Finite Group Databasebeta

← #360618

Subgroups — #360618

Order 512.

1246 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60124C211yesmaximal, characteristic
2#20705C211yesmaximal, characteristic
3#20667C211yesmaximal, characteristic
4#20627C211yesmaximal, characteristic
5C2×#1733C211yesmaximal, characteristic
6#11341C212yesmaximal, complemented
7#11275C212yesmaximal, complemented
8#10901C212yesmaximal, complemented
9#10894C212yesmaximal, complemented
10#10737C211yesmaximal, characteristic
11#10720C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15#2924—24no—
16#2924—24no—
17#2921—22no—
18#2920—22no—
19C2×#430C2212yes—
20#2905C2212yes—
21C2×#423C2212yes—
22C2×#407C2211yescharacteristic
23C2×#407C2211yescharacteristic
24C2×#407C2211yescharacteristic
25#2873C2212yes—
26#2280C2211yescharacteristic
27#1826—24no—
28#1811—22no—
29#1810—22no—
30#1798—24no—
31C2×#344C2212yes—
32C2×#344C2212yes—
33#1738C2212yes—
34#1733C418yescomplemented
35#1733C2212yes—
36#1733C2212yes—
37C2×#339C2212yes—
38C2×#339C2212yes—
39C2×#339C2211yescharacteristic
40C2×#339C2211yescharacteristic
41C2×#339C2211yescharacteristic
42C2×#339C2211yescharacteristic
43#1728C2212yes—
44C24×C4C2311yescharacteristic
45C22×M16—21no—
46C22×M16—22no—
47C22×M16—24no—
48C22×M16—42no—
49C22×M16C2×C414yes—
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2311yescharacteristic
54C22×M16C2312yes—
55C23×C8—21no—
56C2×#117—21no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—21no—
60C2×#117—22no—
61C2×#117—24no—
62C2×#117—42no—
63C2×#117C2×C414yes—
64C2×#117C2×C414yes—
65C2×#117D414yes—
66C2×#117Q812yes—
67C2×#117C2312yes—
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C22×C4⋊C4—21no—
71C22×C4⋊C4Q812yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#433—216no—
78#432—28no—
79#431—28no—
80#430—24no—
81#430—42no—
82C2×#108—24no—
83#424—24no—
84#424—24no—
85#423—24no—
86#423—24no—
87#423—42no—
88C2×C4⋊C8—24no—
89#407—21no—
90#407—21no—
91#407—22no—
92#407—22no—
93#407—21no—
94#407—21no—
95#407—22no—
96#407D412yes—
97#407D412yes—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8D414yes—
101#405—24no—
102#405—24no—
103C4×M16—24no—
104#381—21no—
105#380—22no—
106#379—21no—
107C4×C22⋊C4—22no—
108#344—24no—
109#344—42no—
110#341—24no—
111#339—24no—
112#339—24no—
113#339—24no—
114#339—42no—
115#339C2×C414yes—
116#339C2×C414yes—
117#339C2×C414yes—
118#339C2×C414yes—
119#336—24no—
120C25#4111yescharacteristic
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4C22×C412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×Q811yescharacteristic
134C23×C4C2×Q811yescharacteristic
135C23×C4#4111yescharacteristic
136C23×C4#4112yes—
137C23×C4C2411yescharacteristic
138C2×M16—21no—
139C2×M16—22no—
140C2×M16—24no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—21no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—28no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—44no—
163C2×M16—44no—
164C2×M16—48no—
165C2×M16C22×C412yes—
166C2×M16C22×C412yes—
167C2×M16C2×D414yes—
168C2×M16#4114yes—
169C22×C8—21no—
170C22×C8—24no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—44no—
175C22×C8C2×D414yes—
176#117—24no—
177#117—24no—
178#117—24no—
179#117—24no—
180#117—24no—
181#117—21no—
182#117—21no—
183#117—21no—
184#117—21no—
185#117—28no—
186#117—22no—
187#117—22no—
188#117—22no—
189#117—22no—
190#117—44no—
191#117—48no—
192#117C4⋊C418yes—
193#117C22×C412yes—
194#117C22×C412yes—
195#117C2×D412yes—
196#117C2×D412yes—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4C4⋊C414yes—
204C2×C4⋊C4C4⋊C414yes—
205C2×C4⋊C4C2×Q812yes—
206C2×C4⋊C4C2×Q812yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—42no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42C22×C412yes—
228C2×C42C22×C412yes—
229C2×C42#4114yes—
230#108—48no—
231C4⋊C8—216no—
232C4⋊C8—48no—
233C22⋊C8—24no—
234C22⋊C8—24no—
235C22⋊C8—24no—
236C22⋊C8—24no—
237C22⋊C8—42no—
238C22⋊C8—42no—
239#97—28no—
240C4×C8—28no—
241#95—42no—
242C24—22no—
243C24—22no—
244C24—82no—
245C24C4×D412yes—
246C24C22⋊Q811yescharacteristic
247C24C22⋊Q811yescharacteristic
248C24#12312yes—
249C24C2×#4111yescharacteristic
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—84no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4C2×C4⋊C412yes—
283C22×C4C2×C4⋊C412yes—
284C22×C4C4×D412yes—
285C22×C4C4×D412yes—
286C22×C4C4×D412yes—
287C22×C4#12012yes—
288C22×C4#12012yes—
289C22×C4C22⋊Q812yes—
290C22×C4C22⋊Q812yes—
291C22×C4C22⋊Q811yescharacteristic
292C22×C4C22⋊Q811yescharacteristic
293C22×C4#12312yes—
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×Q811yescharacteristic
299C22×C4C2×#4112yes—
300C22×C4C2×#4112yes—
301C22×C4C2×#4111yescharacteristic
302C22×C4H(5,𝔽2)12yes—
303C22×C42−1+412yes—
304M16—22no—
305M16—24no—
306M16—22no—
307M16—22no—
308M16—24no—
309M16—24no—
310M16—48no—
311M16—42no—
312M16—42no—
313M16—42no—
314M16—42no—
315M16—42no—
316M16—42no—
317M16—42no—
318M16—42no—
319M16—42no—
320M16—42no—
321M16—42no—
322M16#11714yes—
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—22no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—48no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—48no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4C2×C4⋊C414yes—
345C4⋊C4C2×C4⋊C414yes—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—28no—
351C22⋊C4—48no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—84no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—24no—
361C42—28no—
362C42—22no—
363C42—22no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42#11714yes—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—24no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—84no—
381C23—84no—
382C23#38512yes—
383C23#38512yes—
384C23C2×C4×D411yescharacteristic
385C23C2×C22⋊Q811yescharacteristic
386C23C2×#12311yescharacteristic
387C23#55112yes—
388C23#55212yes—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—44no—
410C2×C4—48no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4—82no—
420C2×C4#38512yes—
421C2×C4#38512yes—
422C2×C4C22×C4⋊C411yescharacteristic
423C2×C4C2×#11711yescharacteristic
424C2×C4C2×#11711yescharacteristic
425C2×C4C2×C4×D411yescharacteristic
426C2×C4#51811yescharacteristic
427C2×C4#51911yescharacteristic
428C2×C4C2×#12011yescharacteristic
429C2×C4C2×C22⋊Q811yescharacteristic
430C2×C4#53212yes—
431C2×C4#53312yes—
432C2×C4#53712yes—
433C2×C4#54412yes—
434C2×C4#54612yes—
435C2×C4#54612yes—
436C2×C4#54712yes—
437C2×C4#54712yes—
438C2×C4#55112yes—
439C2×C4#55212yes—
440C8—24no—
441C8—42no—
442C8—44no—
443C8—44no—
444C8—82no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—42no—
455C22—42no—
456C22—44no—
457C22—44no—
458C22—82no—
459C22—84no—
460C22—82no—
461C22C2×#38511yescharacteristic
462C22#234211yescharacteristic
463C22#234411yescharacteristic
464C22#243812yes—
465C22#249812yes—
466C4—22no—
467C4—22no—
468C4—44no—
469C4—44no—
470C4—44no—
471C4—82no—
472C4—162no—
473C4#228511yescharacteristic
474C4#231611yescharacteristic
475C4#234211yescharacteristic
476C4#234411yescharacteristic
477C4#243012yes—
478C4#243812yes—
479C2—21no—
480C2—22no—
481C2—22no—
482C2—21no—
483C2—42no—
484C2—42no—
485C2#1387811yescharacteristic
486C2#2086712yes—
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11341C2semidirect
7. #11275C2semidirect
8. #10901C2semidirect
9. #10894C2semidirect
34. #1733C4semidirect

Maximal subgroups

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

subgroupindex
#601242
#207052
#206672
#206272
C2×#17332
#113412
#112752
#109012
#108942
#107372
#107202