Finite Group Databasebeta

← #497565

Subgroups — #497565

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3334C211yesmaximal, characteristic
2C2×#2168C211yesmaximal, characteristic, complemented
3#22656C211yesmaximal, characteristic, complemented
4#13283C211yesmaximal, characteristic, complemented
5#13276C211yesmaximal, characteristic, complemented
6#13061C211yesmaximal, characteristic, complemented
7#13059C211yesmaximal, characteristic, complemented
8#12992C211yesmaximal, characteristic, complemented
9#12991C211yesmaximal, characteristic, complemented
10#12867C211yesmaximal, characteristic, complemented
11#12866C211yesmaximal, characteristic, complemented
12#12813C211yesmaximal, characteristic, complemented
13#12637C211yesmaximal, characteristic, complemented
14#12557C211yesmaximal, characteristic, complemented
15C2×#1380C211yesmaximal, characteristic, complemented
16C2×#547—21no—
17C22×M32C2211yescharacteristic
18C23×C16C2211yescharacteristic
19#3334—24no—
20#3334C418yescomplemented
21C2×#500C2211yescharacteristic
22C2×#480—21no—
23C2×#480—21no—
24C2×#474—21no—
25C2×#466C2211yescharacteristic
26C2×#466C2211yescharacteristic
27C2×C4×D8C2211yescharacteristic
28C2×C8×D4C2211yescharacteristic
29C2×#427C2211yescharacteristic
30C22×#107C2211yescharacteristic
31C22×#107C2211yescharacteristic
32C22×#107C2211yescharacteristic
33#2352—21no—
34#2168—24no—
35C2×C4⋊C16C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×C22⋊C16C2211yescharacteristic
42C2×#346C2211yescharacteristic
43#1949—21no—
44#1854—21no—
45#1853—21no—
46#1846C2211yescharacteristic
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49C4×#107C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62C22×#41—41no—
63C22×D8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66#547—28no—
67#547—44no—
68C2×C4⋊Q8—21no—
69C2×#123—41no—
70C2×#123—41no—
71C2×C22⋊Q8—41no—
72C2×#121—21no—
73C2×#121—21no—
74C2×C4×D4—21no—
75C2×C4×D4D411yescharacteristic
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C2×M32—22no—
81C2×M32—22no—
82C2×M32—41no—
83C22×C16—22no—
84C22×C16—22no—
85C22×C16—41no—
86C22×C16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90#500—22no—
91#500C2×C414yescomplemented
92#480—28no—
93#480—28no—
94#474—28no—
95#466—22no—
96#466—22no—
97#466C2×C414yes—
98#466C2×C414yescomplemented
99C4×D8—22no—
100C4×D8C2×C414yescomplemented
101C8×D4C2×C414yes—
102C8×D4D414yes—
103#427—22no—
104#427—41no—
105C2×#107—22no—
106C2×#107—22no—
107C2×#107—21no—
108C2×#107—21no—
109C2×#107—21no—
110C2×#107—22no—
111C2×#107—21no—
112C2×#107—41no—
113C2×#107C2×C414yes—
114C2×#107C2×C414yescomplemented
115C2×#107C2311yescharacteristic
116C2×#107C2311yescharacteristic
117C2×#107C2311yescharacteristic
118C2×#107C2311yescharacteristic
119C2×#107C2311yescharacteristic
120C2×#107C2311yescharacteristic
121C2×#107C2311yescharacteristic
122C2×#106—21no—
123C2×#106—21no—
124C2×C4⋊C8D411yescharacteristic
125C2×#102—21no—
126C2×#102—21no—
127C2×#102—21no—
128C2×C22⋊C8D411yescharacteristic
129C2×C22⋊C8D411yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#388—41no—
132#387—41no—
133#386—41no—
134#382—21no—
135#380—21no—
136#380—21no—
137C4×C4⋊C4—21no—
138C4×C22⋊C4—41no—
139C4⋊C16—22no—
140#357—22no—
141#357—22no—
142#357—41no—
143#357—41no—
144C22⋊C16—22no—
145C22⋊C16—22no—
146#346—22no—
147#340—21no—
148#340—21no—
149#340—21no—
150C2×#41—48no—
151C2×#41—84no—
152C2×#41—84no—
153C22×Q8—41no—
154C22×D4—21no—
155C22×D4—41no—
156C22×D4—41no—
157C22×D4D811yescharacteristic
158C23×C4—41no—
159C23×C4—41no—
160C23×C4—41no—
161C23×C4—41no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C2×D8—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8C22×C412yes—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8C22×C412yes—
179C22×C8C22×C412yes—
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D412yes—
184C22×C8C2×D412yes—
185C22×C8C2411yescharacteristic
186C4⋊Q8—24no—
187C4⋊Q8—42no—
188#123—48no—
189#123—48no—
190C22⋊Q8—48no—
191#121—24no—
192#121—24no—
193#121—42no—
194#121—42no—
195C4×D4—24no—
196C4×D4—42no—
197C4×D4C22⋊C414yes—
198C4×D4SD1614yes—
199#117—42no—
200#117—81no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4D811yescharacteristic
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4D811yescharacteristic
226C2×C22⋊C4D811yescharacteristic
227C2×C42—41no—
228C2×C42—41no—
229C2×C42C2×D411yescharacteristic
230M32—44no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—44no—
243C2×C16#4112yescomplemented
244#107—24no—
245#107—24no—
246#107—24no—
247#107—21no—
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#107C22×C412yes—
259#107C22×C412yes—
260#107C22×C412yes—
261#107#4112yescomplemented
262#106—42no—
263#106—81no—
264C4⋊C8C22⋊C414yes—
265#102—24no—
266#102—24no—
267#102—24no—
268C22⋊C8C22⋊C414yes—
269C22⋊C8C22⋊C414yes—
270C4×C8C22×C412yes—
271C4×C8C2×D412yes—
272#95—41no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95—41no—
277C24—41no—
278C24C2×D811yescharacteristic
279C24C2×D811yescharacteristic
280#41—832no—
281C2×Q8—44no—
282C2×Q8—82no—
283C2×Q8—82no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—44no—
288C2×D4—44no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—82no—
295C2×D4—82no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4#10214yes—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—161no—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×D811yescharacteristic
329C22×C4C2×D811yescharacteristic
330C22×C4C2×D811yescharacteristic
331C22×C4C2×D811yescharacteristic
332C22×C4C2×SD1612yes—
333C22×C4C2×SD1612yes—
334C22×C4C22×D411yescharacteristic
335D8—22no—
336D8—22no—
337D8—41no—
338D8—41no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8C2×C22⋊C412yes—
354C2×C8C2×C22⋊C412yes—
355C2×C8C2×C22⋊C412yes—
356C2×C8C23×C411yescharacteristic
357C2×C8C22×D411yescharacteristic
358C2×C8C2×#4111yescharacteristic
359C2×C8C2×#4111yescharacteristic
360C2×C8H(5,𝔽2)11yescharacteristic
361C2×C82−1+411yescharacteristic
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—82no—
380C4⋊C4#10214yes—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—161no—
387C22⋊C4#10214yes—
388C22⋊C4#10214yes—
389C42—82no—
390C42—161no—
391C42C2×C22⋊C412yes—
392C42C2×SD1612yes—
393C16—22no—
394C16—41no—
395C16—41no—
396C16—41no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—42no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23C2×#10212yes—
415C23C2×#10212yes—
416C23#50912yes—
417C23#50912yes—
418C23C22×D811yescharacteristic
419Q8—88no—
420D4—28no—
421D4—42no—
422D4—42no—
423D4—88no—
424D4—88no—
425D4—81no—
426D4—81no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—88no—
448C2×C4—88no—
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—88no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—164no—
468C2×C4C2×#10212yes—
469C2×C4C2×#10212yes—
470C2×C4C2×#10212yes—
471C2×C4C2×#10212yes—
472C2×C4C22×C22⋊C411yescharacteristic
473C2×C4#54611yescharacteristic
474C2×C4#54711yescharacteristic
475C2×C4C22×SD1611yescharacteristic
476C2×C4#57711yescharacteristic
477C2×C4#57811yescharacteristic
478C8—22no—
479C8—22no—
480C8—22no—
481C8C4×#4111yescharacteristic
482C8#52011yescharacteristic
483C8#54611yescharacteristic
484C8#54711yescharacteristic
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—161no—
502C22—161no—
503C22—161no—
504C22—161no—
505C22#214014yes—
506C22C22×#10211yescharacteristic
507C22#329411yescharacteristic
508C22#333411yescharacteristic
509C22C2×#50911yescharacteristic
510C22C2×#50911yescharacteristic
511C22#341611yescharacteristic
512C22#341711yescharacteristic
513C4—22no—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—82no—
518C4—82no—
519C4—82no—
520C4—82no—
521C4—161no—
522C4—161no—
523C4—161no—
524C4#233911yescharacteristic
525C4#289811yescharacteristic
526C4#329811yescharacteristic
527C4#333511yescharacteristic
528C2—24no—
529C2—42no—
530C2—162no—
531C2#2248511yescharacteristic
532C2C2×#214011yescharacteristic
533C2#3338211yescharacteristic
534C2#3387311yescharacteristic
535C2#3387511yescharacteristic
536C2#3391011yescharacteristic
537C2#3391211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#3334C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2168C2semidirect
3. #22656C2semidirect
4. #13283C2semidirect
5. #13276C2semidirect
6. #13061C2semidirect
7. #13059C2semidirect
8. #12992C2semidirect
9. #12991C2semidirect
10. #12867C2semidirect
11. #12866C2semidirect
12. #12813C2semidirect
13. #12637C2semidirect
14. #12557C2semidirect
15. C2×#1380C2semidirect
20. #3334C4semidirect
91. #500C2×C4semidirect
98. #466C2×C4semidirect
100. C4×D8C2×C4semidirect
114. C2×#107C2×C4semidirect
243. C2×C16#41semidirect
261. #107#41semidirect

Maximal subgroups

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

subgroupindex
C2×#33342
C2×#21682
#226562
#132832
#132762
#130612
#130592
#129922
#129912
#128672
#128662
#128132
#126372
#125572
C2×#13802