Finite Group Databasebeta

← #369373

Subgroups — #369373

Order 512.

1560 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3142C211yesmaximal, characteristic
2#34207C211yesmaximal, characteristic
3#21314C211yesmaximal, characteristic, complemented
4C2×#1963C211yesmaximal, characteristic
5#14283C211yesmaximal, characteristic, complemented
6#11195C214yesmaximal, complemented
7#10857C211yesmaximal, characteristic, complemented
8#10781C214yesmaximal, complemented
9#10701C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×#125C2211yescharacteristic
12C2×#520C2212yes—
13#3142—24no—
14#3142C418yescomplemented
15C2×#491—21no—
16C2×#490—21no—
17C2×#488—22no—
18C2×#423C2211yescharacteristic
19C22×#104C2214yes—
20C2×#417C2214yes—
21C2×C4×M16C2211yescharacteristic
22#2361—24no—
23#2354—22no—
24D4×C4⋊C4—22no—
25#2326—22no—
26#2311C2212yes—
27C4×C4⋊Q8—21no—
28C4×#127—21no—
29C2×#384C2211yescharacteristic
30C4×#117C2211yescharacteristic
31C2×#376C2211yescharacteristic
32#1989—22no—
33#1989—22no—
34#1963—24no—
35C2×#343C2214yes—
36#1736C2212yes—
37#1736C2212yes—
38C2×#339C2212yes—
39C2×#339C2212yes—
40#1441—24no—
41C4⋊#95C2214yes—
42C22×#41C2312yes—
43C22×M16C2312yes—
44C22×M16C2312yes—
45#532—22no—
46#532—22no—
47#532C2×C414yes—
48#532D414yes—
49C2×C4⋊Q8D411yescharacteristic
50C2×#127D411yescharacteristic
51C2×#125—21no—
52C2×#125—21no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125C2×C414yescomplemented
58C2×#124D412yes—
59C2×#121—22no—
60#520—28no—
61#520—44no—
62C2×C4×Q8—22no—
63C2×C4×D4—22no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—22no—
67C2×#117—22no—
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4D414yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77#491—28no—
78#490—28no—
79#488—216no—
80#423—22no—
81#423—22no—
82C2×C4⋊C8—22no—
83C2×#104—28no—
84C2×#104—28no—
85C2×#104—44no—
86#417—28no—
87#417C2×C4116yescomplemented
88C2×#103—24no—
89C2×#102—24no—
90C4×M16—22no—
91C4×M16C2×C414yescomplemented
92C2×#97—21no—
93C2×C4×C8—21no—
94#391—22no—
95#390—24no—
96#390—22no—
97#389—24no—
98#388—28no—
99#386—24no—
100#385—28no—
101#384—21no—
102#384—21no—
103#384—22no—
104#380—24no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C4×C22⋊C4—22no—
109#376—21no—
110#376—22no—
111#376—22no—
112C2×#95—22no—
113C2×#95—28no—
114C43—21no—
115#343—44no—
116#340—24no—
117#340—22no—
118#340—22no—
119#339—24no—
120#339—24no—
121C2×#41—22no—
122C2×#41—22no—
123C2×#41—24no—
124C2×#41—24no—
125C2×#41—24no—
126C2×#41—24no—
127C2×#41C22×C414yes—
128C2×#41C2×D414yes—
129C22×Q8C2×D412yes—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4C2×D412yes—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4C2×D414yes—
138C23×C4C2×D412yes—
139C23×C4C2411yescharacteristic
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—24no—
147C2×M16—22no—
148C2×M16—24no—
149C2×M16C22×C414yes—
150C22×C8—22no—
151C22×C8—22no—
152C4⋊Q8—22no—
153C4⋊Q8C22⋊C414yes—
154#127—22no—
155#127C22⋊C414yes—
156#125—24no—
157#125—24no—
158#125—28no—
159#125—44no—
160#125—42no—
161#125—42no—
162#124—24no—
163#124C22⋊C418yes—
164#121—28no—
165#121—28no—
166C4×Q8—48no—
167C4×D4—28no—
168C4×D4—28no—
169C4×D4—48no—
170C4×D4—416no—
171C4×D4—88no—
172#117—22no—
173#117—22no—
174#117—22no—
175#117—44no—
176#117—44no—
177#117#4114yes—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—84no—
198C2×C4⋊C4C22×C414yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—28no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42C22×C412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42#4114yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C4⋊C8—44no—
226#104—416no—
227#103—216no—
228#102—216no—
229#97—24no—
230C4×C8—24no—
231#95—28no—
232#95—22no—
233#95—22no—
234#95—44no—
235#95—88no—
236C24—22no—
237C24C22×D411yescharacteristic
238#41—28no—
239#41—28no—
240#41—28no—
241#41—28no—
242#41—44no—
243#41—44no—
244#41—44no—
245#41—44no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8C2×C22⋊C414yes—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—22no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—28no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4C2×C22⋊C414yes—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—28no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—48no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—48no—
287C22×C4—88no—
288C22×C4C4×D414yes—
289C22×C4#12012yes—
290C22×C4#12012yes—
291C22×C4#12114yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4112yes—
298C22×C4C2×#4112yes—
299C22×C4C2×#4111yescharacteristic
300C22×C4H(5,𝔽2)12yes—
301C22×C42−1+412yes—
302M16—24no—
303M16—24no—
304M16—42no—
305M16—42no—
306M16—44no—
307M16—44no—
308C2×C8—24no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—28no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—48no—
322C4⋊C4—48no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—48no—
329C4⋊C4—48no—
330C4⋊C4—84no—
331C4⋊C4#12412yes—
332C4⋊C4#12412yes—
333C22⋊C4—24no—
334C22⋊C4—216no—
335C22⋊C4—48no—
336C22⋊C4—42no—
337C22⋊C4—88no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—44no—
342C42—48no—
343C42—44no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42C2×C22⋊C412yes—
348C42C2×C22⋊C412yes—
349C42C4×D414yes—
350C42#12114yes—
351C42#12312yes—
352C42#12312yes—
353C23—22no—
354C23—22no—
355C23—24no—
356C23—42no—
357C23—42no—
358C23—44no—
359C23—44no—
360C23C2×C4×D411yescharacteristic
361C23C2×#12011yescharacteristic
362C23C2×#12111yescharacteristic
363C23#54612yes—
364C23#54812yes—
365Q8—28no—
366Q8—44no—
367D4—28no—
368D4—416no—
369D4—48no—
370D4—48no—
371D4—44no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—88no—
405C2×C4—168no—
406C2×C4#38614yes—
407C2×C4#38614yes—
408C2×C4C22×C22⋊C411yescharacteristic
409C2×C4C2×C4×D411yescharacteristic
410C2×C4C4×#4112yes—
411C2×C4#52012yes—
412C2×C4C2×#12111yescharacteristic
413C2×C4C2×#12311yescharacteristic
414C2×C4C2×#12411yescharacteristic
415C2×C4#52912yes—
416C2×C4#53211yescharacteristic
417C2×C4#53511yescharacteristic
418C2×C4#53611yescharacteristic
419C2×C4#54212yes—
420C2×C4#54612yes—
421C2×C4#54612yes—
422C2×C4#54712yes—
423C2×C4#54712yes—
424C2×C4#55612yes—
425C2×C4#55912yes—
426C8—44no—
427C8—82no—
428C22—22no—
429C22—22no—
430C22—24no—
431C22—22no—
432C22—22no—
433C22—44no—
434C22—48no—
435C22—82no—
436C22—82no—
437C22—84no—
438C22C2×#38611yescharacteristic
439C22#236712yes—
440C22#242712yes—
441C22#243211yescharacteristic
442C22#243511yescharacteristic
443C22#301912yes—
444C22#302012yes—
445C4—24no—
446C4—24no—
447C4—42no—
448C4—42no—
449C4—44no—
450C4—84no—
451C4—84no—
452C4—82no—
453C4—84no—
454C4C4×#12411yescharacteristic
455C4#232611yescharacteristic
456C4#233912yes—
457C4#244712yes—
458C4#254411yescharacteristic
459C4#254711yescharacteristic
460C2—22no—
461C2—22no—
462C2—84no—
463C2#1402711yescharacteristic
464C2#2094312yes—
465C2#2566912yes—
466C2#2587811yescharacteristic
467C2#2587911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
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. #21314C2semidirect
5. #14283C2semidirect
6. #11195C2semidirect
7. #10857C2semidirect
8. #10781C2semidirect
9. #10701C2semidirect
14. #3142C4semidirect
57. C2×#125C2×C4semidirect
87. #417C2×C4semidirect
91. C4×M16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31422
#342072
#213142
C2×#19632
#142832
#111952
#108572
#107812
#107012