Finite Group Databasebeta

← #6850641

Subgroups — #6850641

Order 512.

1740 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36187C211yesmaximal, characteristic
2#36180C211yesmaximal, characteristic, complemented
3#33289C211yesmaximal, characteristic, complemented
4#32048C218yesmaximal, complemented
5#29807C212yesmaximal, complemented
6#29141C211yesmaximal, characteristic, complemented
7#28509C211yesmaximal, characteristic, complemented
8#27973C212yesmaximal, complemented
9#27889C212yesmaximal, complemented
10#24680C214yesmaximal, complemented
11#24118C214yesmaximal, complemented
12#21981C212yesmaximal, complemented
13C4×#492C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic, complemented
15C2×#555—21no—
16C2×#554C2212yes—
17C2×#492C2211yescharacteristic
18C2×#476C2214yes—
19C2×C4×SD16C2214yes—
20C2×#431C2212yes—
21#2868C2211yescharacteristic
22#2868C2211yescharacteristic
23#2857C2212yes—
24#2856C2212yes—
25#2567C2212yes—
26#2563—21no—
27#2552—21no—
28#2552—21no—
29#2537—21no—
30#2537C2212yes—
31#2502C2214yes—
32#2500—22no—
33#2491—22no—
34#2491C2214yes—
35#2440—22no—
36#2440—22no—
37#2354—21no—
38#2354C2212yes—
39D4×C4⋊C4—21no—
40D4×C4⋊C4—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×#127C2211yescharacteristic
43#2285C2212yes—
44C2×#384C2212yes—
45C42×D4C2211yescharacteristic
46#2094C2214yes—
47#2071C2218yes—
48#2055C2218yes—
49#2035C2218yes—
50#2024C2214yes—
51#2000C2218yes—
52#1990C2214yes—
53#1990C2214yes—
54#1968C2214yes—
55#1957C2211yescharacteristic
56#1957C2212yes—
57#1956C2212yes—
58#1956C2211yescharacteristic
59#1954C2212yes—
60#1948C2218yes—
61#1939C2212yes—
62#1927C2218yes—
63#1923C2214yes—
64#1874C2218yes—
65#1853C2218yes—
66#1844C2212yes—
67#1844C2211yescharacteristic
68#1834C2212yes—
69#1834C2212yes—
70#1834C2212yes—
71#1803C2214yes—
72C4×C4⋊C8C2211yescharacteristic
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75C22×SD16C2314yes—
76#555—48no—
77#554—48no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×#127C2311yescharacteristic
86C2×#125—22no—
87C2×#124—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8C2314yes—
90C2×#121—22no—
91C2×#121—22no—
92C2×#121—22no—
93C2×#121C2314yes—
94C2×C4×Q8C2312yes—
95C2×C4×D4—22no—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4D411yescharacteristic
99C2×C4×D4D411yescharacteristic
100C2×C4×D4C2312yes—
101C2×C4×D4C2311yescharacteristic
102C2×#117—22no—
103C2×#117—22no—
104C22×C4⋊C4—22no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4C2314yes—
107C22×C42C2312yes—
108#492—24no—
109#476—48no—
110C4×SD16—48no—
111#431—28no—
112C2×#106C2318yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2314yes—
115C2×C4⋊C8C2312yes—
116C2×#103C2318yes—
117C2×#103C2314yes—
118C2×#102C2318yes—
119C2×#102C2314yes—
120C2×C22⋊C8C2314yes—
121C2×C22⋊C8C2314yes—
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2312yes—
124C2×C4×C8C2312yes—
125C2×C4×C8C2312yes—
126C2×C4×C8C2311yescharacteristic
127#398—24no—
128#398—24no—
129#395—24no—
130#394—24no—
131#394—24no—
132#393—24no—
133#391C2312yes—
134#391C2314yes—
135#390—21no—
136#390—21no—
137#390—22no—
138#390—22no—
139#390C2312yes—
140#390C2314yes—
141#389—22no—
142#389—22no—
143#389—21no—
144#389—21no—
145#389—22no—
146#389—22no—
147#388—24no—
148#388—24no—
149#385—24no—
150#385—24no—
151#384—21no—
152#384—21no—
153#384—21no—
154#384—21no—
155#384—22no—
156#384—22no—
157#384C2311yescharacteristic
158#384C2311yescharacteristic
159#384C2314yes—
160#384C2312yes—
161#384C2312yes—
162#384C2312yes—
163#384C2312yes—
164#382—22no—
165#380—22no—
166#380—22no—
167#380—22no—
168#380—22no—
169#380—22no—
170#380C2314yes—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4D411yescharacteristic
175C4×C4⋊C4D411yescharacteristic
176C4×C4⋊C4C2314yes—
177C4×C4⋊C4C2311yescharacteristic
178C4×C22⋊C4D412yes—
179C4×C22⋊C4D412yes—
180C4×C22⋊C4C2312yes—
181C43C2311yescharacteristic
182#340C2318yes—
183#340C2318yes—
184#340C2318yes—
185#336C2314yes—
186#336C2312yes—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8C2412yes—
190C22×D4—24no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4C2×D411yescharacteristic
194C22×D4C2412yes—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C23×C4C2412yes—
200C2×SD16—28no—
201C2×SD16—44no—
202C2×SD16—44no—
203C22×C8C2412yes—
204C22×C8C2414yes—
205C22×C8C2412yes—
206C4⋊Q8—22no—
207C4⋊Q8—28no—
208C4⋊Q8—22no—
209C4⋊Q8—44no—
210C4⋊Q8—44no—
211C4⋊Q8—82no—
212C4⋊Q8—82no—
213C4⋊Q8—82no—
214#127—22no—
215#127—22no—
216#125—48no—
217#124—416no—
218C22⋊Q8—416no—
219C22⋊Q8—48no—
220#121—48no—
221#121—48no—
222#121—48no—
223#121—84no—
224C4×Q8—28no—
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—28no—
228C4×D4—48no—
229C4×D4—44no—
230C4×D4—44no—
231C4×D4SD1618yescomplemented
232#117—48no—
233#117—48no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—48no—
256C2×C4⋊C4—84no—
257C2×C4⋊C4—84no—
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2414yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4C2×D412yes—
274C2×C22⋊C4C2×D412yes—
275C2×C22⋊C4C2×D412yes—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2412yes—
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290#106—44no—
291#106—48no—
292C4⋊C8—28no—
293C4⋊C8—24no—
294#103—48no—
295#103—44no—
296#102—44no—
297#102—48no—
298C22⋊C8—28no—
299C22⋊C8—44no—
300C4×C8—24no—
301C4×C8—24no—
302C4×C8—22no—
303C4×C8—42no—
304C4×C8#4114yes—
305#95—24no—
306#95—22no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95C2×D412yes—
312#95C2×D412yes—
313#95C2×D412yes—
314C24—22no—
315C24C22×D412yes—
316C2×Q8—22no—
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—82no—
323C2×Q8—82no—
324C2×Q8—82no—
325C2×Q8—82no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—24no—
332C2×D4—48no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—84no—
340C2×D4—84no—
341C2×D4C2×SD1614yes—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—48no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4C2×SD1618yes—
363C22×C4C22×D412yes—
364C22×C4C22×D412yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D412yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C2511yescharacteristic
373SD16—416no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—24no—
377C2×C8—22no—
378C2×C8—24no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—24no—
382C2×C8C2×#4114yes—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—28no—
387C4⋊C4—24no—
388C4⋊C4—28no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—84no—
405C4⋊C4—84no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4C2×SD1614yes—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—44no—
416C22⋊C4—48no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—84no—
420C22⋊C4—84no—
421C22⋊C4C2×SD1618yes—
422C42—24no—
423C42—24no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—44no—
430C42—48no—
431C42C2×SD1614yes—
432C42#13714yes—
433C42C2×#4112yes—
434C42H(5,𝔽2)14yes—
435C422−1+412yes—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23C22×SD1614yes—
450C23C23×D411yescharacteristic
451Q8—44no—
452Q8—44no—
453Q8—164no—
454Q8—164no—
455D4—28no—
456D4—44no—
457D4—44no—
458D4—84no—
459D4—84no—
460D4—168no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—216no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—88no—
495C2×C4—168no—
496C2×C4#53512yes—
497C2×C4#53512yes—
498C2×C4#54712yes—
499C2×C4#54712yes—
500C2×C4C22×SD1614yes—
501C2×C4C22×SD1612yes—
502C2×C4C22×SD1612yes—
503C2×C4C22×SD1612yes—
504C2×C4C2×#13712yes—
505C2×C4C2×#13712yes—
506C2×C4C2×#13712yes—
507C2×C4#57612yes—
508C2×C4#57712yes—
509C2×C4C22×#4111yescharacteristic
510C2×C4C2×H(5,𝔽2)12yes—
511C2×C4C2×2−1+411yescharacteristic
512C2×C4#58512yes—
513C8—24no—
514C8—44no—
515C8—42no—
516C22—24no—
517C22—24no—
518C22—24no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22C2×#53511yescharacteristic
531C22C2×#54711yescharacteristic
532C22#350011yescharacteristic
533C22C23×SD1611yescharacteristic
534C22C22×#13711yescharacteristic
535C22C2×#57611yescharacteristic
536C22C2×#57711yescharacteristic
537C4—24no—
538C4—22no—
539C4—22no—
540C4—22no—
541C4—42no—
542C4—42no—
543C4—84no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—84no—
548C4#320614yes—
549C4#333514yes—
550C4#335714yes—
551C4#350612yes—
552C4#354712yes—
553C2—44no—
554C2—162no—
555C2#3949111yescharacteristic
556C2C2×#320611yescharacteristic
557C2#6082211yescharacteristic
558C2C2×#333511yescharacteristic
559C2C2×#335711yescharacteristic
560C2#6252511yescharacteristic
561C2#6254211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
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
2. #36180C2semidirect
3. #33289C2semidirect
4. #32048C2semidirect
5. #29807C2semidirect
6. #29141C2semidirect
7. #28509C2semidirect
8. #27973C2semidirect
9. #27889C2semidirect
10. #24680C2semidirect
11. #24118C2semidirect
12. #21981C2semidirect
13. C4×#492C2semidirect
14. #21264C2semidirect
231. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#361872
#361802
#332892
#320482
#298072
#291412
#285092
#279732
#278892
#246802
#241182
#219812
C4×#4922
#212642