Finite Group Databasebeta

← #374476

Subgroups — #374476

Order 512.

1530 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2566C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#21325C212yesmaximal, complemented
4C2×C4×#100C211yesmaximal, characteristic
5#11161C212yesmaximal
6#11161C212yesmaximal
7#10892C212yesmaximal, complemented
8#10891C212yesmaximal, complemented
9#10851C212yesmaximal, complemented
10C22×#125C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#100C2212yes—
13C2×C4×M16C2212yes—
14#2566—24no—
15#2566C418yescomplemented
16C2×#396—22no—
17#2357—24no—
18#2354—24no—
19#2312—22no—
20#2309C2212yes—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#385—22no—
24#2285C2212yes—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27#2280—22no—
28#2274C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1992—24no—
31#1991—24no—
32C4×#100—24no—
33C2×#343C2212yes—
34#1736C2214yes—
35#1736C2214yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C23×D4C2311yescharacteristic
39C24×C4D412yes—
40C22×M16C2314yes—
41C2×#125—22no—
42C2×#125—22no—
43C2×#125—22no—
44C2×#125—24no—
45C2×#125C2×C414yes—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4—24no—
49C2×C4×D4C2×C414yes—
50C2×C4×D4D414yes—
51C2×#117—24no—
52C2×#117C2314yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4D412yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#100—24no—
67C2×#100—28no—
68C4×M16—24no—
69C4×M16—24no—
70C2×#97—22no—
71C2×C4×C8—22no—
72#396—216no—
73#390—22no—
74#390—22no—
75#390—28no—
76#390C2×C414yes—
77#390C2×C414yes—
78#388—216no—
79#385—216no—
80#384—22no—
81#384C2×C414yes—
82#384D414yes—
83#382—22no—
84#382C2×C414yes—
85#382D414yes—
86#381—24no—
87#380—24no—
88#380—22no—
89#380—22no—
90#379—24no—
91C4×C4⋊C4—28no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4C2×C414yes—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—24no—
98C2×#95—22no—
99C2×#95—21no—
100#343—24no—
101#340—28no—
102#340—28no—
103#339—28no—
104#339—28no—
105C25C2×D412yes—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—24no—
109C22×D4—24no—
110C22×D4C22×C412yes—
111C22×D4C2×D412yes—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—84no—
120C23×C4C22⋊C418yes—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2411yescharacteristic
125C2×M16—24no—
126C2×M16—24no—
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—24no—
130C2×M16#4118yes—
131C22×C8—24no—
132#125—28no—
133#125—28no—
134#125—44no—
135#125—44no—
136C4×D4—28no—
137C4×D4—28no—
138C4×D4—28no—
139C4×D4—28no—
140#117—24no—
141#117—24no—
142#117—24no—
143#117—24no—
144#117—88no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C2×D418yes—
170C2×C4⋊C4C2×D412yes—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—48no—
179C2×C22⋊C4—84no—
180C2×C22⋊C4C22⋊C418yes—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—28no—
187C2×C42C22×C412yes—
188C2×C42C22×C412yes—
189C2×C42C22×C412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42#4114yes—
194#100—216no—
195#97—44no—
196C4×C8—44no—
197#95—28no—
198#95—24no—
199#95—28no—
200#95—24no—
201#95—24no—
202C24—22no—
203C24—24no—
204C24—22no—
205C24—24no—
206C24—44no—
207C24—44no—
208C24C2×C22⋊C414yes—
209C24C22×D411yescharacteristic
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—44no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—28no—
225C22×C4—28no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—28no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—48no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—84no—
246C22×C4—88no—
247C22×C4—88no—
248C22×C4C2×C22⋊C414yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4#12011yescharacteristic
252C22×C4#12012yes—
253C22×C4#12011yescharacteristic
254C22×C4#12312yes—
255C22×C4#12312yes—
256C22×C4C23×C411yescharacteristic
257C22×C4C22×D412yes—
258C22×C4C22×D411yescharacteristic
259C22×C4C2×#4112yes—
260C22×C4C2×#4114yes—
261C22×C42−1+414yes—
262M16—28no—
263M16—28no—
264M16—44no—
265M16—44no—
266C2×C8—28no—
267C2×C8—44no—
268C2×C8—44no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—28no—
274C4⋊C4—28no—
275C4⋊C4—48no—
276C4⋊C4—48no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—48no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—84no—
283C4⋊C4—88no—
284C22⋊C4—216no—
285C22⋊C4—416no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—88no—
289C22⋊C4—88no—
290C22⋊C4—88no—
291C42—24no—
292C42—24no—
293C42—42no—
294C42—42no—
295C42—44no—
296C42—44no—
297C42—88no—
298C42C4×D414yes—
299C42C4×D414yes—
300C42#12114yes—
301C42#12114yes—
302C23—22no—
303C23—22no—
304C23—24no—
305C23—24no—
306C23—24no—
307C23—44no—
308C23—44no—
309C23—416no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23—44no—
314C23—82no—
315C23—82no—
316C23—88no—
317C23C22×C22⋊C411yescharacteristic
318C23C2×#12011yescharacteristic
319C23C2×#12311yescharacteristic
320C23#53212yes—
321C23#53612yes—
322D4—48no—
323D4—48no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—416no—
334C2×C4—42no—
335C2×C4—48no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—48no—
344C2×C4—48no—
345C2×C4—48no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—88no—
350C2×C4—82no—
351C2×C4—84no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4#38614yes—
357C2×C4#38612yes—
358C2×C4#38612yes—
359C2×C4C2×C4×D412yes—
360C2×C4C4×#4112yes—
361C2×C4#51912yes—
362C2×C4C2×#12112yes—
363C2×C4#52512yes—
364C2×C4#53612yes—
365C2×C4#54714yes—
366C2×C4#54814yes—
367C2×C4#55214yes—
368C2×C4#55614yes—
369C8—48no—
370C22—24no—
371C22—24no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—48no—
377C22—44no—
378C22—44no—
379C22—44no—
380C22—84no—
381C22—88no—
382C22—88no—
383C22C2×#38611yescharacteristic
384C22#229711yescharacteristic
385C22#231011yescharacteristic
386C22#243212yes—
387C22#249912yes—
388C22#301812yes—
389C22#301912yes—
390C4—24no—
391C4—22no—
392C4—22no—
393C4—44no—
394C4—44no—
395C4—82no—
396C4—88no—
397C4—82no—
398C4—82no—
399C4#236214yes—
400C4#241514yes—
401C2—21no—
402C2—21no—
403C2—22no—
404C2—84no—
405C2#1413811yescharacteristic
406C2#2095211yescharacteristic
407C2#2095311yescharacteristic
408C2#2569714yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21325C2semidirect
7. #10892C2semidirect
8. #10891C2semidirect
9. #10851C2semidirect
15. #2566C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25662
#341862
#213252
C2×C4×#1002
#111612
#108922
#108912
#108512