊C4C2×D411yescharacteristic
174C2×C22⋊C4C2×D411yescharacteristic
175C2×C42—24no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42C22⋊C412yes—
179C2×C42C22⋊C412yes—
180C2×C42C22×C412yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C4⋊C8—44no—
189C4⋊C8—42no—
190C4⋊C8—42no—
191C4⋊C8—44no—
192C22⋊C8—24no—
193C22⋊C8—24no—
194#97—44no—
195C4×C8—44no—
196C4×C8—42no—
197C4×C8—42no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—21no—
203#95—21no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—21no—
213#95—21no—
214C24C2×C22⋊C411yescharacteristic
215C2×Q8—24no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—24no—
222C2×Q8—42no—
223C2×Q8—416no—
224C2×Q8—44no—
225C2×Q8—82no—
226C2×Q8—82no—
227C2×Q8#10114yes—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—81no—
243C22×C4—82no—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4#13114yes—
252C22×C4C23×C411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C2×C8—44no—
256C2×C8—44no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4#9914yes—
281C22⋊C4—24no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C42—24no—
287C42—22no—
288C42—22no—
289C42—24no—
290C42—24no—
291C42—22no—
292C42—22no—
293C42—24no—
294C42—42no—
295C42—42no—
296C42—42no—
297C42—42no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23#40812yes—
303C23C22×C22⋊C411yescharacteristic
304Q8—28no—
305Q8—48no—
306Q8—44no—
307Q8—164no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—28no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—42no—
323C2×C4—48no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
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—84no—
347C2×C4—162no—
348C2×C4#40812yes—
349C2×C4C2×#9914yes—
350C2×C4C2×#9912yes—
351C2×C4C2×#10114yes—
352C2×C4C2×#10112yes—
353C2×C4C2×#13112yes—
354C8—84no—
355C8—84no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—42no—
360C22—42no—
361C22—41no—
362C22—41no—
363C22C2×#40811yescharacteristic
364C22C22×#9911yescharacteristic
365C22C22×#10111yescharacteristic
366C4—24no—
367C4—22no—
368C4—24no—
369C4—42no—
370C4—41no—
371C4—41no—
372C4—42no—
373C4—41no—
374C4—41no—
375C4—42no—
376C4—82no—
377C4—81no—
378C4—82no—
379C4—81no—
380C4—81no—
381C4#147214yes—
382C2—21no—
383C2—21no—
384C2—42no—
385C2C2×#147211yescharacteristic
386C2#3336211yescharacteristic
387C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #11574C2semidirect
5. #11571C2semidirect

Maximal subgroups

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

subgroupindex
#356682
#173712
#173522
#115742
#115712
#114502
#114472