Finite Group Databasebeta

← #6594172

Subgroups — #6594172

Order 512.

1562 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#36180C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#33049C212yesmaximal, complemented
5#29218C211yesmaximal, characteristic, complemented
6#29141C211yesmaximal, characteristic, complemented
7#27975C212yesmaximal, complemented
8#27960C212yesmaximal, complemented
9#27738C214yesmaximal, complemented
10#27721C218yesmaximal, complemented
11#22502C214yesmaximal
12C4×#487C211yesmaximal, characteristic, complemented
13#20764C212yesmaximal, complemented
14C4×#436C211yesmaximal, characteristic
15C2×#554C2212yes—
16C2×#487C2211yescharacteristic
17C2×#436C2212yes—
18C2×C4×M16C2212yes—
19#2868C2211yescharacteristic
20#2868C2211yescharacteristic
21#2857C2212yes—
22#2857C2212yes—
23#2571C2212yes—
24#2567C2212yes—
25#2563—21no—
26#2552—21no—
27#2552—21no—
28#2538—21no—
29#2502C2214yes—
30#2497—22no—
31#2492—22no—
32#2440—22no—
33#2440—22no—
34#2439C2214yes—
35#2369—21no—
36#2355—21no—
37#2354C2212yes—
38D4×C4⋊C4—21no—
39D4×C4⋊C4—21no—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42#2285C2212yes—
43C2×#384C2212yes—
44C42×D4C2211yescharacteristic
45#2100C2214yes—
46#2099C2218yes—
47#2074C2218yes—
48#2024C2214yes—
49#1990C2214yes—
50#1990C2214yes—
51#1990C2212yes—
52#1990C2211yescharacteristic
53#1987C2214yes—
54#1956C2212yes—
55#1956C2211yescharacteristic
56#1935C2218yes—
57#1933C2214yes—
58#1894C2218yes—
59#1873C2218yes—
60#1836C2218yes—
61#1834C2212yes—
62#1834C2212yes—
63#1834C2211yescharacteristic
64#1834C2212yes—
65#1834C2212yes—
66#1823C2218yes—
67#1803C2214yes—
68#1770C2212yes—
69C4×C4⋊C8C2211yescharacteristic
70C4×#103C2214yes—
71#1753C2214yes—
72C4×C22⋊C8C2212yes—
73#1736C2218yes—
74C42×C8C2211yescharacteristic
75C22×M16C2314yes—
76#554—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×#125—22no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8C2314yes—
89C2×C4×Q8—21no—
90C2×C4×Q8C2312yes—
91C2×C4×D4—22no—
92C2×C4×D4D411yescharacteristic
93C2×C4×D4D411yescharacteristic
94C2×C4×D4C2311yescharacteristic
95C2×#117C2314yes—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4C2314yes—
99C22×C42C2312yes—
100#487—24no—
101#436—28no—
102C2×C4⋊C8C2312yes—
103C2×#103C2314yes—
104C2×#103C2318yes—
105C2×#103C2314yes—
106C2×C22⋊C8C2314yes—
107C4×M16—28no—
108C2×#97C2314yes—
109C2×C4×C8C2312yes—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2312yes—
112C2×C4×C8C2312yes—
113C2×C4×C8C2311yescharacteristic
114#400—24no—
115#398—24no—
116#398—24no—
117#396—24no—
118#395—24no—
119#393—24no—
120#391—21no—
121#391—22no—
122#391—24no—
123#391C2312yes—
124#391C2314yes—
125#390—22no—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389C2312yes—
136#389C2314yes—
137#388—24no—
138#387—24no—
139#385—24no—
140#385—24no—
141#384—21no—
142#384—21no—
143#384—21no—
144#384—21no—
145#384—22no—
146#384—22no—
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#384C2314yes—
150#384C2312yes—
151#384C2312yes—
152#384C2312yes—
153#384C2312yes—
154#382—22no—
155#382C2314yes—
156#380—22no—
157#380—22no—
158#380—22no—
159#380—22no—
160#380—22no—
161#380C2314yes—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
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×C4—24no—
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2412yes—
194C2×M16—28no—
195C2×M16—28no—
196C2×M16—44no—
197C22×C8C2414yes—
198C22×C8C2412yes—
199C22×C8C2412yes—
200C4⋊Q8—28no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207C4⋊Q8—82no—
208C4⋊Q8—82no—
209C4⋊Q8—82no—
210#125—48no—
211C22⋊Q8—216no—
212C22⋊Q8—416no—
213C22⋊Q8—84no—
214C4×Q8—28no—
215C4×Q8—44no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—48no—
219C4×D4#4118yes—
220#117—48no—
221C2×C4⋊C4—216no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—84no—
247C2×C4⋊C4—84no—
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2412yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4C2×D412yes—
260C2×C22⋊C4C2×D412yes—
261C2×C22⋊C4C2×D412yes—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2411yescharacteristic
274C2×C42C2412yes—
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277C4⋊C8—24no—
278#103—28no—
279#103—44no—
280#103—48no—
281C22⋊C8—28no—
282#97—28no—
283C4×C8—24no—
284C4×C8—24no—
285C4×C8—22no—
286C4×C8—42no—
287C4×C8#4114yes—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95C2×D412yes—
299#95C2×D412yes—
300#95C2×D412yes—
301C24C22×D412yes—
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—44no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—44no—
310C2×Q8—42no—
311C2×Q8—82no—
312C2×Q8—82no—
313C2×Q8—82no—
314C2×Q8—82no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4#12414yes—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—48no—
334C22×C4—84no—
335C22×C4—84no—
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C22×D412yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C2×#4118yes—
346C22×C4C2511yescharacteristic
347M16—416no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—24no—
356C2×C8C2×#4114yes—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—28no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—28no—
365C4⋊C4—48no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—48no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4#12414yes—
390C22⋊C4—24no—
391C22⋊C4—24no—
392C22⋊C4—48no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C22⋊C4#12418yes—
396C42—24no—
397C42—24no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—44no—
404C42—44no—
405C42—84no—
406C42#13714yes—
407C42#13714yes—
408C42C2×#4114yes—
409C42C2×#4112yes—
410C422−1+412yes—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23C2×#12414yes—
418C23C23×D411yescharacteristic
419Q8—44no—
420Q8—44no—
421Q8—84no—
422Q8—164no—
423Q8—164no—
424D4—28no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—48no—
442C2×C4—48no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—168no—
462C2×C4C2×#12412yes—
463C2×C4C2×#12414yes—
464C2×C4C2×#12412yes—
465C2×C4#54712yes—
466C2×C4#54712yes—
467C2×C4C2×#13712yes—
468C2×C4C2×#13712yes—
469C2×C4C2×#13712yes—
470C2×C4C2×#13712yes—
471C2×C4C2×#13712yes—
472C2×C4C2×#13712yes—
473C2×C4#57612yes—
474C2×C4#57712yes—
475C2×C4C22×#4111yescharacteristic
476C2×C4C22×#4112yes—
477C2×C4C2×2−1+411yescharacteristic
478C2×C4#58512yes—
479C8—24no—
480C8—44no—
481C8—42no—
482C22—24no—
483C22—24no—
484C22—24no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22C22×#12411yescharacteristic
490C22C2×#54711yescharacteristic
491C22#350011yescharacteristic
492C22C22×#13711yescharacteristic
493C22C22×#13711yescharacteristic
494C22C2×#57611yescharacteristic
495C22C2×#57711yescharacteristic
496C4—24no—
497C4—22no—
498C4—22no—
499C4—22no—
500C4—42no—
501C4—42no—
502C4—82no—
503C4—82no—
504C4—84no—
505C4—82no—
506C4—82no—
507C4—84no—
508C4#313914yes—
509C4#335614yes—
510C4#335714yes—
511C4#347512yes—
512C4#351512yes—
513C2—44no—
514C2#3616411yescharacteristic
515C2C2×#313911yescharacteristic
516C2#6062211yescharacteristic
517C2C2×#335611yescharacteristic
518C2C2×#335711yescharacteristic
519C2#6253711yescharacteristic
520C2#6254211yescharacteristic
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. #33049C2semidirect
5. #29218C2semidirect
6. #29141C2semidirect
7. #27975C2semidirect
8. #27960C2semidirect
9. #27738C2semidirect
10. #27721C2semidirect
12. C4×#487C2semidirect
13. #20764C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#361802
#332892
#330492
#292182
#291412
#279752
#279602
#277382
#277212
#225022
C4×#4872
#207642
C4×#4362