Finite Group Databasebeta

← #6458404

Subgroups — #6458404

Order 512.

1752 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35257C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#27971C214yesmaximal, complemented
4#27970C214yesmaximal, complemented
5#27968C212yesmaximal, complemented
6#27965C212yesmaximal, complemented
7#27963C212yesmaximal, complemented
8#27960C212yesmaximal, complemented
9#23747C211yesmaximal, characteristic, complemented
10#23745C211yesmaximal, characteristic, complemented
11#23744C211yesmaximal, characteristic, complemented
12C2×#2099C211yesmaximal, characteristic, complemented
13#21326C212yesmaximal, complemented
14#20763C212yesmaximal, complemented
15#20703C212yesmaximal
16#20703C212yesmaximal
17#20378C211yesmaximal, characteristic
18C2×#544C2211yescharacteristic
19C2×#520C2212yes—
20C2×C4×#41C2211yescharacteristic
21C2×#408C2212yes—
22C2×#405C2212yes—
23#2857C2212yes—
24#2851C2212yes—
25#2629C2212yes—
26#2628C2212yes—
27#2454—21no—
28#2453—21no—
29#2452—21no—
30#2451—21no—
31#2450—22no—
32#2449—22no—
33Q8×C4⋊C4—21no—
34Q8×C4⋊C4—21no—
35D4×C4⋊C4—22no—
36D4×C4⋊C4—22no—
37D4×C4⋊C4—21no—
38D4×C4⋊C4—21no—
39#2317C2211yescharacteristic
40#2317C2212yes—
41#2317C2214yes—
42#2311C2211yescharacteristic
43#2311C2211yescharacteristic
44#2296C2211yescharacteristic
45C2×#389C2211yescharacteristic
46#2285C2212yes—
47#2285C2211yescharacteristic
48C2×#384C2212yes—
49C2×#384C2211yescharacteristic
50#2101C2212yes—
51#2101C2214yes—
52#2101C2212yes—
53#2100C2211yescharacteristic
54#2100C2212yes—
55#2100C2211yescharacteristic
56#2100C2212yes—
57#2100C2212yes—
58#2099C2211yescharacteristic
59#2099C2212yes—
60#2099C2212yes—
61#2099C2212yes—
62#2099C2212yes—
63#2099C2211yescharacteristic
64#2099C2212yes—
65#2099C2214yes—
66#1993C2214yes—
67#1992C2214yes—
68#1991C2214yes—
69#1990C2214yes—
70#1836C2214yes—
71#1836C2214yes—
72#1835C2214yes—
73#1834C2214yes—
74#1825C2214yes—
75#1825C2214yes—
76#1825C2214yes—
77#1825C2214yes—
78#1824C2214yes—
79#1824C2214yes—
80#1823C2214yes—
81#1823C2214yes—
82#1750C2214yes—
83#1749C2212yes—
84C4×C22⋊C8C2212yes—
85#1736C2212yes—
86#1736C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1735C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98#544—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×#125—21no—
102C2×#125C2312yes—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105#520—216no—
106C4×#41—28no—
107C2×C4×Q8—22no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8D411yescharacteristic
110C2×C4×D4—24no—
111C2×C4×D4—22no—
112C2×C4×D4D412yes—
113C2×C4×D4D412yes—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2314yes—
119C2×#117C2312yes—
120C2×#117C2312yes—
121C2×#117C2311yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#408—28no—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136#405—28no—
137C2×#97C2314yes—
138C2×C4×C8C2314yes—
139#400—22no—
140#399—22no—
141#398—22no—
142#398—22no—
143#396—22no—
144#396—22no—
145#395—22no—
146#393—22no—
147#391—21no—
148#391—21no—
149#391—22no—
150#390—21no—
151#390—21no—
152#389—24no—
153#389—24no—
154#389—22no—
155#389—22no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389C2311yescharacteristic
162#389C2312yes—
163#389C2311yescharacteristic
164#389C2312yes—
165#389C2311yescharacteristic
166#389C2311yescharacteristic
167#387—22no—
168#387—22no—
169#386—22no—
170#386—22no—
171#385—22no—
172#385—22no—
173#385—21no—
174#385—21no—
175#385—24no—
176#385—24no—
177#385—24no—
178#385—22no—
179#384—21no—
180#384—21no—
181#384—22no—
182#384—22no—
183#384C2314yes—
184#384C2312yes—
185#384C2312yes—
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2311yescharacteristic
189#382C2314yes—
190#382C2312yes—
191#382C2312yes—
192#380—22no—
193#380—21no—
194#380—22no—
195#380—21no—
196#380—22no—
197#380C2312yes—
198#380C2312yes—
199#380C2312yes—
200#380C2312yes—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C2×#95—21no—
208#340C2312yes—
209#340C2312yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2312yes—
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233C2×#41—24no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41#4118yes—
237C22×Q8—22no—
238C22×Q8C2×D411yescharacteristic
239C22×D4C2×D412yes—
240C22×D4C2×D411yescharacteristic
241C23×C4—24no—
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C2×M16—24no—
252C2×M16—24no—
253C2×M16—24no—
254C22×C8—24no—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C4⋊Q8—28no—
261C4⋊Q8—44no—
262#125—28no—
263#125—44no—
264C22⋊Q8—48no—
265C22⋊Q8—84no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—48no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—28no—
272C4×D4—28no—
273C4×D4—416no—
274C4×D4—48no—
275#117—28no—
276#117—216no—
277#117—28no—
278#117—24no—
279#117—22no—
280#117—22no—
281#117#4118yes—
282C2×C4⋊C4—216no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4—84no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—24no—
342C2×C42—28no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354C22⋊C8—28no—
355C22⋊C8—28no—
356#97—28no—
357C4×C8—28no—
358#95—22no—
359#95—22no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374C24C22×D412yes—
375C24C22×D411yescharacteristic
376#41—216no—
377#41—216no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—44no—
381C2×Q8—82no—
382C2×Q8—82no—
383C2×Q8#12414yes—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4#12418yes—
389C2×D4#12414yes—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—48no—
417C22×C4—44no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—82no—
422C22×C4—84no—
423C22×C4—82no—
424C22×C4—84no—
425C22×C4—81no—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C2×#4118yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C42−1+412yes—
438C22×C42−1+412yes—
439C22×C4C2511yescharacteristic
440M16—48no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—48no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—28no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—82no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4#12314yes—
491C4⋊C4#12314yes—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—42no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4#12318yes—
503C42—24no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—42no—
510C42—48no—
511C42—44no—
512C42—42no—
513C42C2×#4118yes—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23C2×#12312yes—
525C23C2×#12414yes—
526C23C2×#12412yes—
527C23#53612yes—
528C23C23×D411yescharacteristic
529Q8—28no—
530Q8—164no—
531D4—216no—
532D4—28no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—84no—
583C2×C4—162no—
584C2×C4—164no—
585C2×C4—168no—
586C2×C4C2×#12314yes—
587C2×C4C2×#12314yes—
588C2×C4C2×#12312yes—
589C2×C4C2×#12414yes—
590C2×C4C2×#12412yes—
591C2×C4#53612yes—
592C2×C4#55218yes—
593C2×C4#55218yes—
594C2×C4#57612yes—
595C2×C4#57712yes—
596C2×C4C22×#4112yes—
597C2×C4C22×#4112yes—
598C2×C4C2×2−1+411yescharacteristic
599C2×C4C2×2−1+411yescharacteristic
600C8—42no—
601C8—42no—
602C8—44no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22#250314yes—
617C22C22×#12311yescharacteristic
618C22C22×#12411yescharacteristic
619C22C2×#53611yescharacteristic
620C22C2×#57611yescharacteristic
621C22C2×#57711yescharacteristic
622C22#358611yescharacteristic
623C22#358611yescharacteristic
624C4—22no—
625C4—24no—
626C4—22no—
627C4—24no—
628C4—24no—
629C4—41no—
630C4—41no—
631C4—82no—
632C4—81no—
633C4—81no—
634C4—84no—
635C4—82no—
636C4—82no—
637C4—81no—
638C4—82no—
639C4—81no—
640C4#250314yes—
641C4C2×#55212yes—
642C4C2×#55212yes—
643C2—24no—
644C2—44no—
645C2C2×#250311yescharacteristic
646C2#6052111yescharacteristic
647C2#6052211yescharacteristic
648C2#6062211yescharacteristic
649C2#6062711yescharacteristic
650C2#6088811yescharacteristic
651C2#6088911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27971C2semidirect
4. #27970C2semidirect
5. #27968C2semidirect
6. #27965C2semidirect
7. #27963C2semidirect
8. #27960C2semidirect
9. #23747C2semidirect
10. #23745C2semidirect
11. #23744C2semidirect
12. C2×#2099C2semidirect
13. #21326C2semidirect
14. #20763C2semidirect

Maximal subgroups

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

subgroupindex
#352572
#341832
#279712
#279702
#279682
#279652
#279632
#279602
#237472
#237452
#237442
C2×#20992
#213262
#207632
#207032
#203782