Finite Group Databasebeta

← #224240

Subgroups — #224240

Order 512.

919 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36559C211yesmaximal, characteristic
2#22823C211yesmaximal, characteristic, complemented
3#22821C211yesmaximal, characteristic, complemented
4#9906C211yesmaximal, characteristic, complemented
5#9903C211yesmaximal, characteristic, complemented
6#9901C211yesmaximal, characteristic, complemented
7#9901C211yesmaximal, characteristic, complemented
8#9901C211yesmaximal, characteristic, complemented
9#9898C211yesmaximal, characteristic, complemented
10#8592C211yesmaximal, characteristic
11#8588C211yesmaximal, characteristic
12#8588C211yesmaximal, characteristic
13#8234C211yesmaximal, characteristic
14#8234C211yesmaximal, characteristic
15C4×#331C211yesmaximal, characteristic
16C2×#554C412yes—
17#2868C2211yescharacteristic
18#2859C2211yescharacteristic
19#2859C2211yescharacteristic
20#2566C412yes—
21#2566C412yes—
22#2563C412yes—
23#2538—21no—
24#2538—21no—
25#2452—21no—
26#2452—21no—
27#2452—21no—
28#2452—21no—
29D4×C4⋊C4—21no—
30D4×C4⋊C4—21no—
31C4×#127C2211yescharacteristic
32C4×#127C2211yescharacteristic
33C4×#127C2211yescharacteristic
34C42×Q8C2211yescharacteristic
35#1985—21no—
36#1984—21no—
37#1917—21no—
38#1916—21no—
39#1916—21no—
40#1916—21no—
41#1842C2211yescharacteristic
42#1842C2211yescharacteristic
43#1831C2211yescharacteristic
44#1831C2211yescharacteristic
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1771—21no—
48C8×C4⋊C4—21no—
49#1767C2211yescharacteristic
50#1767C2211yescharacteristic
51C4×C4⋊C8C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#331C2211yescharacteristic
56C2×#331C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×#331C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71#554—48no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8C2×C412yes—
75C2×#127C2311yescharacteristic
76C2×#125C2×C412yes—
77C2×#125C2×C412yes—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—22no—
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8D411yescharacteristic
83C2×C4×Q8D411yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4—21no—
90C2×C4×D4C2×C412yes—
91C2×C4×D4C2×C412yes—
92C2×C4×D4C2×C412yes—
93C22×C4⋊C4—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#97—21no—
107C2×#97—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112#396—22no—
113#396—22no—
114#396—22no—
115#396—22no—
116#393—22no—
117#393—22no—
118#390—21no—
119#390—21no—
120#390—21no—
121#390—21no—
122#390—21no—
123#390—21no—
124#390C2×C412yes—
125#390C2×C412yes—
126#390C2×C412yes—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—21no—
135#385—22no—
136#385—22no—
137#385—22no—
138#385—22no—
139#384—21no—
140#384—21no—
141#384C2×C412yes—
142#384C2311yescharacteristic
143#384C2311yescharacteristic
144#384C2311yescharacteristic
145#382C2×C412yes—
146#382C2×C412yes—
147#380—22no—
148#380—22no—
149C4×C4⋊C4C2×C411yescharacteristic
150C4×C4⋊C4C2×C411yescharacteristic
151C4×C4⋊C4C2×C411yescharacteristic
152C4×C4⋊C4D411yescharacteristic
153C4×C4⋊C4D411yescharacteristic
154C4×C4⋊C4D411yescharacteristic
155C4×C4⋊C4D411yescharacteristic
156C4×C4⋊C4D411yescharacteristic
157C4×C4⋊C4D411yescharacteristic
158C43C2311yescharacteristic
159C43C2311yescharacteristic
160C43C2311yescharacteristic
161#336—21no—
162#336—21no—
163#336—21no—
164#336—21no—
165#336—21no—
166#336—21no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#331—22no—
176#331—22no—
177#331—22no—
178#331Q814yescomplemented
179C22×Q8—22no—
180C22×Q8C22⋊C411yescharacteristic
181C22×D4—21no—
182C22×D4—21no—
183C22×D4C22×C412yes—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C4⋊Q8—44no—
196C4⋊Q8—81no—
197C4⋊Q8—81no—
198C4⋊Q8—81no—
199C4⋊Q8—81no—
200#127—21no—
201#127—21no—
202#127C2×Q812yes—
203#127#4112yes—
204#125—44no—
205#125—44no—
206C22⋊Q8—48no—
207C22⋊Q8—48no—
208C22⋊Q8—48no—
209C4×Q8—24no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4—44no—
217C4×D4—44no—
218C4×D4—44no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—81no—
222C4×D4—81no—
223C4×D4—81no—
224C4×D4—81no—
225C4×D4—81no—
226C4×D4—81no—
227C4×D4—81no—
228C4×D4—81no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C22⋊C411yescharacteristic
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C22×C412yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2411yescharacteristic
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—42no—
298C4⋊C8—42no—
299C4⋊C8—81no—
300C4⋊C8—81no—
301C4⋊C8C2×Q812yes—
302C4⋊C8C2×Q812yes—
303#97—42no—
304#97—81no—
305C4×C8—42no—
306C4×C8—81no—
307C4×C8—81no—
308C4×C8—81no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95C22⋊C411yescharacteristic
314#95C22⋊C411yescharacteristic
315#95C22⋊C411yescharacteristic
316#95C22⋊C411yescharacteristic
317#95C22⋊C411yescharacteristic
318#95C22⋊C411yescharacteristic
319C24—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—41no—
323C2×Q8—44no—
324C2×Q8—82no—
325C2×Q8—82no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4C4×Q814yes—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4#10014yes—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—82no—
430C22⋊C4—82no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42#13512yes—
462C42#13512yes—
463C42C22×Q811yescharacteristic
464C42C2×#4111yescharacteristic
465C422−1+411yescharacteristic
466C422−1+411yescharacteristic
467C23—24no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23C22×C22⋊C411yescharacteristic
475Q8—44no—
476Q8—164no—
477D4—44no—
478D4—44no—
479D4—42no—
480D4—42no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—42no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—88no—
530C2×C4—164no—
531C2×C4—164no—
532C2×C4C2×#10012yes—
533C2×C4C2×#10012yes—
534C2×C4C2×#10012yes—
535C2×C4#41612yes—
536C2×C4#41612yes—
537C2×C4C2×C4×Q811yescharacteristic
538C2×C4#51911yescharacteristic
539C2×C4#54811yescharacteristic
540C2×C4#54811yescharacteristic
541C2×C4D4×Q811yescharacteristic
542C2×C4D4×Q811yescharacteristic
543C2×C4C2×#13511yescharacteristic
544C2×C4C2×#13511yescharacteristic
545C2×C4#57511yescharacteristic
546C2×C4#57511yescharacteristic
547C2×C4#57611yescharacteristic
548C2×C4#57611yescharacteristic
549C2×C4#57711yescharacteristic
550C2×C4#57711yescharacteristic
551C8—42no—
552C8—42no—
553C8—81no—
554C8—81no—
555C8—81no—
556C8—81no—
557C8—81no—
558C8—81no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22Q8×C22⋊C411yescharacteristic
567C22C22×#10011yescharacteristic
568C22#288911yescharacteristic
569C22C2×#41611yescharacteristic
570C22#289611yescharacteristic
571C22#289711yescharacteristic
572C22#289711yescharacteristic
573C4—22no—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—21no—
579C4—21no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4#336011yescharacteristic
591C4#336111yescharacteristic
592C4#337011yescharacteristic
593C4#337611yescharacteristic
594C4#338011yescharacteristic
595C4#338111yescharacteristic
596C4#338811yescharacteristic
597C4#338911yescharacteristic
598C2—84no—
599C2#1771411yescharacteristic
600C2#1772711yescharacteristic
601C2#1776911yescharacteristic
602C2#1777011yescharacteristic
603C2Q8×#10011yescharacteristic
604C2#2251911yescharacteristic
605C2#2252611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22823C2semidirect
3. #22821C2semidirect
4. #9906C2semidirect
5. #9903C2semidirect
6. #9901C2semidirect
7. #9901C2semidirect
8. #9901C2semidirect
9. #9898C2semidirect
178. #331Q8semidirect

Maximal subgroups

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

subgroupindex
#365592
#228232
#228212
#99062
#99032
#99012
#98982
#85922
#85882
#82342
C4×#3312