Finite Group Databasebeta

← #369272

Subgroups — #369272

Order 512.

1706 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#490C211yesmaximal, characteristic
2C2×#2330C211yesmaximal, characteristic, complemented
3C2×#1991C211yesmaximal, characteristic, complemented
4C2×#1957C211yesmaximal, characteristic
5#14272C211yesmaximal, characteristic
6#11066C214yesmaximal, complemented
7#10931C214yesmaximal, complemented
8#10843C211yesmaximal, characteristic, complemented
9#10694C211yesmaximal, characteristic, complemented
10C22×#127C2211yescharacteristic
11C22×#125C2211yescharacteristic
12C2×#490—24no—
13C2×#490—24no—
14C2×#490C418yescomplemented
15C22×#102C2214yes—
16C22×#102C2214yes—
17C22×#97C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2330—24no—
20#2330—24no—
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#389—22no—
24C2×#389C2212yes—
25C2×#387—22no—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×#376C2211yescharacteristic
30#1991—24no—
31#1957—24no—
32C2×#340C2214yes—
33C2×#340C2214yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1438—22no—
37#1438—22no—
38C23×D4—22no—
39C23×D4C2312yes—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×#127—22no—
43C2×#127—22no—
44C2×#127—22no—
45C2×#127—22no—
46C2×#127—22no—
47C2×#127C2×C414yes—
48C2×#125—22no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125C2×C414yes—
52C2×#125D414yes—
53C2×#125D414yes—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—24no—
63C22×C42—22no—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67#490—232no—
68C2×#102—28no—
69C2×#102—28no—
70C2×#102—28no—
71C2×#102—28no—
72C2×#102—44no—
73C2×#102C2×C4116yes—
74C2×#97—22no—
75C2×#97—22no—
76C2×#97C2×C414yes—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80#390—28no—
81#390—44no—
82#389—216no—
83#389—24no—
84#389—24no—
85#387—216no—
86#384—22no—
87#384D414yes—
88#382—22no—
89#382—22no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—24no—
92#376—22no—
93#376—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D412yes—
98C2×#95D412yes—
99#340—28no—
100#340—44no—
101#336—24no—
102#336—24no—
103C25—22no—
104C22×D4—28no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—24no—
110C22×D4—24no—
111C22×D4—22no—
112C22×D4—24no—
113C22×D4—24no—
114C22×D4—44no—
115C22×D4—44no—
116C22×D4—84no—
117C22×D4—84no—
118C22×D4C22×C414yes—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2411yescharacteristic
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8C22×C414yes—
140C22×C8#4114yes—
141#127—28no—
142#127—28no—
143#127—44no—
144#127—44no—
145#125—28no—
146#125—28no—
147#125—28no—
148#125C22⋊C4116yescomplemented
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—84no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4C22×C414yes—
167C2×C4⋊C4C2×D414yes—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2×D414yes—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—48no—
173C2×C22⋊C4—84no—
174C2×C22⋊C4—84no—
175C2×C22⋊C4—84no—
176C2×C22⋊C4—84no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—28no—
185C2×C42—84no—
186C2×C42C22×C412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42#4114yes—
191C2×C42#4112yes—
192C2×C42#4112yes—
193#102—232no—
194#102—416no—
195#97—28no—
196C4×C8—44no—
197#95—28no—
198#95—28no—
199#95—24no—
200#95—24no—
201#95—44no—
202C24—24no—
203C24—42no—
204C24—42no—
205C24—42no—
206C24—42no—
207C24—42no—
208C24—42no—
209C24—44no—
210C24—44no—
211C24—44no—
212C24—44no—
213C24C22×D411yescharacteristic
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—28no—
217C2×D4—28no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—44no—
228C2×D4—44no—
229C2×D4—44no—
230C2×D4—416no—
231C2×D4—816no—
232C2×D4—88no—
233C2×D4—88no—
234C2×D4—816no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4C4×D414yes—
265C22×C4#12014yes—
266C22×C4#12114yes—
267C22×C4#12414yes—
268C22×C4C23×C411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4112yes—
274C22×C4C2×#4112yes—
275C22×C4C2×#4111yescharacteristic
276C22×C4H(5,𝔽2)12yes—
277C22×C42−1+412yes—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—232no—
296C4⋊C4—28no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—88no—
301C4⋊C4—88no—
302C4⋊C4C2×C22⋊C418yes—
303C22⋊C4—816no—
304C22⋊C4—88no—
305C22⋊C4—88no—
306C42—24no—
307C42—24no—
308C42—42no—
309C42—42no—
310C42—88no—
311C42C2×C22⋊C414yes—
312C42C4×D414yes—
313C42#12114yes—
314C42#12314yes—
315C23—44no—
316C23—48no—
317C23—48no—
318C23—44no—
319C23—44no—
320C23—84no—
321C23—84no—
322C23—84no—
323C23—84no—
324C23—82no—
325C23—82no—
326C23—82no—
327C23—82no—
328C23—82no—
329C23—82no—
330C23—84no—
331C23—84no—
332C23—84no—
333C23—84no—
334C23—84no—
335C23—84no—
336C23—82no—
337C23—82no—
338C23C2×C4×D411yescharacteristic
339C23C2×#12011yescharacteristic
340C23C2×#12111yescharacteristic
341C23C2×#12411yescharacteristic
342C23#53211yescharacteristic
343C23#53511yescharacteristic
344C23#53611yescharacteristic
345C23#57614yes—
346C23#57714yes—
347D4—48no—
348D4—48no—
349D4—832no—
350D4—84no—
351D4—84no—
352C2×C4—28no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—28no—
356C2×C4—28no—
357C2×C4—24no—
358C2×C4—28no—
359C2×C4—48no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—48no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—84no—
383C2×C4—82no—
384C2×C4#38618yes—
385C2×C4C22×C22⋊C411yescharacteristic
386C2×C4C2×C4×D411yescharacteristic
387C2×C4C4×#4112yes—
388C2×C4#52012yes—
389C2×C4C2×#12111yescharacteristic
390C2×C4C2×#12311yescharacteristic
391C2×C4#52912yes—
392C2×C4#54212yes—
393C2×C4#54612yes—
394C2×C4#54812yes—
395C2×C4#54812yes—
396C2×C4#54812yes—
397C2×C4#55012yes—
398C2×C4#55012yes—
399C2×C4#55612yes—
400C2×C4#55912yes—
401C8—44no—
402C8—44no—
403C22—88no—
404C22—88no—
405C22—84no—
406C22—84no—
407C22—84no—
408C22—162no—
409C22—162no—
410C22—164no—
411C22—162no—
412C22—162no—
413C22—164no—
414C22C2×#38611yescharacteristic
415C22C4×#12411yescharacteristic
416C22#232611yescharacteristic
417C22#243111yescharacteristic
418C22#243611yescharacteristic
419C22#254511yescharacteristic
420C22#254811yescharacteristic
421C22#294614yes—
422C22#300812yes—
423C22#301012yes—
424C22#304514yes—
425C22#313614yes—
426C22#317412yes—
427C22#317612yes—
428C22#322012yes—
429C22#322212yes—
430C22#324114yes—
431C4—28no—
432C4—44no—
433C4—48no—
434C4—81no—
435C4—81no—
436C4—84no—
437C4—162no—
438C4#234112yes—
439C4#236712yes—
440C4#242612yes—
441C4#244812yes—
442C2—164no—
443C2#1402611yescharacteristic
444C2#2093212yes—
445C2#2165212yes—
446C2#2223212yes—
447C2#2560711yescharacteristic
448C2#2560811yescharacteristic
449C2#2591512yes—
450C2#2866512yes—
451C2#2876112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2330C2semidirect
3. C2×#1991C2semidirect
6. #11066C2semidirect
7. #10931C2semidirect
8. #10843C2semidirect
9. #10694C2semidirect
14. C2×#490C4semidirect
148. #125C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4902
C2×#23302
C2×#19912
C2×#19572
#142722
#110662
#109312
#108432
#106942