Finite Group Databasebeta

← #375688

Subgroups — #375688

Order 512.

1157 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#35619C211yesmaximal, characteristic
2#20397C211yesmaximal, characteristic
3#20393C211yesmaximal, characteristic
4#11149C212yesmaximal
5#10985C212yesmaximal
6#10957C212yesmaximal, complemented
7#10947C212yesmaximal, complemented
8#10893C212yesmaximal, complemented
9#10888C212yesmaximal, complemented
10C2×#536C2211yescharacteristic
11C2×#520C2211yescharacteristic
12C2×#419C2214yes—
13C2×#413C2212yes—
14C2×#410C2212yes—
15#2873C2212yes—
16#2503C412yescomplemented
17#2502C412yescomplemented
18#2499C412yescomplemented
19#2499C412yescomplemented
20#2447—22no—
21#2446—22no—
22#2419—24no—
23#2317C2211yescharacteristic
24#2310C2211yescharacteristic
25#2297C2211yescharacteristic
26C2×#391C2211yescharacteristic
27#2285C2211yescharacteristic
28#2274C2212yes—
29C4×#101—22no—
30C4×#100—22no—
31C4×#99—24no—
32C2×#343C2212yes—
33#1740C2212yes—
34#1733C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C22×#41C2311yescharacteristic
38C23×Q8C2311yescharacteristic
39C22×M16C2314yes—
40#536—44no—
41C2×C4⋊Q8—21no—
42C2×#124—21no—
43C2×#123—22no—
44C2×C22⋊Q8C2×C412yes—
45C2×C22⋊Q8C2×C412yes—
46#520—44no—
47C2×C4×Q8D412yes—
48C2×C4×D4D414yes—
49C2×C4×D4D412yes—
50C2×#117C2×C412yes—
51C2×#117C2314yes—
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#419—28no—
59#419—28no—
60C2×#103—44no—
61#413—28no—
62C2×#101—22no—
63C2×#100—22no—
64#410—28no—
65C2×#99—24no—
66#405—44no—
67C4×M16—24no—
68#400—24no—
69#398—24no—
70#397—24no—
71#397—24no—
72#391—21no—
73#391—22no—
74#391—22no—
75#391C2×C412yes—
76#391C2×C412yes—
77#391C2×C412yes—
78#391C2×C412yes—
79#390—21no—
80#389—22no—
81#387—24no—
82#387—24no—
83#386—22no—
84#385—24no—
85#385—22no—
86#384C2×C412yes—
87#384C2×C411yescharacteristic
88#384C2×C411yescharacteristic
89#382C2×C412yes—
90#382C2×C411yescharacteristic
91#382C2×C411yescharacteristic
92#381—24no—
93#381—22no—
94#380—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—44no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4C2×C412yes—
99C4×C22⋊C4C2×C412yes—
100C2×#95—22no—
101#343—24no—
102#342—24no—
103#339—28no—
104#339—44no—
105H(3,ℤ/4)—28no—
106C2×#41—24no—
107C2×#41—24no—
108C2×#41—24no—
109C22×Q8—21no—
110C22×Q8—24no—
111C22×Q8—24no—
112C22×Q8—24no—
113C22×Q8C22×C412yes—
114C22×Q8C2×D411yescharacteristic
115C22×D4C2×D412yes—
116C22×D4C2×D411yescharacteristic
117C23×C4—21no—
118C23×C4—21no—
119C23×C4C22⋊C414yes—
120C23×C4C22×C412yes—
121C23×C4C2×D412yes—
122C23×C4C2411yescharacteristic
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—24no—
126C2×M16—24no—
127C2×M16—24no—
128C2×M16#4118yes—
129C22×C8—24no—
130C4⋊Q8—44no—
131#124—44no—
132#123—48no—
133C22⋊Q8—44no—
134C22⋊Q8—44no—
135C4×Q8—44no—
136C4×D4—44no—
137C4×D4—48no—
138#117—24no—
139#117—24no—
140#117—24no—
141#117—24no—
142#117—24no—
143#117—24no—
144#117—44no—
145#117#4118yes—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4C22⋊C414yes—
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C22×C411yescharacteristic
158C2×C4⋊C4C22×C411yescharacteristic
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C411yescharacteristic
161C2×C4⋊C4C2×D412yes—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4C22⋊C414yes—
170C2×C22⋊C4C2×D412yes—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—44no—
176C2×C42C22⋊C414yes—
177C2×C42C22×C412yes—
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C2×D412yes—
181#103—416no—
182#101—28no—
183#100—28no—
184#99—216no—
185#97—44no—
186C4×C8—44no—
187#95—24no—
188#95—24no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—44no—
197C24C2×C22⋊C412yes—
198C24C22×D411yescharacteristic
199#41—44no—
200#41—44no—
201#41—44no—
202#41—44no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—44no—
213C2×Q8—44no—
214C2×Q8—44no—
215C2×Q8—44no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C22×C4—21no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—81no—
240C22×C4—82no—
241C22×C4—84no—
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C414yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C23×C411yescharacteristic
247C22×C4C22×D411yescharacteristic
248C22×C4C2×#4114yes—
249C22×C4C2×#4114yes—
250C22×C4H(5,𝔽2)14yes—
251M16—28no—
252M16—28no—
253M16—44no—
254M16—44no—
255C2×C8—28no—
256C2×C8—44no—
257C2×C8—44no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—28no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—82no—
272C4⋊C4—84no—
273C22⋊C4—24no—
274C22⋊C4—28no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—82no—
283C42—24no—
284C42—22no—
285C42—22no—
286C42—24no—
287C42—28no—
288C42—44no—
289C42—42no—
290C42—42no—
291C42—44no—
292C42—164no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—41no—
297C23—41no—
298C23—42no—
299C23—42no—
300C23C22×C22⋊C411yescharacteristic
301C23#52512yes—
302C23#53212yes—
303C23#53412yes—
304Q8—44no—
305Q8—48no—
306Q8—84no—
307D4—48no—
308D4—44no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—48no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—48no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4—84no—
341C2×C4—164no—
342C2×C4C4×#4112yes—
343C2×C4C4×#4112yes—
344C2×C4#51812yes—
345C2×C4#52512yes—
346C2×C4#53212yes—
347C2×C4#53412yes—
348C2×C4#55114yes—
349C2×C4#55314yes—
350C2×C4#55314yes—
351C2×C4#55914yes—
352C8—48no—
353C22—21no—
354C22—22no—
355C22—21no—
356C22—22no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—82no—
364C22#229711yescharacteristic
365C22#229711yescharacteristic
366C22#230811yescharacteristic
367C22#260214yes—
368C4—42no—
369C4—42no—
370C4—44no—
371C4—42no—
372C4—42no—
373C4—82no—
374C4—81no—
375C4—88no—
376C4—81no—
377C4—81no—
378C4—82no—
379C4#237414yes—
380C4#260214yes—
381C2—21no—
382C2—21no—
383C2—24no—
384C2—82no—
385C2#1416711yescharacteristic
386C2#2178311yescharacteristic
387C2#2178411yescharacteristic
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
6. #10957C2semidirect
7. #10947C2semidirect
8. #10893C2semidirect
9. #10888C2semidirect
16. #2503C4semidirect
17. #2502C4semidirect
18. #2499C4semidirect
19. #2499C4semidirect

Maximal subgroups

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

subgroupindex
#356192
#203972
#203932
#111492
#109852
#109572
#109472
#108932
#108882