Finite Group Databasebeta

← #481877

Subgroups — #481877

Order 512.

832 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35711C211yesmaximal, characteristic
2#20691C211yesmaximal, characteristic
3#17371C211yesmaximal, characteristic
4#11813C214yesmaximal
5#11810C214yesmaximal
6#11718C212yesmaximal
7#11715C212yesmaximal
8C2×Q82C411yescharacteristic, complemented
9C2×#519C2211yescharacteristic
10#2983—21no—
11#2628C2212yes—
12#2628C2211yescharacteristic
13#2581C412yescomplemented
14#2581C411yescharacteristic, complemented
15#2568C411yescharacteristic, complemented
16#2568C412yescomplemented
17#2451—21no—
18#2451—22no—
19#2439—22no—
20#2439—21no—
21#2396C2212yes—
22#2396C2211yescharacteristic
23#2351—21no—
24#2351C411yescharacteristic, complemented
25Q8×C22⋊C4—21no—
26#2310—21no—
27#1820—22no—
28#1820C2214yes—
29#1740—21no—
30#1740C2212yes—
31#1467—22no—
32#1464C2214yes—
33#1461—21no—
34#1458C2212yes—
35#1352C2218yes—
36#1352C2214yes—
37#1351C2214yes—
38C23×Q8D411yescharacteristic
39Q82—44no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44#519—44no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D412yes—
54C2×C4×Q8D411yescharacteristic
55C2×#117—22no—
56C2×#117D412yes—
57C2×#117D411yescharacteristic
58C2×#117C2312yes—
59C2×#117C2311yescharacteristic
60#446—44no—
61#432—24no—
62#431—22no—
63#400C2×C412yes—
64#400C2×C411yescharacteristic
65#400C2×C412yes—
66#400C2×C412yes—
67#400C2×C412yes—
68#399C2311yescharacteristic
69#399C2312yes—
70#398—22no—
71#398—21no—
72#396—22no—
73#396—21no—
74#395—22no—
75#395—21no—
76#395C2×C412yes—
77#395C2×C411yescharacteristic
78#393—22no—
79#393—21no—
80#393C2311yescharacteristic
81#391—21no—
82#391—22no—
83#391—21no—
84#391—22no—
85#391—22no—
86#391—22no—
87#391—22no—
88#391—44no—
89#389—22no—
90#389—21no—
91#389—22no—
92#389—21no—
93#389—21no—
94#389—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#382—22no—
99#382—21no—
100#382C2×C412yes—
101#382C2×C411yescharacteristic
102#381—24no—
103#380—22no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C22⋊C4—22no—
110C4×C22⋊C4—21no—
111#342—22no—
112#342—24no—
113#342C2314yes—
114#324—22no—
115#324—24no—
116#323C2314yes—
117C22×Q8—24no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—24no—
121C22×Q8—42no—
122C22×Q8C22⋊C411yescharacteristic
123C22×Q8C22⋊C412yes—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22×C411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C4⋊Q8—44no—
129C4⋊Q8—42no—
130C4⋊Q8—48no—
131C4⋊Q8—44no—
132C4×Q8—24no—
133C4×Q8—24no—
134C4×Q8—44no—
135C4×Q8—42no—
136C4×Q8—42no—
137C4×Q8—44no—
138C4×Q8—44no—
139C4×Q8—42no—
140C4×Q8—44no—
141C4×Q8—44no—
142#117—42no—
143#117—44no—
144#117—42no—
145#117—44no—
146#117—44no—
147#117—44no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C22⋊C4—21no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4C2×D411yescharacteristic
176C2×C22⋊C4C2×D412yes—
177C2×C22⋊C4C2411yescharacteristic
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42—42no—
183C2×C42C22⋊C412yes—
184C2×C42C22⋊C412yes—
185C2×C42C22⋊C412yes—
186C2×C42C22×C412yes—
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D412yes—
190C4⋊C8—42no—
191C4⋊C8—44no—
192C22⋊C8—24no—
193#97—44no—
194C4×C8—42no—
195#95—22no—
196#95—21no—
197#95—22no—
198#95—21no—
199#95—21no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—21no—
209#95—21no—
210#95—48no—
211#95C22×C411yescharacteristic
212#95C22×C412yes—
213#95C22×C411yescharacteristic
214C24C22×D411yescharacteristic
215C2×Q8—24no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—24no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—42no—
223C2×Q8—48no—
224C2×Q8—48no—
225C2×Q8—42no—
226C2×Q8—44no—
227C2×Q8—42no—
228C2×Q8—41no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8#9914yes—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4#13714yes—
254C22×C4C23×C411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C2×C8—44no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—84no—
276C4⋊C4—81no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—81no—
280C22⋊C4—41no—
281C22⋊C4—42no—
282C22⋊C4—41no—
283C22⋊C4—42no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—82no—
287C22⋊C4—81no—
288C42—22no—
289C42—24no—
290C42—24no—
291C42—22no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42—44no—
296C42—44no—
297C42—42no—
298C42—42no—
299C42—84no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23C22×C22⋊C411yescharacteristic
305C23C2×#13712yes—
306Q8—28no—
307Q8—48no—
308Q8—42no—
309Q8—42no—
310Q8—44no—
311Q8—164no—
312Q8—162no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—48no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4C2×#9914yes—
357C2×C4C2×#9912yes—
358C2×C4#41912yes—
359C2×C4#41912yes—
360C2×C4#42112yes—
361C8—84no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—81no—
369C22C22×#9911yescharacteristic
370C22C2×#41911yescharacteristic
371C22C2×#42111yescharacteristic
372C4—22no—
373C4—24no—
374C4—41no—
375C4—42no—
376C4—42no—
377C4—41no—
378C4—41no—
379C4—42no—
380C4—41no—
381C4—42no—
382C4—81no—
383C4—82no—
384C4—81no—
385C4—82no—
386C4—81no—
387C4—164no—
388C4#151114yes—
389C2—21no—
390C2—21no—
391C2—42no—
392C2C2×#151111yescharacteristic
393C2#3334911yescharacteristic
394C2#3336211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×Q82C4semidirect
13. #2581C4semidirect
14. #2581C4semidirect
15. #2568C4semidirect
16. #2568C4semidirect
24. #2351C4semidirect

Maximal subgroups

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

subgroupindex
#357112
#206912
#173712
#118132
#118102
#117182
#117152