Finite Group Databasebeta

← #6541986

Subgroups — #6541986

Order 512.

1471 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2#34337C211yesmaximal, characteristic
3#29989C212yesmaximal
4#29984C211yesmaximal, characteristic
5#29259C212yesmaximal
6#29255C211yesmaximal, characteristic
7#28518C212yesmaximal
8#28511C211yesmaximal, characteristic
9#26474C214yesmaximal
10#26465C212yesmaximal
11#26055C214yesmaximal
12#26047C212yesmaximal
13#23782C212yesmaximal
14#23776C211yesmaximal, characteristic
15#22736C211yesmaximal, characteristic
16#22728C212yesmaximal
17#22547C212yesmaximal
18C2×#557C2211yescharacteristic
19C2×#479C2214yes—
20C2×#479C2214yes—
21C2×#477C2214yes—
22C2×#446C2212yes—
23#2868C2211yescharacteristic
24#2859C2212yes—
25#2582—21no—
26#2567—21no—
27#2552C2212yes—
28#2550C2214yes—
29#2507—22no—
30#2451—22no—
31#2370—21no—
32#2370—22no—
33#2369—21no—
34#2365C2211yescharacteristic
35#2365C2212yes—
36#2355C2212yes—
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4C2211yescharacteristic
40Q8×C4⋊C4C2211yescharacteristic
41Q8×C4⋊C4C2212yes—
42#2330—21no—
43#2330—22no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48#2295C2211yescharacteristic
49#2295C2212yes—
50#2276—21no—
51C42×Q8C2211yescharacteristic
52#2100C2212yes—
53#2099C2211yescharacteristic
54#2096C2212yes—
55#2096C2211yescharacteristic
56#2095C2212yes—
57#2095C2212yes—
58#2095C2212yes—
59#2088C2212yes—
60#2086C2214yes—
61#2066C2214yes—
62#2066C2212yes—
63#2059C2214yes—
64#2058C2211yescharacteristic
65#2057C2212yes—
66#2026C2214yes—
67#2024C2212yes—
68#2024C2212yes—
69#2024C2211yescharacteristic
70#1981C2212yes—
71#1981C2214yes—
72#1960C2214yes—
73#1956C2212yes—
74#1949C2214yes—
75#1948C2212yes—
76#1947C2212yes—
77#1947C2214yes—
78#1940C2214yes—
79#1939C2212yes—
80#1865C2214yes—
81#1863C2214yes—
82#1863C2214yes—
83#1847C2214yes—
84#1845C2212yes—
85#1831C2211yescharacteristic
86#1831C2212yes—
87#1805C2214yes—
88#1803C2212yes—
89#1776C2214yes—
90C4×#106C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
94C4×#103C2212yes—
95#557—24no—
96#557Q818yescomplemented
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#125—21no—
104C2×#125—22no—
105C2×#125C2314yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118#479—48no—
119#479—48no—
120#477—48no—
121#446—28no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106C2312yes—
126C2×#106C2312yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141#398—24no—
142#398—24no—
143#395—24no—
144#391—22no—
145#391—22no—
146#391—21no—
147#391—24no—
148#391C2312yes—
149#391C2311yescharacteristic
150#391C2311yescharacteristic
151#391C2312yes—
152#389—22no—
153#389—21no—
154#389—22no—
155#389—21no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—24no—
171#389C2314yes—
172#389C2312yes—
173#389C2311yescharacteristic
174#389C2311yescharacteristic
175#389C2312yes—
176#387—24no—
177#387—22no—
178#387—21no—
179#387—22no—
180#387—21no—
181#387—22no—
182#387—22no—
183#387—22no—
184#384—22no—
185#384—21no—
186#384—22no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382C2314yes—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218#376—22no—
219#376—21no—
220C43C2312yes—
221C43C2311yescharacteristic
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2411yescharacteristic
238C22×Q8C2412yes—
239C22×C8C2412yes—
240C22×C8C2414yes—
241C22×C8C2412yes—
242C4⋊Q8—24no—
243C4⋊Q8—22no—
244C4⋊Q8—24no—
245C4⋊Q8—44no—
246C4⋊Q8—44no—
247C4⋊Q8—82no—
248C4⋊Q8C2×Q814yes—
249C4⋊Q8C2×Q818yes—
250#125—216no—
251#125—28no—
252#125—44no—
253#125—44no—
254C4×Q8—28no—
255C4×Q8—24no—
256C4×Q8—22no—
257C4×Q8—24no—
258C4×Q8—42no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—44no—
264C4×Q8—84no—
265C4×Q8C2×Q814yes—
266C4×Q8C2×Q818yes—
267C4×Q8C2×Q814yes—
268C4×Q8#4114yes—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C42—22no—
311C2×C42—21no—
312C2×C42—22no—
313C2×C42—21no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2412yes—
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330C2×C42C2412yes—
331C2×C42C2411yescharacteristic
332#107—28no—
333#107—44no—
334#107—44no—
335#106—24no—
336#106—24no—
337#106—44no—
338C4⋊C8—24no—
339C4⋊C8—28no—
340C4⋊C8—24no—
341C4⋊C8—28no—
342#103—28no—
343#103—24no—
344#103—24no—
345#103—28no—
346#103—44no—
347#103—44no—
348#103—44no—
349#97—28no—
350C4×C8—24no—
351#95—21no—
352#95—22no—
353#95—21no—
354#95—22no—
355#95—21no—
356#95—22no—
357#95—21no—
358#95—21no—
359#95—22no—
360#95—24no—
361#95—21no—
362#95—22no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×Q8C22⋊Q814yes—
388C2×Q8C22×Q812yes—
389C2×Q8C22×Q814yes—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C2511yescharacteristic
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8H(5,𝔽2)14yes—
419C4⋊C4—24no—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—28no—
424C4⋊C4—24no—
425C4⋊C4—28no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—82no—
449C4⋊C4—81no—
450C4⋊C4—82no—
451C4⋊C4—81no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4C22⋊Q818yes—
461C4⋊C4C22⋊Q814yes—
462C4⋊C4C22×Q812yes—
463C4⋊C42−1+414yes—
464C42—28no—
465C42—24no—
466C42—24no—
467C42—22no—
468C42—22no—
469C42—24no—
470C42—24no—
471C42—22no—
472C42—22no—
473C42—42no—
474C42—42no—
475C42—41no—
476C42—44no—
477C42—44no—
478C42—42no—
479C42—82no—
480C42—82no—
481C42—82no—
482C42—84no—
483C42#13714yes—
484C42C22×Q814yes—
485C42C22×Q812yes—
486C42C2×#4112yes—
487C42C2×#4114yes—
488C23C23×D411yescharacteristic
489Q8—28no—
490Q8—28no—
491Q8—24no—
492Q8—48no—
493Q8—44no—
494Q8—42no—
495Q8—164no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4C2×C22⋊Q814yes—
545C2×C4C2×C22⋊Q812yes—
546C2×C4C2×C22⋊Q814yes—
547C2×C4C2×C22⋊Q812yes—
548C2×C4C2×#13712yes—
549C2×C4C2×#13712yes—
550C2×C4C2×#13712yes—
551C2×C4#57512yes—
552C2×C4#57812yes—
553C2×C4C23×Q811yescharacteristic
554C2×C4C22×#4111yescharacteristic
555C2×C4C2×H(5,𝔽2)12yes—
556C2×C4C2×2−1+412yes—
557C8—24no—
558C8—42no—
559C8—44no—
560C22C22×C22⋊Q811yescharacteristic
561C22#348611yescharacteristic
562C22#349811yescharacteristic
563C22C22×#13711yescharacteristic
564C22C2×#57511yescharacteristic
565C22C2×#57811yescharacteristic
566C22#358511yescharacteristic
567C4—24no—
568C4—24no—
569C4—22no—
570C4—22no—
571C4—22no—
572C4—22no—
573C4—42no—
574C4—44no—
575C4—44no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—41no—
580C4—81no—
581C4—82no—
582C4—81no—
583C4—82no—
584C4—84no—
585C4#307814yes—
586C4#350312yes—
587C4#352212yes—
588C2#3574811yescharacteristic
589C2C2×#307811yescharacteristic
590C2#6049811yescharacteristic
591C2#6182611yescharacteristic
592C2#6184211yescharacteristic
593C2#6242411yescharacteristic
594C2#6245111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
96. #557Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#343372
#299892
#299842
#292592
#292552
#285182
#285112
#264742
#264652
#260552
#260472
#237822
#237762
#227362
#227282
#225472