Finite Group Databasebeta

← #365736

Subgroups — #365736

Order 512.

1794 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2328C211yesmaximal, characteristic
2C2×#1961C211yesmaximal, characteristic
3C2×#1957C211yesmaximal, characteristic
4C2×#1836C211yesmaximal, characteristic, complemented
5#14277C211yesmaximal, characteristic, complemented
6#11066C214yesmaximal, complemented
7#10842C211yesmaximal, characteristic, complemented
8#10788C214yesmaximal, complemented
9#10694C211yesmaximal, characteristic, complemented
10C22×#127C2211yescharacteristic
11C22×C4×D4C2212yes—
12C22×#102C2214yes—
13C22×#102C2214yes—
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2328—24no—
17#2328—24no—
18C2×#390C2212yes—
19C2×#389C2212yes—
20C2×#386—22no—
21C2×#386—22no—
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#1961C418yescomplemented
27#1957—24no—
28#1836—24no—
29C2×#340C2214yes—
30C2×#340C2214yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1444—22no—
34#1442—22no—
35C4⋊#95C2212yes—
36C23×D4—22no—
37C23×D4C2312yes—
38C24×C4—22no—
39C23×C8C2312yes—
40C23×C8C2312yes—
41C2×#127—22no—
42C2×#127—22no—
43C2×#127—22no—
44C2×#127C2×C414yes—
45C2×#127D414yes—
46C2×#127Q814yes—
47C2×C4×D4—24no—
48C2×C4×D4—24no—
49C2×C4×D4—28no—
50C2×C4×D4—44no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#102—28no—
63C2×#102—28no—
64C2×#102—44no—
65C2×#102—44no—
66C2×#102—44no—
67C2×#102C2×C4116yescomplemented
68C2×#97—22no—
69C2×#97—22no—
70C2×#97C2×C414yescomplemented
71C2×C4×C8—22no—
72C2×C4×C8—22no—
73C2×C4×C8—22no—
74#390—24no—
75#390—24no—
76#389—44no—
77#386—216no—
78#386—216no—
79#384—22no—
80#384—22no—
81#382—22no—
82#382—22no—
83#382C2×C414yescomplemented
84#382D414yes—
85#376—22no—
86#376—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—24no—
91C2×#95D412yes—
92C2×#95D412yes—
93#340—28no—
94#340—44no—
95#336—24no—
96#336—24no—
97C25—22no—
98C22×D4—28no—
99C22×D4—28no—
100C22×D4—28no—
101C22×D4—22no—
102C22×D4—24no—
103C22×D4—24no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—24no—
107C22×D4—24no—
108C22×D4—84no—
109C22×D4—84no—
110C22×D4C22×C414yes—
111C22×D4C2×D414yes—
112C22×D4C2×Q814yes—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—28no—
116C23×C4—22no—
117C23×C4—44no—
118C23×C4—44no—
119C23×C4—84no—
120C23×C4—84no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2411yescharacteristic
126C22×C8—22no—
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×C8C22×C414yes—
139C22×C8#4114yes—
140#127—28no—
141#127—28no—
142#127—28no—
143#127C4⋊C4116yescomplemented
144C4×D4—416no—
145C4×D4—416no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—84no—
159C2×C4⋊C4—84no—
160C2×C4⋊C4C22×C414yes—
161C2×C4⋊C4C2×D414yes—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—44no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—84no—
168C2×C22⋊C4—84no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42C22×C412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×Q812yes—
180C2×C42#4114yes—
181C2×C42#4112yes—
182C2×C42#4112yes—
183#102—416no—
184#102—416no—
185#97—28no—
186C4×C8—44no—
187#95—28no—
188#95—28no—
189#95—24no—
190#95—24no—
191#95—44no—
192#95—84no—
193C24—24no—
194C24—24no—
195C24—24no—
196C24—42no—
197C24—42no—
198C24—42no—
199C24—42no—
200C24—44no—
201C24—44no—
202C24—44no—
203C24—44no—
204C24C22×D411yescharacteristic
205C2×D4—24no—
206C2×D4—28no—
207C2×D4—28no—
208C2×D4—24no—
209C2×D4—24no—
210C2×D4—28no—
211C2×D4—28no—
212C2×D4—28no—
213C2×D4—28no—
214C2×D4—232no—
215C2×D4—44no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—44no—
221C2×D4—816no—
222C2×D4—816no—
223C2×D4—88no—
224C2×D4—88no—
225C2×D4C2×C4⋊C418yes—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—24no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—416no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—816no—
257C22×C4—88no—
258C22×C4—88no—
259C22×C4—816no—
260C22×C4C4×D414yes—
261C22×C4#12014yes—
262C22×C4C22⋊Q814yes—
263C22×C4#12414yes—
264C22×C4C23×C411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×Q811yescharacteristic
268C22×C4C2×#4111yescharacteristic
269C22×C4C2×#4112yes—
270C22×C4C2×#4112yes—
271C22×C4C2×#4111yescharacteristic
272C22×C4H(5,𝔽2)12yes—
273C22×C4H(5,𝔽2)12yes—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C4⋊C4—28no—
289C4⋊C4—28no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—88no—
295C4⋊C4—88no—
296C22⋊C4—416no—
297C22⋊C4—88no—
298C22⋊C4—88no—
299C42—24no—
300C42—24no—
301C42—42no—
302C42—42no—
303C42C2×C4⋊C414yes—
304C42C4×D414yes—
305C42C22⋊Q814yes—
306C42#12314yes—
307C23—28no—
308C23—44no—
309C23—44no—
310C23—48no—
311C23—48no—
312C23—44no—
313C23—44no—
314C23—48no—
315C23—48no—
316C23—44no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—84no—
323C23—84no—
324C23—84no—
325C23—84no—
326C23—84no—
327C23—84no—
328C23—84no—
329C23—84no—
330C23C2×C4×D411yescharacteristic
331C23C2×#12011yescharacteristic
332C23C2×C22⋊Q811yescharacteristic
333C23C2×#12411yescharacteristic
334C23#53211yescharacteristic
335C23#53511yescharacteristic
336C23#53511yescharacteristic
337C23#57614yes—
338C23#57714yes—
339D4—216no—
340D4—48no—
341D4—48no—
342D4—48no—
343D4—832no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—28no—
349C2×C4—24no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—48no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—832no—
367C2×C4—84no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—84no—
377C2×C4#38518yes—
378C2×C4C22×C4⋊C411yescharacteristic
379C2×C4C2×C4×D411yescharacteristic
380C2×C4C4×#4112yes—
381C2×C4#51812yes—
382C2×C4C2×C22⋊Q811yescharacteristic
383C2×C4C2×#12311yescharacteristic
384C2×C4#52912yes—
385C2×C4#54012yes—
386C2×C4#54612yes—
387C2×C4#54612yes—
388C2×C4#54612yes—
389C2×C4#55012yes—
390C2×C4#55312yes—
391C2×C4#55412yes—
392C2×C4#55412yes—
393C2×C4#55512yes—
394C8—44no—
395C8—44no—
396C22—48no—
397C22—48no—
398C22—48no—
399C22—84no—
400C22—88no—
401C22—84no—
402C22—84no—
403C22—84no—
404C22—88no—
405C22—88no—
406C22—88no—
407C22C2×#38511yescharacteristic
408C22C4×#12411yescharacteristic
409C22#232811yescharacteristic
410C22#243111yescharacteristic
411C22#243511yescharacteristic
412C22#256311yescharacteristic
413C22#256611yescharacteristic
414C22#294614yes—
415C22#300812yes—
416C22#300912yes—
417C22#308114yes—
418C22#313614yes—
419C22#317412yes—
420C22#317612yes—
421C22#322012yes—
422C22#322112yes—
423C22#322414yes—
424C4—28no—
425C4—44no—
426C4—81no—
427C4—84no—
428C4—81no—
429C4—84no—
430C4—162no—
431C4#234912yes—
432C4#236312yes—
433C4#242012yes—
434C4#245412yes—
435C2—88no—
436C2#1397911yescharacteristic
437C2#2087612yes—
438C2#2165212yes—
439C2#2227912yes—
440C2#2560611yescharacteristic
441C2#2560811yescharacteristic
442C2#2586512yes—
443C2#2915212yes—
444C2#2920112yes—
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
4. C2×#1836C2semidirect
5. #14277C2semidirect
6. #11066C2semidirect
7. #10842C2semidirect
8. #10788C2semidirect
9. #10694C2semidirect
26. #1961C4semidirect
67. C2×#102C2×C4semidirect
70. C2×#97C2×C4semidirect
83. #382C2×C4semidirect
143. #127C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23282
C2×#19612
C2×#19572
C2×#18362
#142772
#110662
#108422
#107882
#106942