Finite Group Databasebeta

← #6798409

Subgroups — #6798409

Order 512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#31354C218yesmaximal, complemented
5#29877C212yesmaximal, complemented
6#29141C211yesmaximal, characteristic, complemented
7#28687C211yesmaximal, characteristic, complemented
8#27960C212yesmaximal, complemented
9#26108C214yesmaximal, complemented
10#25672C212yesmaximal, complemented
11#24969C214yesmaximal
12#21860C212yesmaximal, complemented
13C4×#487C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic
15C2×#554—21no—
16C2×#533C2212yes—
17C2×#498C2212yes—
18C2×#487C2211yescharacteristic
19C2×#431C2212yes—
20#2868C2211yescharacteristic
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2857C2212yes—
24#2567—21no—
25#2552—21no—
26#2552—21no—
27#2552C2212yes—
28#2542—21no—
29#2538C2212yes—
30#2502—22no—
31#2441—22no—
32#2440—22no—
33#2440C2214yes—
34#2429—22no—
35#2423C2214yes—
36#2365—21no—
37#2354—21no—
38Q8×C4⋊C4C2212yes—
39D4×C4⋊C4—21no—
40D4×C4⋊C4C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C2×#384C2212yes—
44C42×D4C2211yescharacteristic
45#2099C2218yes—
46#2088C2218yes—
47#2084C2214yes—
48#2077C2218yes—
49#2035C2218yes—
50#2024C2214yes—
51#1990C2212yes—
52#1990C2212yes—
53#1990C2214yes—
54#1990C2212yes—
55#1990C2211yescharacteristic
56#1956C2212yes—
57#1956C2211yescharacteristic
58#1954C2212yes—
59#1948C2214yes—
60#1939C2212yes—
61#1933C2214yes—
62#1923C2214yes—
63#1872C2218yes—
64#1865C2218yes—
65#1834C2214yes—
66#1834C2211yescharacteristic
67#1834C2212yes—
68#1823C2218yes—
69#1803C2218yes—
70#1803C2214yes—
71C4×C4⋊C8C2211yescharacteristic
72C4×#103C2214yes—
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75#554—48no—
76#533—28no—
77#533—44no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#125—22no—
87C2×#125—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8C2314yes—
93C2×C4×Q8—21no—
94C2×C4×Q8—22no—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2314yes—
97C2×C4×D4—22no—
98C2×C4×D4D411yescharacteristic
99C2×C4×D4D411yescharacteristic
100C2×C4×D4C2311yescharacteristic
101C2×#117—22no—
102C22×C4⋊C4—22no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4C2314yes—
105C22×C42C2312yes—
106#498—44no—
107#487—24no—
108#431—44no—
109C2×#106C2318yes—
110C2×C4⋊C8C2312yes—
111C2×C4⋊C8C2314yes—
112C2×#103C2314yes—
113C2×#103C2318yes—
114C2×#103C2314yes—
115C2×C22⋊C8C2314yes—
116C2×C22⋊C8C2314yes—
117C2×C4×C8C2312yes—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2312yes—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2312yes—
122#399—24no—
123#398—24no—
124#398—24no—
125#398—24no—
126#396—24no—
127#395—24no—
128#391—21no—
129#391—22no—
130#391—22no—
131#391C2312yes—
132#390—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—21no—
137#389—21no—
138#389—21no—
139#389—22no—
140#389—22no—
141#389—24no—
142#389C2312yes—
143#389C2314yes—
144#389C2314yes—
145#388—24no—
146#387—24no—
147#385—24no—
148#385—24no—
149#384—22no—
150#384—22no—
151#384—21no—
152#384—21no—
153#384—21no—
154#384—21no—
155#384—22no—
156#384C2311yescharacteristic
157#384C2311yescharacteristic
158#384C2312yes—
159#384C2314yes—
160#384C2312yes—
161#384C2312yes—
162#384C2312yes—
163#380—22no—
164#380—22no—
165#380C2314yes—
166#380C2314yes—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4D411yescharacteristic
172C4×C4⋊C4D411yescharacteristic
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2311yescharacteristic
175C4×C22⋊C4D412yes—
176C4×C22⋊C4D412yes—
177C4×C22⋊C4C2312yes—
178C43C2311yescharacteristic
179#340C2318yes—
180#340C2318yes—
181#340C2318yes—
182#340C2314yes—
183#340C2314yes—
184#340C2318yes—
185#336C2312yes—
186#336C2314yes—
187C22×Q8—22no—
188C22×Q8—21no—
189C22×Q8—22no—
190C22×Q8C2412yes—
191C22×D4C2×D411yescharacteristic
192C23×C4—24no—
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2412yes—
196C22×C8C2412yes—
197C22×C8C2414yes—
198C22×C8C2412yes—
199C4⋊Q8—24no—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202C4⋊Q8—24no—
203C4⋊Q8—22no—
204C4⋊Q8—44no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207C4⋊Q8—82no—
208C4⋊Q8—82no—
209C4⋊Q8#4114yes—
210#125—48no—
211#125—48no—
212C22⋊Q8—28no—
213C22⋊Q8—44no—
214C22⋊Q8—48no—
215C22⋊Q8—48no—
216C22⋊Q8—416no—
217C22⋊Q8—84no—
218C4×Q8—28no—
219C4×Q8—28no—
220C4×Q8—44no—
221C4×Q8—48no—
222C4×Q8—44no—
223C4×D4—24no—
224C4×D4—24no—
225C4×D4—48no—
226C4×D4SD1618yes—
227#117—48no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4—48no—
256C2×C4⋊C4—84no—
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4C2×D412yes—
269C2×C22⋊C4C2×D412yes—
270C2×C22⋊C4C2×D412yes—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2412yes—
283C2×C42C2411yescharacteristic
284C2×C42C2411yescharacteristic
285C2×C42C2411yescharacteristic
286#106—48no—
287C4⋊C8—24no—
288C4⋊C8—44no—
289#103—28no—
290#103—44no—
291#103—48no—
292C22⋊C8—44no—
293C22⋊C8—44no—
294C4×C8—22no—
295C4×C8—22no—
296C4×C8—24no—
297C4×C8—24no—
298C4×C8—42no—
299#95—24no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95C2×D412yes—
308#95C2×D412yes—
309#95C2×D412yes—
310C24C22×D412yes—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—44no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—42no—
321C2×Q8—82no—
322C2×Q8—82no—
323C2×Q8C2×#4114yes—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4C2×SD1614yes—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—48no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4C2×SD1618yes—
346C22×C4C22×D412yes—
347C22×C4C22×D412yes—
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D412yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C2511yescharacteristic
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—24no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—24no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—24no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—28no—
376C4⋊C4—48no—
377C4⋊C4—48no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—48no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4C2×SD1614yes—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—48no—
404C22⋊C4—48no—
405C22⋊C4—84no—
406C22⋊C4C2×SD1618yes—
407C42—24no—
408C42—24no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—48no—
415C42—44no—
416C42—44no—
417C42C2×SD1614yes—
418C42#13714yes—
419C42C2×#4112yes—
420C42H(5,𝔽2)12yes—
421C422−1+414yes—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23C22×SD1614yes—
429C23C23×D411yescharacteristic
430Q8—28no—
431Q8—44no—
432Q8—88no—
433Q8—84no—
434Q8—164no—
435D4—28no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—216no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—48no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—168no—
473C2×C4#53612yes—
474C2×C4#53612yes—
475C2×C4#54612yes—
476C2×C4#54612yes—
477C2×C4C22×SD1612yes—
478C2×C4C22×SD1614yes—
479C2×C4C22×SD1612yes—
480C2×C4C22×SD1612yes—
481C2×C4C2×#13712yes—
482C2×C4C2×#13712yes—
483C2×C4C2×#13712yes—
484C2×C4#57612yes—
485C2×C4#57812yes—
486C2×C4C22×#4111yescharacteristic
487C2×C4C2×H(5,𝔽2)11yescharacteristic
488C2×C4C2×2−1+412yes—
489C2×C4#58512yes—
490C8—42no—
491C8—42no—
492C8—44no—
493C22—24no—
494C22—24no—
495C22—24no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22C2×#53611yescharacteristic
501C22C2×#54611yescharacteristic
502C22#350011yescharacteristic
503C22C23×SD1611yescharacteristic
504C22C22×#13711yescharacteristic
505C22C2×#57611yescharacteristic
506C22C2×#57811yescharacteristic
507C4—22no—
508C4—22no—
509C4—24no—
510C4—22no—
511C4—42no—
512C4—44no—
513C4—42no—
514C4—82no—
515C4—84no—
516C4—82no—
517C4—82no—
518C4—84no—
519C4#322914yes—
520C4#329814yes—
521C4#333314yes—
522C4#351512yes—
523C4#354712yes—
524C2—44no—
525C2#3895511yescharacteristic
526C2C2×#322911yescharacteristic
527C2#6088011yescharacteristic
528C2C2×#329811yescharacteristic
529C2C2×#333311yescharacteristic
530C2#6252511yescharacteristic
531C2#6253611yescharacteristic
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. #31354C2semidirect
5. #29877C2semidirect
6. #29141C2semidirect
7. #28687C2semidirect
8. #27960C2semidirect
9. #26108C2semidirect
10. #25672C2semidirect
12. #21860C2semidirect
13. C4×#487C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#361342
#332892
#313542
#298772
#291412
#286872
#279602
#261082
#256722
#249692
#218602
C4×#4872
#212642