Finite Group Databasebeta

← #375504

Subgroups — #375504

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35620C211yesmaximal, characteristic
2#20393C211yesmaximal, characteristic
3#20358C211yesmaximal, characteristic
4#11388C212yesmaximal, complemented
5#11348C212yesmaximal, complemented
6#10892C212yesmaximal, complemented
7#10888C212yesmaximal, complemented
8#10751C212yesmaximal
9#10742C212yesmaximal
10C2×#568C2212yes—
11C2×#544C2211yescharacteristic
12C2×#520C2211yescharacteristic
13C2×#410C2212yes—
14#2568C412yescomplemented
15#2566C412yescomplemented
16#2565C412yescomplemented
17#2563C412yescomplemented
18#2450—24no—
19#2442—22no—
20#2439—22no—
21#2317C2211yescharacteristic
22#2311C2211yescharacteristic
23#2297C2211yescharacteristic
24#2296C2211yescharacteristic
25C2×#382C2211yescharacteristic
26#2274C2212yes—
27C4×#99—24no—
28#1742—22no—
29#1742C2212yes—
30C2×#343—22no—
31C2×#343C2212yes—
32#1740C2212yes—
33#1736C2214yes—
34#1733C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C22×#41C2311yescharacteristic
38#568—28no—
39#568—24no—
40#568—24no—
41C2×#131—44no—
42C22×M16—24no—
43C22×M16C2314yes—
44#544—44no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2×C412yes—
47C2×#125—21no—
48C2×#125C2×C412yes—
49C2×C22⋊Q8—22no—
50#520—44no—
51C2×C4×Q8C2×C412yes—
52C2×C4×D4C2×C412yes—
53C2×C4×D4D414yes—
54C2×#117D412yes—
55C2×#117C2314yes—
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C2×#117C2311yescharacteristic
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C42C2311yescharacteristic
63#410—24no—
64#410Q818yescomplemented
65C2×#99—24no—
66#400—24no—
67#398—24no—
68#396—24no—
69#393—24no—
70#391—24no—
71#391C2×C411yescharacteristic
72#391C2×C412yes—
73#391C2×C411yescharacteristic
74#390C2×C411yescharacteristic
75#390C2×C412yes—
76#390C2×C411yescharacteristic
77#389—22no—
78#389—21no—
79#389—21no—
80#387—24no—
81#386—22no—
82#386—22no—
83#385—24no—
84#385—22no—
85#385—22no—
86#384C2×C412yes—
87#382—22no—
88#382C2×C412yes—
89#382C2×C412yes—
90#382C2×C412yes—
91#380—24no—
92#380—22no—
93C4×C4⋊C4—44no—
94C4×C4⋊C4C2×C412yes—
95C4×C4⋊C4C2×C412yes—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—24no—
98C2×#95—22no—
99#343—22no—
100#343—24no—
101#343—22no—
102#342—24no—
103#340—44no—
104#339—28no—
105#339—44no—
106H(3,ℤ/4)—28no—
107C2×#41—24no—
108C2×#41—24no—
109C2×#41C2×Q814yes—
110C2×#41#4114yes—
111C22×Q8—22no—
112C22×Q8C22×C411yescharacteristic
113C22×D4C22×C411yescharacteristic
114C22×D4C2×D412yes—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4C22⋊C414yes—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2411yescharacteristic
121#131—416no—
122#131—416no—
123C2×M16—24no—
124C2×M16—216no—
125C2×M16—24no—
126C2×M16—24no—
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—44no—
130C2×M16—48no—
131C2×M16#4118yes—
132C22×C8—24no—
133C22×C8—44no—
134C4⋊Q8—44no—
135C4⋊Q8—44no—
136#125—44no—
137#125—44no—
138C22⋊Q8—48no—
139C4×Q8—44no—
140C4×D4—44no—
141C4×D4—48no—
142#117—24no—
143#117—24no—
144#117—24no—
145#117—24no—
146#117—24no—
147#117—24no—
148#117—44no—
149#117C2×Q818yes—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4C22⋊C414yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C22⋊C4—24no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4C22⋊C414yes—
177C2×C22⋊C4C2×D412yes—
178C2×C42—24no—
179C2×C42—24no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—44no—
183C2×C42C22⋊C414yes—
184C2×C42C22×C412yes—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C412yes—
188#99—216no—
189#95—24no—
190#95—24no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—24no—
196#95—24no—
197#95—44no—
198C24C2×C22⋊C412yes—
199C24C22×D411yescharacteristic
200#41—28no—
201#41—28no—
202#41—28no—
203#41—28no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—44no—
207C2×Q8—84no—
208C2×Q8C4×Q814yes—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4C4×Q814yes—
214C2×D4C22⋊Q818yes—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—84no—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C414yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4#12414yes—
238C22×C4C4⋊Q814yes—
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×Q812yes—
242C22×C4C2×#4112yes—
243C22×C4C2×#4114yes—
244C22×C4H(5,𝔽2)14yes—
245M16—28no—
246M16—28no—
247M16—416no—
248M16—44no—
249M16—44no—
250C2×C8—28no—
251C2×C8—416no—
252C2×C8—44no—
253C2×C8—44no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—28no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—44no—
269C4⋊C4—84no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C22⋊C4—24no—
273C22⋊C4—28no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—84no—
279C22⋊C4—82no—
280C22⋊C4—82no—
281C42—24no—
282C42—22no—
283C42—22no—
284C42—24no—
285C42—28no—
286C42—44no—
287C42—42no—
288C42—42no—
289C42—44no—
290C42—164no—
291C23—22no—
292C23—22no—
293C23—41no—
294C23—41no—
295C23—42no—
296C23—42no—
297C23#39114yes—
298C23C22×C22⋊C411yescharacteristic
299C23C2×C22⋊Q812yes—
300C23#53212yes—
301C23#53412yes—
302Q8—28no—
303Q8—164no—
304D4—216no—
305D4—28no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—28no—
309C2×C4—28no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—48no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4—84no—
336C2×C4—164no—
337C2×C4—162no—
338C2×C4#39114yes—
339C2×C4C2×C4×Q812yes—
340C2×C4C4×#4112yes—
341C2×C4#51812yes—
342C2×C4#52512yes—
343C2×C4C2×#12411yescharacteristic
344C2×C4C2×C4⋊Q811yescharacteristic
345C2×C4#53412yes—
346C2×C4#55114yes—
347C2×C4#55114yes—
348C2×C4#55414yes—
349C2×C4#55414yes—
350C8—48no—
351C22—21no—
352C22—22no—
353C22—21no—
354C22—22no—
355C22—24no—
356C22—42no—
357C22—42no—
358C22—44no—
359C22—42no—
360C22—42no—
361C22—82no—
362C22C2×#39111yescharacteristic
363C22#229711yescharacteristic
364C22#230811yescharacteristic
365C22#256414yes—
366C4—24no—
367C4—44no—
368C4—42no—
369C4—42no—
370C4—82no—
371C4—81no—
372C4—81no—
373C4—88no—
374C4—81no—
375C4—82no—
376C4—81no—
377C4#236814yes—
378C4#256112yes—
379C4#258312yes—
380C2—21no—
381C2—24no—
382C2—21no—
383C2—44no—
384C2#1415911yescharacteristic
385C2#2178411yescharacteristic
386C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11388C2semidirect
5. #11348C2semidirect
6. #10892C2semidirect
7. #10888C2semidirect
14. #2568C4semidirect
15. #2566C4semidirect
16. #2565C4semidirect
17. #2563C4semidirect
64. #410Q8semidirect

Maximal subgroups

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

subgroupindex
#356202
#203932
#203582
#113882
#113482
#108922
#108882
#107512
#107422