Finite Group Databasebeta

← #6382084

Subgroups — #6382084

Order 512.

2040 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic, complemented
3#34186C211yesmaximal, characteristic
4#23071C214yesmaximal, complemented
5#23067C212yesmaximal, complemented
6#23066C212yesmaximal, complemented
7#23065C214yesmaximal, complemented
8#23061C212yesmaximal, complemented
9#23060C212yesmaximal, complemented
10#21318C211yesmaximal, characteristic, complemented
11#21170C211yesmaximal, characteristic, complemented
12#21091C212yesmaximal
13#21063C212yesmaximal
14#21035C212yesmaximal, complemented
15#20453C212yesmaximal, complemented
16#20437C211yesmaximal, characteristic, complemented
17C2×C4×#102C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#3077C4116yescomplemented
21C2×#478C2214yes—
22C2×#476C2212yes—
23C2×#474C2212yes—
24C2×#428C2212yes—
25C2×#428C2212yes—
26C2×#423C2211yescharacteristic
27C2×#418C2212yes—
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C2×#405C2212yes—
31C22×C4×C8C2211yescharacteristic
32#2370C2214yescomplemented
33#2369C2212yescomplemented
34#2365C2212yescomplemented
35#2357—22no—
36#2357C2214yes—
37#2355—21no—
38#2355—22no—
39#2354—21no—
40#2354—21no—
41#2354C2212yes—
42#2354C2212yes—
43Q8×C4⋊C4—21no—
44#2329—22no—
45#2317C2212yes—
46#2312—22no—
47#2309C2212yes—
48C4×C22⋊Q8—22no—
49#2285C2211yescharacteristic
50#2285C2211yescharacteristic
51#2280—22no—
52#2274C2212yes—
53#2274C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#1991C2212yescomplemented
56#1991C2212yescomplemented
57#1990C2211yescharacteristic, complemented
58#1990C2211yescharacteristic, complemented
59#1990C2211yescharacteristic, complemented
60#1990C2211yescharacteristic, complemented
61#1949C2214yescomplemented
62#1948C2212yescomplemented
63#1947C2212yescomplemented
64#1936C2214yes—
65#1936C2214yes—
66#1934C2214yes—
67#1934C2214yes—
68#1926C2214yes—
69#1926C2214yes—
70#1925C2214yes—
71#1925C2214yes—
72#1921C2214yescomplemented
73#1920C2214yescomplemented
74#1919C2214yescomplemented
75#1918C2214yescomplemented
76#1776C2214yescomplemented
77#1776C2214yescomplemented
78C4×#107C2214yescomplemented
79C4×#106C2214yescomplemented
80#1770C2212yescomplemented
81#1770C2212yescomplemented
82C8×C4⋊C4C2212yescomplemented
83C8×C4⋊C4C2212yescomplemented
84C4×#102C2212yes—
85C4×#102C2212yes—
86C4×#102C2212yes—
87C4×#102C2212yes—
88#1736C2212yes—
89#1735C2212yes—
90#1735C2212yes—
91C2×#340C2211yescharacteristic
92C2×#340C2211yescharacteristic
93#1729C2212yes—
94C23×D4C2311yescharacteristic
95C24×C4—21no—
96C22×M16C2312yes—
97C23×C8C2311yescharacteristic
98C23×C8C2311yescharacteristic
99#533—24no—
100#533C2×C418yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—24no—
112C2×C4×D4C2×C418yes—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×#117—22no—
116C2×#117—22no—
117C2×#117C2312yes—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126#478C2×C4132yescomplemented
127#476C2×C4116yescomplemented
128#474C2×C4116yescomplemented
129#428—28no—
130#428C2×C4116yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135#423C2×C418yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138#418C2×C4116yes—
139C2×#102—24no—
140C2×#102—24no—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147#405—28no—
148C2×#97C2314yes—
149C2×C4×C8—24no—
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2314yes—
155#391—22no—
156#391—24no—
157#390—22no—
158#390—22no—
159#390C2314yes—
160#390C2314yes—
161#389—24no—
162#389—22no—
163#389C2314yes—
164#389C2314yes—
165#388—24no—
166#388—24no—
167#388—24no—
168#388—24no—
169#387—24no—
170#387—24no—
171#387—24no—
172#385—24no—
173#384—22no—
174#384D411yescharacteristic
175#384D411yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#382—22no—
179#382D412yes—
180#382C2312yes—
181#381—24no—
182#380—21no—
183#380—21no—
184#380—22no—
185#380—22no—
186#380D412yes—
187#380D412yes—
188#379—24no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—24no—
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2314yes—
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C22⋊C4—24no—
208C4×C22⋊C4—24no—
209C4×C22⋊C4—24no—
210C2×#95—21no—
211#340C2312yes—
212#340C2312yes—
213#340C2314yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#340C2314yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#336C2314yes—
226#336C2314yes—
227C25—21no—
228C22×Q8—21no—
229C22×Q8—21no—
230C22×D4—21no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4C22×C414yes—
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4—28no—
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—82no—
245C23×C4—82no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×M16—24no—
250C2×M16—24no—
251C2×M16C22×C418yes—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8#4114yes—
258C22×C8C2411yescharacteristic
259C22×C8C2411yescharacteristic
260C22×C8C2414yes—
261C22×C8C2411yescharacteristic
262C22×C8C2411yescharacteristic
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8C22×C414yes—
266C4⋊Q8C22×C414yes—
267#125—24no—
268#125C22×C418yes—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271C4×Q8—28no—
272C4×Q8—28no—
273C4×Q8—84no—
274C4×D4—28no—
275C4×D4—28no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4C22×C418yes—
279C4×D4C22×C418yes—
280#117—24no—
281#117—24no—
282#117—24no—
283#117—24no—
284#117—24no—
285#117—84no—
286#117—84no—
287#117C22×C418yes—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4C22×C414yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4#4114yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4C2×D414yes—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—82no—
339C2×C42C22×C414yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42#4114yes—
343C2×C42C2414yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346#107—28no—
347#107C22×C4116yes—
348#106—28no—
349#106C22×C4116yes—
350C4⋊C8C22×C418yes—
351C4⋊C8C22×C418yes—
352#102—24no—
353#102—24no—
354#102—24no—
355#102—24no—
356#102C22×C4116yes—
357#102C22×C4116yes—
358#97—28no—
359C4×C8—28no—
360C4×C8—22no—
361C4×C8—22no—
362C4×C8—22no—
363C4×C8—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95C2×D414yes—
375#95C2×D414yes—
376C24—22no—
377C24—24no—
378C24—42no—
379C24—42no—
380C24—81no—
381C24—81no—
382C24—81no—
383C24—81no—
384C24C22×D411yescharacteristic
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—81no—
390C2×Q8—81no—
391C2×D4—24no—
392C2×D4—28no—
393C2×D4—28no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4C23×C414yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—816no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4C4×D414yes—
432C22×C4C4×D414yes—
433C22×C4C23×C412yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C42−1+412yes—
445C22×C42−1+412yes—
446C22×C4C2511yescharacteristic
447M16—216no—
448C2×C8—28no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—44no—
462C2×C8C23×C418yes—
463C2×C8C2×#4112yes—
464C2×C8C2×#4112yes—
465C4⋊C4—24no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4C23×C414yes—
495C4⋊C4C23×C414yes—
496C4⋊C4C23×C414yes—
497C4⋊C4C2×#4114yes—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—44no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C42—28no—
508C42—24no—
509C42—24no—
510C42—44no—
511C42—44no—
512C42—84no—
513C42—82no—
514C42—84no—
515C42—84no—
516C42—84no—
517C42C4×D414yes—
518C42C4×D414yes—
519C42C23×C412yes—
520C42C23×C412yes—
521C42C2×#4112yes—
522C42C2×#4112yes—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—24no—
528C23—42no—
529C23—42no—
530C23—48no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23C2×C4×D412yes—
542C23#54712yes—
543C23#54812yes—
544C23C23×D411yescharacteristic
545Q8—48no—
546Q8—84no—
547D4—24no—
548D4—24no—
549D4—48no—
550D4—42no—
551D4—42no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—416no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—88no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4C2×C4×D412yes—
593C2×C4C2×C4×D412yes—
594C2×C4C2×C4×D412yes—
595C2×C4C2×C4×D412yes—
596C2×C4C2×C4×D412yes—
597C2×C4C4×#4114yes—
598C2×C4#51914yes—
599C2×C4#54712yes—
600C2×C4#54812yes—
601C2×C4#55214yes—
602C2×C4#55614yes—
603C2×C4#57612yes—
604C2×C4#57712yes—
605C2×C4C24×C411yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×2−1+411yescharacteristic
610C2×C4C2×2−1+411yescharacteristic
611C8—28no—
612C8—22no—
613C8—22no—
614C8—41no—
615C8—41no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—816no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22#236214yes—
631C22C22×C4×D411yescharacteristic
632C22C2×#54711yescharacteristic
633C22C2×#54811yescharacteristic
634C22C2×#57611yescharacteristic
635C22C2×#57711yescharacteristic
636C22#358411yescharacteristic
637C22#358411yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—44no—
642C4—44no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4#236214yes—
651C4#294614yes—
652C4C2×C4×#4111yescharacteristic
653C4C2×#51911yescharacteristic
654C4C2×#55211yescharacteristic
655C4C2×#55611yescharacteristic
656C2—24no—
657C2—84no—
658C2C2×#236211yescharacteristic
659C2C2×#294611yescharacteristic
660C2#6016111yescharacteristic
661C2#6128411yescharacteristic
662C2#6128511yescharacteristic
663C2#6132511yescharacteristic
664C2#6132711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34213C2semidirect
4. #23071C2semidirect
5. #23067C2semidirect
6. #23066C2semidirect
7. #23065C2semidirect
8. #23061C2semidirect
9. #23060C2semidirect
10. #21318C2semidirect
11. #21170C2semidirect
14. #21035C2semidirect
15. #20453C2semidirect
16. #20437C2semidirect
20. #3077C4semidirect
32. #2370C22semidirect
33. #2369C22semidirect
34. #2365C22semidirect
55. #1991C22semidirect
56. #1991C22semidirect
57. #1990C22semidirect
58. #1990C22semidirect
59. #1990C22semidirect
60. #1990C22semidirect
61. #1949C22semidirect
62. #1948C22semidirect
63. #1947C22semidirect
72. #1921C22semidirect
73. #1920C22semidirect
74. #1919C22semidirect
75. #1918C22semidirect
76. #1776C22semidirect
77. #1776C22semidirect
78. C4×#107C22semidirect
79. C4×#106C22semidirect
80. #1770C22semidirect
81. #1770C22semidirect
82. C8×C4⋊C4C22semidirect
83. C8×C4⋊C4C22semidirect
126. #478C2×C4semidirect
127. #476C2×C4semidirect
128. #474C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30772
#342132
#341862
#230712
#230672
#230662
#230652
#230612
#230602
#213182
#211702
#210912
#210632
#210352
#204532
#204372
C2×C4×#1022