Finite Group Databasebeta

← #6797947

Subgroups — #6797947

Order 512.

1714 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36187C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic, complemented
3#33289C211yesmaximal, characteristic, complemented
4#31931C218yesmaximal, complemented
5#29842C212yesmaximal, complemented
6#28716C211yesmaximal, characteristic, complemented
7#28511C211yesmaximal, characteristic, complemented
8#27949C212yesmaximal, complemented
9#25598C212yesmaximal, complemented
10#24814C214yesmaximal, complemented
11#24121C214yesmaximal, complemented
12#21864C212yesmaximal, complemented
13C4×#492C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic, complemented
15C2×#555C2212yes—
16C2×#533C2212yes—
17C2×#492C2211yescharacteristic
18C2×#486C2212yes—
19C2×#477C2214yes—
20C2×C4×SD16C2214yes—
21C2×#431C2212yes—
22#2868C2211yescharacteristic
23#2868C2211yescharacteristic
24#2857C2212yes—
25#2856C2212yes—
26#2563C2212yes—
27#2552—21no—
28#2542—21no—
29#2538C2212yes—
30#2537—21no—
31#2537—21no—
32#2500C2214yes—
33#2491—22no—
34#2491—22no—
35#2441—22no—
36#2429—22no—
37#2423C2214yes—
38#2365—21no—
39#2354—21no—
40#2354—21no—
41#2354—21no—
42Q8×C4⋊C4C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2285C2212yes—
46C42×D4C2211yescharacteristic
47#2095C2214yes—
48#2085C2218yes—
49#2071C2214yes—
50#2067C2218yes—
51#2033C2218yes—
52#2000C2218yes—
53#1990C2212yes—
54#1990C2214yes—
55#1990C2212yes—
56#1990C2212yes—
57#1968C2214yes—
58#1957C2211yescharacteristic
59#1956C2212yes—
60#1956C2211yescharacteristic
61#1954C2212yes—
62#1939C2212yes—
63#1923C2214yes—
64#1880C2218yes—
65#1873C2218yes—
66#1854C2218yes—
67#1845C2212yes—
68#1844C2211yescharacteristic
69#1834C2214yes—
70#1805C2218yes—
71#1802C2214yes—
72C4×C4⋊C8C2211yescharacteristic
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75C22×SD16C2314yes—
76#555—48no—
77#533—28no—
78#533—44no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#127C2311yescharacteristic
87C2×#125—22no—
88C2×#124—24no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8C2314yes—
92C2×#121—22no—
93C2×#121—22no—
94C2×#121—22no—
95C2×#121C2314yes—
96C2×C4×Q8—22no—
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2314yes—
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4D411yescharacteristic
102C2×C4×D4D411yescharacteristic
103C2×C4×D4C2312yes—
104C2×C4×D4C2311yescharacteristic
105C2×#117—22no—
106C2×#117—22no—
107C2×#117C2314yes—
108C22×C42C2312yes—
109#492—24no—
110#486—44no—
111#477—48no—
112C4×SD16—48no—
113#431—44no—
114C2×#107C2318yes—
115C2×#106C2314yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2314yes—
118C2×#103C2318yes—
119C2×#103C2314yes—
120C2×#102C2318yes—
121C2×#102C2314yes—
122C2×C22⋊C8C2314yes—
123C2×C22⋊C8C2314yes—
124C2×C4×C8C2312yes—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2312yes—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2312yes—
129#399—24no—
130#398—24no—
131#396—24no—
132#394—24no—
133#394—24no—
134#393—24no—
135#391—22no—
136#391C2312yes—
137#390—22no—
138#390—21no—
139#390—21no—
140#390—22no—
141#390—22no—
142#390C2312yes—
143#390C2314yes—
144#389—22no—
145#389—21no—
146#389—21no—
147#389—22no—
148#389—24no—
149#389C2314yes—
150#388—24no—
151#388—24no—
152#388—24no—
153#387—24no—
154#384—22no—
155#384—21no—
156#384—21no—
157#384—21no—
158#384—21no—
159#384C2311yescharacteristic
160#384C2311yescharacteristic
161#384C2312yes—
162#384C2312yes—
163#384C2312yes—
164#384C2312yes—
165#382C2314yes—
166#380—22no—
167#380—22no—
168#380—22no—
169#380—22no—
170#380—22no—
171#380C2314yes—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4D411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C4⋊C4C2311yescharacteristic
180C4×C22⋊C4D412yes—
181C4×C22⋊C4D412yes—
182C4×C22⋊C4C2312yes—
183C43C2311yescharacteristic
184#340C2318yes—
185#340C2318yes—
186#340C2318yes—
187#336C2312yes—
188#336C2314yes—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8C2412yes—
192C22×D4—24no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4C2×D411yescharacteristic
196C22×D4C2412yes—
197C23×C4—22no—
198C23×C4C2×D412yes—
199C23×C4C2×D412yes—
200C23×C4C2412yes—
201C2×SD16—28no—
202C2×SD16—28no—
203C2×SD16—44no—
204C22×C8C2414yes—
205C22×C8C2412yes—
206C22×C8C2412yes—
207C4⋊Q8—24no—
208C4⋊Q8—22no—
209C4⋊Q8—24no—
210C4⋊Q8—28no—
211C4⋊Q8—44no—
212C4⋊Q8—82no—
213C4⋊Q8—82no—
214C4⋊Q8—82no—
215C4⋊Q8#4114yes—
216#127—22no—
217#127—22no—
218#125—48no—
219#124—416no—
220C22⋊Q8—28no—
221C22⋊Q8—44no—
222C22⋊Q8—48no—
223C22⋊Q8—84no—
224#121—48no—
225#121—48no—
226#121—84no—
227#121—84no—
228C4×Q8—28no—
229C4×Q8—28no—
230C4×Q8—44no—
231C4×Q8—48no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—28no—
236C4×D4—44no—
237C4×D4—44no—
238#117—48no—
239#117—48no—
240#117—48no—
241C2×C4⋊C4—21no—
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—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2412yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D412yes—
279C2×C22⋊C4C2×D412yes—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2412yes—
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295#107—48no—
296#106—44no—
297C4⋊C8—24no—
298C4⋊C8—44no—
299#103—44no—
300#103—48no—
301#102—44no—
302#102—48no—
303C22⋊C8—28no—
304C22⋊C8—44no—
305C4×C8—24no—
306C4×C8—22no—
307C4×C8—22no—
308C4×C8—24no—
309C4×C8—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—22no—
314#95—22no—
315#95—24no—
316#95C2×D412yes—
317#95C2×D412yes—
318#95C2×D412yes—
319C24—22no—
320C24C22×D412yes—
321C2×Q8—22no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—82no—
329C2×Q8—82no—
330C2×Q8C2×#4114yes—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—48no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—84no—
346C2×D4—84no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2511yescharacteristic
375SD16—416no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—24no—
380C2×C8—22no—
381C2×C8—24no—
382C2×C8—22no—
383C2×C8—24no—
384C2×C82−1+414yes—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—28no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—84no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—84no—
423C22⋊C4—84no—
424C22⋊C4—84no—
425C42—24no—
426C42—24no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—48no—
433C42—48no—
434C42—44no—
435C42#13514yes—
436C42#13714yes—
437C42C2×#4112yes—
438C42H(5,𝔽2)14yes—
439C42H(5,𝔽2)12yes—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23C23×D411yescharacteristic
455Q8—28no—
456Q8—44no—
457Q8—88no—
458Q8—164no—
459D4—44no—
460D4—44no—
461D4—44no—
462D4—84no—
463D4—84no—
464D4—168no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—48no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—48no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—88no—
500C2×C4#53512yes—
501C2×C4#53512yes—
502C2×C4#54612yes—
503C2×C4#54612yes—
504C2×C4C2×#13512yes—
505C2×C4C2×#13512yes—
506C2×C4C2×#13512yes—
507C2×C4C2×#13712yes—
508C2×C4C2×#13712yes—
509C2×C4C2×#13712yes—
510C2×C4#57612yes—
511C2×C4#57712yes—
512C2×C4C22×#4111yescharacteristic
513C2×C4C2×H(5,𝔽2)12yes—
514C2×C4C2×H(5,𝔽2)11yescharacteristic
515C2×C4C2×2−1+412yes—
516C8—24no—
517C8—42no—
518C8—44no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22C2×#53511yescharacteristic
534C22C2×#54611yescharacteristic
535C22#349911yescharacteristic
536C22C22×#13511yescharacteristic
537C22C22×#13711yescharacteristic
538C22C2×#57611yescharacteristic
539C22C2×#57711yescharacteristic
540C4—22no—
541C4—22no—
542C4—22no—
543C4—42no—
544C4—44no—
545C4—42no—
546C4—42no—
547C4—82no—
548C4—82no—
549C4—84no—
550C4—84no—
551C4#321014yes—
552C4#330514yes—
553C4#332914yes—
554C4#352012yes—
555C4#354812yes—
556C2—82no—
557C2—162no—
558C2#3895111yescharacteristic
559C2C2×#321011yescharacteristic
560C2#6082211yescharacteristic
561C2C2×#330511yescharacteristic
562C2C2×#332911yescharacteristic
563C2#6247011yescharacteristic
564C2#6248011yescharacteristic
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. #36134C2semidirect
3. #33289C2semidirect
4. #31931C2semidirect
5. #29842C2semidirect
6. #28716C2semidirect
7. #28511C2semidirect
8. #27949C2semidirect
9. #25598C2semidirect
10. #24814C2semidirect
11. #24121C2semidirect
12. #21864C2semidirect
13. C4×#492C2semidirect
14. #21264C2semidirect

Maximal subgroups

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

subgroupindex
#361872
#361342
#332892
#319312
#298422
#287162
#285112
#279492
#255982
#248142
#241212
#218642
C4×#4922
#212642