Finite Group Databasebeta

← #6594111

Subgroups — #6594111

Order 512.

1546 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#36188C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#33043C212yesmaximal, complemented
5#29219C211yesmaximal, characteristic, complemented
6#29141C211yesmaximal, characteristic, complemented
7#27975C212yesmaximal, complemented
8#27962C212yesmaximal, complemented
9#27730C214yesmaximal, complemented
10#27698C218yesmaximal, complemented
11#22490C214yesmaximal
12C4×#487C211yesmaximal, characteristic, complemented
13#20760C212yesmaximal, complemented
14C4×C8×D4C211yesmaximal, characteristic
15C2×#556C2212yes—
16C2×#487C2211yescharacteristic
17C2×C8×D4C2212yes—
18C22×C4×C8C2212yes—
19#2868C2211yescharacteristic
20#2868C2211yescharacteristic
21#2857C2212yes—
22#2857C2212yes—
23#2571C2212yes—
24#2568C2212yes—
25#2563—21no—
26#2547—21no—
27#2547—21no—
28#2538—21no—
29#2499C2214yes—
30#2497—22no—
31#2496—22no—
32#2496—22no—
33#2492—22no—
34#2439C2214yes—
35#2369—21no—
36#2365—21no—
37#2365—21no—
38#2355—21no—
39#2354C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42#2285C2212yes—
43#2285C2212yes—
44C42×D4C2211yescharacteristic
45#2101C2218yes—
46#2099C2214yes—
47#2074C2218yes—
48#2024C2214yes—
49#1990C2214yes—
50#1990C2214yes—
51#1990C2212yes—
52#1990C2211yescharacteristic
53#1988C2214yes—
54#1956C2212yes—
55#1956C2211yescharacteristic
56#1933C2218yes—
57#1933C2214yes—
58#1892C2218yes—
59#1873C2218yes—
60#1835C2218yes—
61#1834C2212yes—
62#1834C2212yes—
63#1834C2211yescharacteristic
64#1834C2212yes—
65#1834C2212yes—
66#1825C2218yes—
67#1803C2214yes—
68C8×C4⋊C4C2212yes—
69C4×C4⋊C8C2211yescharacteristic
70C4×#103C2214yes—
71C8×C22⋊C4C2214yes—
72C4×C22⋊C8C2212yes—
73#1735C2218yes—
74C42×C8C2211yescharacteristic
75C23×C8C2314yes—
76#556—216no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×#126—24no—
86C2×#124—22no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8C2314yes—
89C2×C4×Q8—22no—
90C2×C4×Q8—21no—
91C2×C4×Q8C2312yes—
92C2×C4×D4D411yescharacteristic
93C2×C4×D4D411yescharacteristic
94C2×C4×D4C2311yescharacteristic
95C2×#117C2314yes—
96C2×#117C2314yes—
97C22×C42C2312yes—
98#487—24no—
99C8×D4—28no—
100C2×C4⋊C8C2312yes—
101C2×#103C2314yes—
102C2×#103C2318yes—
103C2×#103C2314yes—
104C2×C22⋊C8C2314yes—
105C2×C4×C8—28no—
106C2×C4×C8C2312yes—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2314yes—
109C2×C4×C8C2312yes—
110C2×C4×C8C2312yes—
111C2×C4×C8C2311yescharacteristic
112#400—24no—
113#400—24no—
114#397—24no—
115#397—24no—
116#396—24no—
117#393—24no—
118#391—22no—
119#391—22no—
120#391—21no—
121#391—22no—
122#391—24no—
123#391C2312yes—
124#391C2314yes—
125#390—22no—
126#389—22no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389C2312yes—
134#389C2314yes—
135#388—24no—
136#387—24no—
137#387—24no—
138#387—24no—
139#384—21no—
140#384—21no—
141#384—21no—
142#384—21no—
143#384C2311yescharacteristic
144#384C2311yescharacteristic
145#384C2312yes—
146#384C2312yes—
147#384C2312yes—
148#384C2312yes—
149#382—22no—
150#382C2314yes—
151#382C2314yes—
152#380—22no—
153#380—22no—
154#380—22no—
155#380—22no—
156#380—22no—
157#380—22no—
158#380—22no—
159#380C2314yes—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4D411yescharacteristic
169C4×C4⋊C4D411yescharacteristic
170C4×C4⋊C4C2312yes—
171C4×C4⋊C4C2311yescharacteristic
172C4×C22⋊C4D412yes—
173C4×C22⋊C4D412yes—
174C4×C22⋊C4C2312yes—
175C43C2311yescharacteristic
176#340C2318yes—
177#340C2318yes—
178#340C2318yes—
179#340C2314yes—
180#340C2318yes—
181#340C2314yes—
182#340C2318yes—
183#336C2314yes—
184#336C2312yes—
185C22×Q8—21no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8C2412yes—
189C22×D4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2412yes—
193C22×C8—28no—
194C22×C8—28no—
195C22×C8—44no—
196C22×C8C2414yes—
197C22×C8C2412yes—
198C22×C8C2412yes—
199C4⋊Q8—28no—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8—44no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208C4⋊Q8—82no—
209#126—416no—
210#124—48no—
211C22⋊Q8—216no—
212C22⋊Q8—84no—
213C4×Q8—28no—
214C4×Q8—48no—
215C4×Q8—44no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4#4118yes—
219#117—216no—
220#117—48no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2412yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4C2×D412yes—
255C2×C22⋊C4C2×D412yes—
256C2×C22⋊C4C2×D412yes—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2411yescharacteristic
270C2×C42C2412yes—
271C2×C42C2411yescharacteristic
272C2×C42C2411yescharacteristic
273C4⋊C8—24no—
274#103—28no—
275#103—44no—
276#103—48no—
277C22⋊C8—28no—
278C4×C8—28no—
279C4×C8—24no—
280C4×C8—24no—
281C4×C8—22no—
282C4×C8—42no—
283C4×C8#4114yes—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95C2×D412yes—
297#95C2×D412yes—
298#95C2×D412yes—
299C24C22×D412yes—
300C2×Q8—22no—
301C2×Q8—24no—
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—44no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—42no—
311C2×Q8—82no—
312C2×Q8—82no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4#12414yes—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4C22×D412yes—
332C22×C4C22×D412yes—
333C22×C4C22×D412yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C2×#4118yes—
341C22×C4C2511yescharacteristic
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—22no—
346C2×C8—24no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—416no—
351C2×C8C2×#4114yes—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—28no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—28no—
360C4⋊C4—44no—
361C4⋊C4—48no—
362C4⋊C4—44no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—84no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4#12414yes—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—48no—
385C22⋊C4—48no—
386C22⋊C4—84no—
387C22⋊C4—84no—
388C22⋊C4#12418yes—
389C42—24no—
390C42—24no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—48no—
397C42—44no—
398C42—44no—
399C42—84no—
400C42#13514yes—
401C42#13514yes—
402C42C2×#4114yes—
403C42C2×#4112yes—
404C422−1+412yes—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23C2×#12414yes—
412C23C23×D411yescharacteristic
413Q8—44no—
414Q8—44no—
415Q8—84no—
416Q8—88no—
417Q8—164no—
418D4—28no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—48no—
436C2×C4—48no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4C2×#12414yes—
456C2×C4C2×#12412yes—
457C2×C4C2×#12412yes—
458C2×C4#54712yes—
459C2×C4#54712yes—
460C2×C4C2×#13512yes—
461C2×C4C2×#13512yes—
462C2×C4C2×#13512yes—
463C2×C4C2×#13512yes—
464C2×C4C2×#13512yes—
465C2×C4C2×#13512yes—
466C2×C4#57712yes—
467C2×C4#57812yes—
468C2×C4C22×#4111yescharacteristic
469C2×C4C22×#4112yes—
470C2×C4C2×2−1+411yescharacteristic
471C2×C4#58512yes—
472C8—24no—
473C8—44no—
474C8—42no—
475C22—24no—
476C22—24no—
477C22—24no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22C22×#12411yescharacteristic
483C22C2×#54711yescharacteristic
484C22#350011yescharacteristic
485C22C22×#13511yescharacteristic
486C22C22×#13511yescharacteristic
487C22C2×#57711yescharacteristic
488C22C2×#57811yescharacteristic
489C4—22no—
490C4—22no—
491C4—24no—
492C4—22no—
493C4—42no—
494C4—42no—
495C4—82no—
496C4—82no—
497C4—84no—
498C4—82no—
499C4—84no—
500C4—82no—
501C4#313114yes—
502C4#334014yes—
503C4#334214yes—
504C4#347512yes—
505C4#351512yes—
506C2—44no—
507C2#3616411yescharacteristic
508C2C2×#313111yescharacteristic
509C2#6062311yescharacteristic
510C2C2×#334011yescharacteristic
511C2C2×#334211yescharacteristic
512C2#6252811yescharacteristic
513C2#6253111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33289C2semidirect
4. #33043C2semidirect
5. #29219C2semidirect
6. #29141C2semidirect
7. #27975C2semidirect
8. #27962C2semidirect
9. #27730C2semidirect
10. #27698C2semidirect
12. C4×#487C2semidirect
13. #20760C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#361882
#332892
#330432
#292192
#291412
#279752
#279622
#277302
#276982
#224902
C4×#4872
#207602
C4×C8×D42