Finite Group Databasebeta

← #197518

Subgroups — #197518

Order 512.

1151 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#17575C212yesmaximal
4C2×#1461C211yesmaximal, characteristic
5#9202C218yesmaximal
6#7981C212yesmaximal
7C2×#533C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#431C2212yes—
10#2855—22no—
11#2627C2214yes—
12#2582—22no—
13#2581C414yescomplemented
14#2568C412yescomplemented
15#2567C412yescomplemented
16#2565—22no—
17#2356—24no—
18#2355—24no—
19#2317—21no—
20#2313C2214yes—
21#2310—21no—
22#2310—21no—
23#2285C2211yescharacteristic
24#2285C2212yes—
25#2283—22no—
26#2274—21no—
27#1520C2218yes—
28#1518—24no—
29#1503—24no—
30#1461—22no—
31#1461—22no—
32C2×#324C2214yes—
33#1289C2218yes—
34C23×Q8D411yescharacteristic
35#533—22no—
36#533—22no—
37#533—22no—
38#533—22no—
39C2×C4⋊Q8C2×C411yescharacteristic
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#125—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8C2312yes—
44C2×C4×Q8—22no—
45C2×C4×Q8—22no—
46C2×C4×Q8—21no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8D412yes—
53C2×C4×Q8D412yes—
54C2×#117—21no—
55C2×#117—21no—
56C2×#117—22no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C42D412yes—
60C22×C42C2311yescharacteristic
61#431—22no—
62#431—22no—
63#431—22no—
64#431—22no—
65C2×C4⋊C8—22no—
66C2×C22⋊C8C2314yes—
67C2×C4×C8—22no—
68#400C2×C418yes—
69#398—28no—
70#395C2×C418yes—
71#391—28no—
72#391—22no—
73#391—22no—
74#391—24no—
75#391—24no—
76#389—24no—
77#389—24no—
78#387—28no—
79#387—216no—
80#385—24no—
81#384—22no—
82#384—21no—
83#384—22no—
84#383—24no—
85#382—24no—
86#382—22no—
87#382C2×C412yes—
88#382D414yes—
89#381—24no—
90#381—24no—
91#380—21no—
92#380—22no—
93#380C2312yes—
94#380C2314yes—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—28no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4D414yes—
101C4×C22⋊C4—24no—
102#376—24no—
103#328—216no—
104#324—28no—
105C22×Q8—21no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—24no—
109C22×Q8—24no—
110C22×Q8C22⋊C412yes—
111C22×Q8C22⋊C412yes—
112C22×Q8C22×C412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2411yescharacteristic
118C22×C8—24no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123#125—42no—
124#125—42no—
125C22⋊Q8—24no—
126C22⋊Q8—24no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C4×Q8—24no—
132C4×Q8—24no—
133C4×Q8—24no—
134C4×Q8—24no—
135C4×Q8—24no—
136C4×Q8—24no—
137C4×Q8—24no—
138C4×Q8—24no—
139C4×Q8—24no—
140C4×Q8—24no—
141C4×Q8—42no—
142C4×Q8—42no—
143#117—42no—
144#117—42no—
145#117—44no—
146#117—44no—
147#117—44no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C414yes—
168C2×C4⋊C4C2×D414yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—24no—
179C2×C42—28no—
180C2×C42C22⋊C414yes—
181C2×C42C22⋊C414yes—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42#4112yes—
189C2×C42#4112yes—
190C4⋊C8—42no—
191C4⋊C8—42no—
192C22⋊C8—24no—
193C22⋊C8—24no—
194C4×C8—42no—
195C4×C8—42no—
196#95—28no—
197#95—28no—
198#95—28no—
199#95—22no—
200#95—28no—
201#95—28no—
202#95—28no—
203#95—24no—
204#95C22×C414yes—
205#95C2×D418yes—
206C24C22×D411yescharacteristic
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—24no—
212C2×Q8—24no—
213C2×Q8—24no—
214C2×Q8—44no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—44no—
218C2×Q8—44no—
219C2×Q8—48no—
220C2×Q8—82no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—28no—
231C22×C4—42no—
232C22×C4—81no—
233C22×C4—82no—
234C22×C4C2×C22⋊C414yes—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C414yes—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4#12011yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12012yes—
243C22×C4#12311yescharacteristic
244C22×C4#12311yescharacteristic
245C22×C4#12312yes—
246C22×C4#13514yes—
247C22×C4#13714yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D412yes—
250C22×C4C22×D411yescharacteristic
251C22×C4C2×#4112yes—
252C2×C8—44no—
253C2×C8—84no—
254C4⋊C4—28no—
255C4⋊C4—28no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—48no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—44no—
269C4⋊C4—48no—
270C4⋊C4—84no—
271C4⋊C4—82no—
272C4⋊C4—88no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C42—24no—
279C42—24no—
280C42—24no—
281C42—24no—
282C42—44no—
283C42—44no—
284C42—48no—
285C42—88no—
286C42C4×D412yes—
287C42C4×D412yes—
288C42C4×D412yes—
289C42C4×D412yes—
290C42#12112yes—
291C42#12112yes—
292C42#12112yes—
293C42#12112yes—
294C23—41no—
295C23—41no—
296C23C22×C22⋊C411yescharacteristic
297C23C2×#12011yescharacteristic
298C23C2×#12311yescharacteristic
299C23C2×#13512yes—
300C23C2×#13712yes—
301Q8—48no—
302Q8—48no—
303Q8—84no—
304Q8—82no—
305Q8—82no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—44no—
326C2×C4—48no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—48no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4#38611yescharacteristic
336C2×C4#38611yescharacteristic
337C2×C4#38611yescharacteristic
338C2×C4#38612yes—
339C2×C4#38612yes—
340C2×C4#38611yescharacteristic
341C2×C4#41612yes—
342C2×C4#41612yes—
343C2×C4#41912yes—
344C2×C4#41912yes—
345C2×C4#45214yes—
346C2×C4#45214yes—
347C2×C4#48518yes—
348C2×C4C2×C4×D412yes—
349C2×C4C2×#12112yes—
350C8—84no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—42no—
358C22—42no—
359C22C2×#38611yescharacteristic
360C22C2×#41611yescharacteristic
361C22C2×#41911yescharacteristic
362C22C2×#45212yes—
363C22C2×#48512yes—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—22no—
368C4—22no—
369C4—44no—
370C4—44no—
371C4—84no—
372C4—88no—
373C4—82no—
374C4—82no—
375C4#187314yes—
376C4#187314yes—
377C4#191212yes—
378C2—21no—
379C2—22no—
380C2—22no—
381C2—21no—
382C2C2×#187311yescharacteristic
383C2C2×#191211yescharacteristic
384C2#2101411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2581C4semidirect
14. #2568C4semidirect
15. #2567C4semidirect

Maximal subgroups

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

subgroupindex
#351792
#341872
#175752
C2×#14612
#92022
#79812