Finite Group Databasebeta

← #505198

Subgroups — #505198

Order 512.

742 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2241C211yesmaximal, characteristic, complemented
2C22×#358C211yesmaximal, characteristic
3#26065C211yesmaximal, characteristic
4#25780C211yesmaximal, characteristic
5#13481C211yesmaximal, characteristic, complemented
6#13409C211yesmaximal, characteristic, complemented
7#13409C211yesmaximal, characteristic, complemented
8#13364C211yesmaximal, characteristic, complemented
9#13203C211yesmaximal, characteristic, complemented
10#13148C211yesmaximal, characteristic, complemented
11#13148C211yesmaximal, characteristic, complemented
12#13060C211yesmaximal, characteristic
13#12993C211yesmaximal, characteristic
14#12732C211yesmaximal, characteristic
15C16×C22⋊C4C211yesmaximal, characteristic
16C23×Q16C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#494C2211yescharacteristic
20C2×#494C2211yescharacteristic
21C2×#467C2211yescharacteristic
22C2×#451—21no—
23C22×#107C2211yescharacteristic
24#2440—21no—
25#2434—21no—
26#2241—24no—
27C2×#358—24no—
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38C2×#358C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×C4×C16C2211yescharacteristic
42C2×C4×C16C2211yescharacteristic
43#2083—21no—
44#2057—21no—
45#2016—21no—
46#2016—21no—
47#1969C2211yescharacteristic
48#1969C2211yescharacteristic
49#1942C2211yescharacteristic
50#1942C2211yescharacteristic
51#1852—21no—
52#1845C2211yescharacteristic
53#1808C2211yescharacteristic
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C23×Q8—21no—
62C22×Q16—22no—
63C22×Q16—22no—
64C22×Q16—21no—
65C22×Q16—21no—
66C22×Q16—21no—
67C22×Q16—22no—
68C22×Q16—22no—
69C22×Q16—41no—
70C22×Q16—41no—
71C22×Q16C2311yescharacteristic
72C22×Q16C2311yescharacteristic
73C22×Q16C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—41no—
80C22×C4⋊C4—21no—
81C22×C16—22no—
82C22×C16—22no—
83C22×C16—41no—
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88#500—22no—
89#500—41no—
90#494—22no—
91#494—22no—
92#494—41no—
93#494—41no—
94#467—22no—
95#467—22no—
96#451—44no—
97C2×#107—22no—
98C2×#107—21no—
99C2×#107—21no—
100C2×#107—21no—
101C2×#107—41no—
102C2×#107C2311yescharacteristic
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×C22⋊C8D411yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#398—41no—
114#395—41no—
115#395—41no—
116#395—41no—
117#391—21no—
118#391—21no—
119#389—21no—
120#389—21no—
121#385—41no—
122#385—41no—
123#385—41no—
124#384—21no—
125#380—21no—
126C4×C22⋊C4D411yescharacteristic
127#358—22no—
128#358—22no—
129#358—22no—
130#358—22no—
131#358—22no—
132#358—22no—
133#358—22no—
134#358—41no—
135#358—41no—
136#358—41no—
137#358—41no—
138#358D414yes—
139C22⋊C16—22no—
140C4×C16—22no—
141C4×C16—41no—
142#340—21no—
143#340—21no—
144#340—21no—
145#336D411yescharacteristic
146#336D411yescharacteristic
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—81no—
157C22×Q8—81no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4C2×D411yescharacteristic
161C2×Q16—28no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×Q16—21no—
171C2×Q16—21no—
172C2×Q16—21no—
173C2×Q16—21no—
174C2×Q16—21no—
175C2×Q16—21no—
176C2×Q16—21no—
177C2×Q16—21no—
178C2×Q16—44no—
179C2×Q16—44no—
180C2×Q16—42no—
181C2×Q16—42no—
182C2×Q16—44no—
183C2×Q16—42no—
184C2×Q16—42no—
185C2×Q16—42no—
186C2×Q16—42no—
187C2×Q16—42no—
188C2×Q16—81no—
189C2×Q16—81no—
190C2×Q16C2×D412yes—
191C2×Q16#4112yes—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8#4112yes—
200C22×C8#4112yes—
201C22×C8C2411yescharacteristic
202C4⋊Q8—42no—
203C4⋊Q8—42no—
204C4⋊Q8—42no—
205C4⋊Q8—81no—
206C4⋊Q8—81no—
207C4⋊Q8—81no—
208C22⋊Q8—42no—
209C22⋊Q8—42no—
210C22⋊Q8—84no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—81no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4D811yescharacteristic
232C2×C22⋊C4D811yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16—44no—
247C2×C16C2×D412yes—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#107—42no—
258#107—42no—
259#107—81no—
260#107C2×D412yes—
261#103—42no—
262#103—42no—
263#103—42no—
264C22⋊C8SD1614yes—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95D811yescharacteristic
277#95D811yescharacteristic
278C24C2×D811yescharacteristic
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—48no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—82no—
299C2×Q8—82no—
300C2×Q8—82no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×Q8—82no—
305C2×Q8—82no—
306C2×Q8—161no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—161no—
325C22×C4#12412yes—
326C22×C4C2×D811yescharacteristic
327C22×C4C2×D811yescharacteristic
328C22×C4C2×D811yescharacteristic
329C22×C4C2×D811yescharacteristic
330C22×C4C2×D811yescharacteristic
331C22×C4C2×SD1612yes—
332C22×C4C22×D411yescharacteristic
333Q16—22no—
334Q16—22no—
335Q16—22no—
336Q16—22no—
337Q16—48no—
338Q16—41no—
339Q16—41no—
340Q16—41no—
341Q16—41no—
342Q16—41no—
343Q16—41no—
344Q16—41no—
345Q16—41no—
346Q16—84no—
347Q16—84no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8#12412yes—
359C2×C8C2×SD1612yes—
360C2×C8C2×SD1612yes—
361C2×C8C22×D411yescharacteristic
362C2×C8C2×#4111yescharacteristic
363C2×C8C2×#4111yescharacteristic
364C2×C8C2×#4111yescharacteristic
365C2×C8H(5,𝔽2)11yescharacteristic
366C2×C8H(5,𝔽2)11yescharacteristic
367C4⋊C4—22no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—161no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4SD3214yes—
391C22⋊C4Q3214yes—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C16—22no—
397C16—41no—
398C16—41no—
399C16—41no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23#48612yes—
405C23C2×SD3212yes—
406C23C2×Q3212yes—
407C23C22×D811yescharacteristic
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412Q8—88no—
413Q8—81no—
414Q8—81no—
415Q8—81no—
416Q8—81no—
417Q8—81no—
418Q8—81no—
419Q8—81no—
420Q8—81no—
421Q8—164no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—88no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—164no—
462C2×C4#48612yes—
463C2×C4C2×SD3212yes—
464C2×C4C2×SD3212yes—
465C2×C4C2×Q3212yes—
466C2×C4C2×Q3212yes—
467C2×C4C2×#12411yescharacteristic
468C2×C4D4211yescharacteristic
469C2×C4#54611yescharacteristic
470C2×C4C22×SD1611yescharacteristic
471C2×C4#57611yescharacteristic
472C2×C4#57711yescharacteristic
473C8—22no—
474C8—22no—
475C8—22no—
476C8#52511yescharacteristic
477C8#53811yescharacteristic
478C8#54611yescharacteristic
479C8#55111yescharacteristic
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22#224114yescomplemented
487C22C2×#48611yescharacteristic
488C22D4×D811yescharacteristic
489C22#329411yescharacteristic
490C22C22×SD3211yescharacteristic
491C22C22×Q3211yescharacteristic
492C22#341611yescharacteristic
493C22#341711yescharacteristic
494C4—22no—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—82no—
500C4—82no—
501C4—82no—
502C4—161no—
503C4—161no—
504C4—161no—
505C4—161no—
506C4—161no—
507C4#247211yescharacteristic
508C4#313611yescharacteristic
509C4D4×SD1611yescharacteristic
510C4#329511yescharacteristic
511C2—24no—
512C2#2725411yescharacteristic
513C2C2×#224111yescharacteristic
514C2#3370311yescharacteristic
515C2D4×SD3211yescharacteristic
516C2D4×Q3211yescharacteristic
517C2#3386511yescharacteristic
518C2#3386611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2241C2semidirect
5. #13481C2semidirect
6. #13409C2semidirect
7. #13409C2semidirect
8. #13364C2semidirect
9. #13203C2semidirect
10. #13148C2semidirect
11. #13148C2semidirect
486. C22#2241semidirect

Maximal subgroups

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

subgroupindex
C2×#22412
C22×#3582
#260652
#257802
#134812
#134092
#133642
#132032
#131482
#130602
#129932
#127322
C16×C22⋊C42