Finite Group Databasebeta

← #360661

Subgroups — #360661

Order 512.

1254 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34189C211yesmaximal, characteristic
2#20764C211yesmaximal, characteristic, complemented
3#20648C211yesmaximal, characteristic
4C2×#1759C211yesmaximal, characteristic
5#14354C211yesmaximal, characteristic, complemented
6#11079C212yesmaximal
7#11078C212yesmaximal
8#10892C212yesmaximal, complemented
9#10891C212yesmaximal, complemented
10#10858C211yesmaximal, characteristic, complemented
11#10840C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×#104C2212yes—
15C2×#417C2212yes—
16C2×C4×M16C2211yescharacteristic
17#2357—22no—
18#2356—22no—
19#2353—24no—
20#2314—22no—
21#2311C2211yescharacteristic
22#2311C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#384C2212yes—
26#2283—22no—
27C2×#382C2212yes—
28C2×#382C2211yescharacteristic
29#2274C2211yescharacteristic
30#2274C2211yescharacteristic
31C2×#376C2211yescharacteristic
32#1836—22no—
33#1834—22no—
34#1803—22no—
35#1802—22no—
36#1759C418yescomplemented
37C2×#343C2212yes—
38C2×#343C2212yes—
39#1736C2212yes—
40#1736C2212yes—
41#1736C2212yes—
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45#1444—24no—
46C22×#41C2311yescharacteristic
47C22×M16C2312yes—
48C22×M16C2312yes—
49C4×#41—22no—
50C4×#41—22no—
51C4×#41C2×C414yes—
52C4×#41Q814yes—
53C2×C4×Q8D412yes—
54C2×C4×D4—22no—
55C2×C4×D4D412yes—
56C2×#117—21no—
57C2×#117C2311yescharacteristic
58C2×#117C2311yescharacteristic
59C2×#117C2311yescharacteristic
60C22×C4⋊C4—24no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C42—21no—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68#423—22no—
69#423—22no—
70C2×C4⋊C8—22no—
71C2×#104—24no—
72C2×#104—24no—
73C2×#104C2×C418yes—
74#417—42no—
75#417—42no—
76C2×#103—22no—
77C2×#102—22no—
78C4×M16—22no—
79C4×M16C2×C414yes—
80C2×#97—21no—
81C2×C4×C8—21no—
82#391—21no—
83#391—21no—
84#390—21no—
85#390—21no—
86#388—24no—
87#388—24no—
88#387—24no—
89#387—24no—
90#386—22no—
91#386—22no—
92#384—22no—
93#384—21no—
94#384—21no—
95#384—24no—
96#384—42no—
97#384D412yes—
98#383—24no—
99#382—21no—
100#382—22no—
101#382—21no—
102#382—24no—
103#382—42no—
104#382D412yes—
105#381—24no—
106#380—22no—
107#380—22no—
108#380—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C22⋊C4—24no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117#376—24no—
118#376—22no—
119#376C2×C414yes—
120C2×#95—22no—
121C2×#95—24no—
122C2×#95D414yes—
123#343—42no—
124#343—42no—
125#340—24no—
126#340—24no—
127#340—24no—
128#339—24no—
129#339—24no—
130#339C2×C418yes—
131C2×#41—21no—
132C2×#41—21no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41C22×C412yes—
138C2×#41C2×Q812yes—
139C22×Q8C2×D411yescharacteristic
140C22×D4—21no—
141C22×D4C2×D411yescharacteristic
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2411yescharacteristic
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16C22×C414yes—
162C22×C8—22no—
163C22×C8—22no—
164C4×Q8—24no—
165C4×Q8C4⋊C418yescomplemented
166C4×D4—28no—
167C4×D4—28no—
168C4×D4—24no—
169C4×D4C4⋊C418yescomplemented
170#117—21no—
171#117—21no—
172#117—24no—
173#117—24no—
174#117—22no—
175#117—22no—
176#117—22no—
177#117—22no—
178#117C22×C412yes—
179#117C2×Q812yes—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—84no—
201C2×C4⋊C4—88no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42C22×C414yes—
223C2×C42C22×C412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×Q812yes—
228C2×C42#4114yes—
229C2×C42#4114yes—
230C2×C42#4112yes—
231C2×C42#4112yes—
232C4⋊C8—44no—
233#104—216no—
234#103—44no—
235#102—44no—
236#97—24no—
237C4×C8—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—28no—
243#95—24no—
244#95—24no—
245#95—22no—
246#95—22no—
247#95—48no—
248C24—21no—
249C24C22×D411yescharacteristic
250#41—24no—
251#41—24no—
252#41—24no—
253#41—24no—
254#41—42no—
255#41—42no—
256#41—42no—
257#41—42no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8C2×C4⋊C412yes—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4C2×C4⋊C412yes—
274C22×C4—24no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—28no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—48no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4C4×D414yes—
303C22×C4#12012yes—
304C22×C4#12012yes—
305C22×C4C22⋊Q814yes—
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×Q811yescharacteristic
310C22×C4C2×#4112yes—
311C22×C4C2×#4112yes—
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4H(5,𝔽2)12yes—
315C22×C42−1+412yes—
316M16—24no—
317M16—24no—
318M16—42no—
319M16—42no—
320M16—42no—
321M16—42no—
322M16—42no—
323M16—42no—
324C2×C8—24no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C4⋊C4—22no—
331C4⋊C4—28no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—84no—
347C4⋊C4—84no—
348C4⋊C4—816no—
349C4⋊C4C2×C4⋊C414yes—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—28no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—84no—
356C22⋊C4—84no—
357C42—22no—
358C42—24no—
359C42—22no—
360C42—24no—
361C42—24no—
362C42—28no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—84no—
368C42—84no—
369C42C2×C4⋊C414yes—
370C42#11718yes—
371C42C4×D412yes—
372C42C4×D412yes—
373C42C22⋊Q812yes—
374C42C22⋊Q812yes—
375C42#12314yes—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—41no—
380C23—41no—
381C23—42no—
382C23—42no—
383C23C2×C4×D411yescharacteristic
384C23C2×#12011yescharacteristic
385C23C2×C22⋊Q811yescharacteristic
386C23#54612yes—
387C23#54812yes—
388Q8—22no—
389Q8—22no—
390Q8—41no—
391Q8—41no—
392D4—22no—
393D4—22no—
394D4—44no—
395D4—44no—
396D4—41no—
397D4—41no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—28no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—48no—
409C2×C4—48no—
410C2×C4—44no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—88no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—82no—
433C2×C4—84no—
434C2×C4#38514yes—
435C2×C4#38514yes—
436C2×C4C22×C4⋊C411yescharacteristic
437C2×C4C2×#11712yes—
438C2×C4C2×C4×D411yescharacteristic
439C2×C4#52012yes—
440C2×C4C2×C22⋊Q811yescharacteristic
441C2×C4C2×#12311yescharacteristic
442C2×C4#53211yescharacteristic
443C2×C4#53211yescharacteristic
444C2×C4#53312yes—
445C2×C4#53511yescharacteristic
446C2×C4#53611yescharacteristic
447C2×C4#54412yes—
448C2×C4#54712yes—
449C2×C4#55012yes—
450C2×C4#55212yes—
451C2×C4#55312yes—
452C2×C4#55512yes—
453C2×C4#55612yes—
454C8—44no—
455C8—82no—
456C22—24no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—42no—
462C22—44no—
463C22—82no—
464C22—81no—
465C22—81no—
466C22C2×#38511yescharacteristic
467C22#235212yes—
468C22#243111yescharacteristic
469C22#243211yescharacteristic
470C22#243912yes—
471C22#302112yes—
472C22#302212yes—
473C4—24no—
474C4—24no—
475C4—48no—
476C4—42no—
477C4—44no—
478C4—82no—
479C4—82no—
480C4—82no—
481C4—82no—
482C4—162no—
483C4#228511yescharacteristic
484C4#231711yescharacteristic
485C4#235412yes—
486C4#244112yes—
487C4#249911yescharacteristic
488C4#250011yescharacteristic
489C2—22no—
490C2—22no—
491C2—82no—
492C2#1387911yescharacteristic
493C2#2086512yes—
494C2#2559511yescharacteristic
495C2#2559611yescharacteristic
496C2#2567012yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
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
2. #20764C2semidirect
5. #14354C2semidirect
8. #10892C2semidirect
9. #10891C2semidirect
10. #10858C2semidirect
11. #10840C2semidirect
36. #1759C4semidirect
165. C4×Q8C4⋊C4semidirect
169. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#341892
#207642
#206482
C2×#17592
#143542
#110792
#110782
#108922
#108912
#108582
#108402