Finite Group Databasebeta

← #381105

Subgroups — #381105

Order 512.

1433 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2623C211yesmaximal, characteristic
2#14339C211yesmaximal, characteristic, complemented
3#13776C211yesmaximal, characteristic, complemented
4#11203C212yesmaximal, complemented
5#11197C212yesmaximal, complemented
6#11043C212yesmaximal
7#10966C212yesmaximal
8#10960C212yesmaximal
9#10912C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×#99C2214yes—
13#2623—24no—
14#2623—24no—
15C2×#400C412yescomplemented
16C2×#399C412yescomplemented
17C2×#398C412yescomplemented
18C2×#397C412yescomplemented
19C2×#386C2211yescharacteristic
20C2×#386C2211yescharacteristic
21C2×#385C2211yescharacteristic
22C2×#385C2211yescharacteristic
23C2×#381C2212yes—
24C2×#381C2212yes—
25C2×#381C2211yescharacteristic
26C2×#380C2212yes—
27C2×C4×C22⋊C4C2212yes—
28C2×#342C2214yes—
29C2×#342C2214yes—
30C2×#342C2214yes—
31#1441—24no—
32#1436—24no—
33C22⋊#95C2212yes—
34C22⋊#95C2212yes—
35C23×D4C2311yescharacteristic
36C24×C4C2311yescharacteristic
37C24×C4C2311yescharacteristic
38C24×C4C2311yescharacteristic
39C2×#123—22no—
40C2×#123—22no—
41C2×#123—44no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—44no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4D414yes—
51C22×C4⋊C4D414yes—
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C412yes—
56C22×C22⋊C4C2314yes—
57C22×C22⋊C4C2311yescharacteristic
58C22×C22⋊C4C2311yescharacteristic
59C22×C22⋊C4C2311yescharacteristic
60C22×C22⋊C4C2314yes—
61C22×C42—22no—
62C2×#99—28no—
63C2×#99—44no—
64C2×#99—44no—
65#400—28no—
66#399—28no—
67#398—28no—
68#397—28no—
69#386—22no—
70#386—22no—
71#386—22no—
72#386—22no—
73#385—22no—
74#385—22no—
75#385—22no—
76#385—22no—
77#381—22no—
78#381—22no—
79#381—28no—
80#381—44no—
81#380—44no—
82C4×C22⋊C4—44no—
83C2×#95—24no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—24no—
87C2×#95—24no—
88C2×#95—24no—
89C2×#95C2×C412yes—
90C2×#95C2×C412yes—
91C2×#95C2×C412yes—
92C2×#95C2×C412yes—
93C2×#95C2×C412yes—
94C2×#95C2×C412yes—
95#342—28no—
96#342—44no—
97#342—44no—
98C25C22×C411yescharacteristic
99C25C2411yescharacteristic
100C22×D4—21no—
101C22×D4—21no—
102C22×D4—24no—
103C22×D4—24no—
104C22×D4—22no—
105C22×D4—22no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—82no—
115C23×C4—82no—
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C412yes—
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C412yes—
123C23×C4C22×C412yes—
124C23×C4C22×C412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131#123—48no—
132#123—48no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—84no—
147C2×C4⋊C4—82no—
148C2×C4⋊C4—82no—
149C2×C4⋊C4C22⋊C418yes—
150C2×C4⋊C4C22⋊C418yes—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—82no—
179C2×C22⋊C4—82no—
180C2×C22⋊C4#4118yes—
181C2×C42—44no—
182C2×C42—84no—
183#99—416no—
184#95—216no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95—44no—
192#95—44no—
193#95—48no—
194#95—84no—
195#95—84no—
196C24—21no—
197C24—22no—
198C24—22no—
199C24—22no—
200C24—42no—
201C24—42no—
202C24—42no—
203C24—42no—
204C24C23×C411yescharacteristic
205C24C22×D411yescharacteristic
206C24C22×D411yescharacteristic
207C24C2×#4114yes—
208C24H(5,𝔽2)14yes—
209C24H(5,𝔽2)12yes—
210C242−1+412yes—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—42no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—44no—
221C2×D4—44no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—28no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—48no—
279C22×C4—48no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—88no—
284C22×C4—88no—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C414yes—
288C22×C4C2×C22⋊C414yes—
289C4⋊C4—48no—
290C4⋊C4—48no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C4—88no—
296C22⋊C4—28no—
297C22⋊C4—28no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—84no—
314C22⋊C4—84no—
315C42—88no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—44no—
323C23—48no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—41no—
328C23—41no—
329C23—42no—
330C23—42no—
331C23—41no—
332C23—44no—
333C23—84no—
334C23—84no—
335C23—82no—
336C23—82no—
337C23C22×C22⋊C411yescharacteristic
338C23C4×#4112yes—
339C23#51812yes—
340C23#52012yes—
341C23#52512yes—
342C23#53212yes—
343C23#53412yes—
344C23#53412yes—
345C23#53511yescharacteristic
346C23#53611yescharacteristic
347C23#53712yes—
348C23#53814yes—
349C23#53914yes—
350C23#54014yes—
351C23#54214yes—
352D4—84no—
353D4—82no—
354D4—82no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—48no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—416no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—48no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—84no—
393C2×C4—84no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—88no—
398C2×C4—84no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—84no—
404C2×C4—88no—
405C2×C4—88no—
406C2×C4—84no—
407C2×C4—84no—
408C22—22no—
409C22—24no—
410C22—24no—
411C22—22no—
412C22—24no—
413C22—24no—
414C22—24no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—44no—
422C22—44no—
423C22—44no—
424C22—44no—
425C22—82no—
426C22—82no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—164no—
431C22#229711yescharacteristic
432C22#230811yescharacteristic
433C22#231111yescharacteristic
434C22#233514yes—
435C22#263014yes—
436C22#266112yes—
437C22#266312yes—
438C22#288312yes—
439C22#288412yes—
440C4—42no—
441C4—42no—
442C4—44no—
443C4—44no—
444C4—88no—
445C4—82no—
446C4—82no—
447C4—81no—
448C4—81no—
449C4—81no—
450C4—164no—
451C2—24no—
452C2—22no—
453C2—24no—
454C2—22no—
455C2—161no—
456C2#1428711yescharacteristic
457C2#2142312yes—
458C2#2175011yescharacteristic
459C2#2175111yescharacteristic
460C2#2187712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14339C2semidirect
3. #13776C2semidirect
4. #11203C2semidirect
5. #11197C2semidirect
9. #10912C2semidirect
15. C2×#400C4semidirect
16. C2×#399C4semidirect
17. C2×#398C4semidirect
18. C2×#397C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26232
#143392
#137762
#112032
#111972
#110432
#109662
#109602
#109122