Finite Group Databasebeta

← #6445716

Subgroups — #6445716

Order 512.

2978 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3504C211yesmaximal, characteristic
2C2×#3165C211yesmaximal, characteristic, complemented
3C22×#501C211yesmaximal, characteristic, complemented
4#35279C211yesmaximal, characteristic
5#25821C214yesmaximal, complemented
6#25820C214yesmaximal, complemented
7#25819C214yesmaximal, complemented
8#25818C214yesmaximal, complemented
9#23556C212yesmaximal, complemented
10#23555C212yesmaximal, complemented
11#21325C211yesmaximal, characteristic, complemented
12#21321C211yesmaximal, characteristic, complemented
13#20848C212yesmaximal
14#20847C212yesmaximal
15#20375C211yesmaximal, characteristic
16#3504—28no—
17#3504—28no—
18C2×#557—21no—
19C2×#557—21no—
20C2×#554—21no—
21C2×#554—22no—
22C2×#554—22no—
23C2×#554—21no—
24C2×#544C2211yescharacteristic
25C2×#544C2211yescharacteristic
26C2×#544C2212yes—
27C2×#544C2212yes—
28C2×#533C2212yes—
29C2×#533C2211yescharacteristic
30C22×C4⋊Q8C2211yescharacteristic
31C22×#125C2212yes—
32C22×#125C2211yescharacteristic
33C2×#520C2211yescharacteristic
34C2×#520C2211yescharacteristic
35C2×C4×#41C2211yescharacteristic
36#3165—28no—
37C2×#501—28no—
38C2×#501C2212yes—
39C2×#501C2214yes—
40C2×#501C2212yes—
41C2×#501C2214yes—
42C2×#500C2212yes—
43C2×#498C2212yes—
44C2×#428C2212yes—
45C2×#428C2212yes—
46C22×#107C2212yes—
47C22×#106C2212yes—
48C2×#408C2212yes—
49C2×C4×M16C2211yescharacteristic
50C22×#97C2211yescharacteristic
51#2857C2214yes—
52#2628C2214yes—
53#2568—21no—
54#2567—21no—
55#2563—21no—
56#2542—21no—
57#2487—22no—
58#2409—22no—
59#2317C2212yes—
60#2311C2212yes—
61#2285C2211yescharacteristic
62#2059C2218yes—
63#2059C2218yes—
64#2058C2218yes—
65#2057C2218yes—
66#1993C2212yes—
67#1993C2212yes—
68#1992C2212yes—
69#1992C2212yes—
70#1992C2214yes—
71#1991C2214yes—
72#1854C2218yes—
73#1854C2218yes—
74#1853C2218yes—
75#1852C2218yes—
76#1750C2214yes—
77#1749C2214yes—
78#1736C2212yes—
79#1736C2212yes—
80#1735C2212yes—
81#1735C2212yes—
82#1729C2212yes—
83C22×#41C2311yescharacteristic
84C23×Q8—21no—
85C22×M16C2312yes—
86C23×C8C2312yes—
87#557—216no—
88#557—216no—
89#554—216no—
90#554—216no—
91#554—232no—
92#554—232no—
93#544—28no—
94#544—28no—
95#544—28no—
96#544—28no—
97#544—24no—
98#544—24no—
99#544—24no—
100#544—24no—
101#533—28no—
102#533—28no—
103#533—24no—
104#533D418yes—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—22no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—24no—
112C2×C4⋊Q8—24no—
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2312yes—
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×#125—28no—
120C2×#125—28no—
121C2×#125—21no—
122C2×#125—21no—
123C2×#125—22no—
124C2×#125—22no—
125C2×#125—22no—
126C2×#125—22no—
127C2×#125—21no—
128C2×#125—21no—
129C2×#125—24no—
130C2×#125D418yes—
131C2×#125C2312yes—
132C2×#125C2312yes—
133C2×#125C2311yescharacteristic
134C2×#125C2312yes—
135C2×#125C2312yes—
136C2×#125C2311yescharacteristic
137C2×C22⋊Q8—22no—
138C2×C22⋊Q8—22no—
139C2×C22⋊Q8—21no—
140C2×C22⋊Q8—21no—
141C2×C22⋊Q8—24no—
142C2×C22⋊Q8—22no—
143C2×C22⋊Q8—22no—
144C2×C22⋊Q8—22no—
145C2×C22⋊Q8—22no—
146C2×C22⋊Q8—22no—
147C2×C22⋊Q8—22no—
148#520—24no—
149#520—24no—
150#520—24no—
151#520—24no—
152C4×#41—24no—
153C4×#41—24no—
154C2×C4×Q8—21no—
155C2×C4×Q8—21no—
156C2×C4×Q8—22no—
157C2×C4×Q8—22no—
158C2×C4×Q8—21no—
159C2×C4×Q8—21no—
160C2×C4×Q8D411yescharacteristic
161C2×C4×Q8D411yescharacteristic
162C2×C4×D4—22no—
163C2×C4×D4—22no—
164C2×C4×D4—21no—
165C2×C4×D4—21no—
166C2×C4×D4D411yescharacteristic
167C2×C4×D4D412yes—
168C2×C4×D4D412yes—
169C2×C4×D4D411yescharacteristic
170C2×#117C2312yes—
171C2×#117C2312yes—
172C2×#117C2312yes—
173C2×#117C2311yescharacteristic
174C2×#117C2311yescharacteristic
175C2×#117C2311yescharacteristic
176C2×#117C2312yes—
177C22×C4⋊C4—22no—
178C22×C4⋊C4—22no—
179C22×C4⋊C4—21no—
180C22×C4⋊C4—21no—
181C22×C4⋊C4—21no—
182C22×C4⋊C4C2312yes—
183C22×C4⋊C4C2312yes—
184C22×C4⋊C4C2312yes—
185C22×C4⋊C4C2311yescharacteristic
186C22×C4⋊C4C2311yescharacteristic
187C22×C42C2312yes—
188C22×C42C2311yescharacteristic
189#501—216no—
190#501—216no—
191#501—28no—
192#501—28no—
193#500—28no—
194#498—28no—
195#428—28no—
196#428—28no—
197C2×#107—28no—
198C2×#107C2314yes—
199C2×#107C2318yes—
200C2×#107C2314yes—
201C2×#106—28no—
202C2×#106C2314yes—
203C2×#106C2318yes—
204C2×#106C2314yes—
205#408Q8116yescomplemented
206C2×C22⋊C8C2318yes—
207C4×M16—24no—
208C2×#97—24no—
209C2×#97C2312yes—
210C2×#97C2312yes—
211C2×#97C2312yes—
212C2×C4×C8C2312yes—
213#400—24no—
214#399—24no—
215#395—24no—
216#393—24no—
217#391—22no—
218#391—24no—
219#390—22no—
220#389—24no—
221#389C2318yes—
222#386—24no—
223#385—24no—
224#384—22no—
225#384C2312yes—
226#382—22no—
227#382C2312yes—
228#380—24no—
229#380—21no—
230#380—21no—
231#380C2318yes—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234#340C2314yes—
235#340C2318yes—
236#340C2314yes—
237#340C2318yes—
238#340C2314yes—
239#340C2314yes—
240#336C2318yes—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—24no—
244C2×#41C2×Q818yes—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—21no—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—44no—
252C22×Q8—84no—
253C22×Q8C2×D411yescharacteristic
254C22×D4C2×D412yes—
255C22×D4C2×D411yescharacteristic
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2412yes—
266C23×C4C2411yescharacteristic
267C2×M16—24no—
268C2×M16—24no—
269C2×M16C2×Q818yes—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8C2×Q818yes—
273C22×C8C2414yes—
274C22×C8C2414yes—
275C4⋊Q8—216no—
276C4⋊Q8—216no—
277C4⋊Q8—28no—
278C4⋊Q8—28no—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—24no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—24no—
286C4⋊Q8—24no—
287C4⋊Q8—24no—
288C4⋊Q8—24no—
289C4⋊Q8—44no—
290C4⋊Q8—44no—
291C4⋊Q8C2×D414yes—
292C4⋊Q8C2×D414yes—
293#125—216no—
294#125—216no—
295#125—216no—
296#125—216no—
297#125—28no—
298#125—28no—
299#125—24no—
300#125—24no—
301#125—24no—
302#125—24no—
303#125—28no—
304#125—28no—
305#125—22no—
306#125—24no—
307#125—24no—
308#125—22no—
309#125C2×D414yes—
310#125C2×D418yes—
311#125C2×D418yes—
312#125C2×D414yes—
313C22⋊Q8—216no—
314C22⋊Q8—216no—
315C22⋊Q8—216no—
316C22⋊Q8—216no—
317C22⋊Q8—216no—
318C22⋊Q8—216no—
319C22⋊Q8—216no—
320C22⋊Q8—216no—
321C22⋊Q8—28no—
322C22⋊Q8—28no—
323C22⋊Q8—88no—
324C4×Q8—28no—
325C4×Q8—28no—
326C4×Q8—216no—
327C4×Q8—216no—
328C4×Q8—28no—
329C4×Q8—28no—
330C4×Q8—24no—
331C4×Q8—24no—
332C4×D4—216no—
333C4×D4—216no—
334C4×D4—28no—
335C4×D4—28no—
336C4×D4—28no—
337C4×D4—28no—
338C4×D4—24no—
339C4×D4—24no—
340#117—22no—
341#117—22no—
342#117—22no—
343#117—22no—
344#117—24no—
345#117—24no—
346#117—24no—
347#117—24no—
348#117—24no—
349#117—24no—
350#117—28no—
351#117C2×D414yes—
352#117C2×D414yes—
353C2×C4⋊C4—216no—
354C2×C4⋊C4—216no—
355C2×C4⋊C4—28no—
356C2×C4⋊C4—28no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—24no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—28no—
391C2×C4⋊C4—44no—
392C2×C4⋊C4C2×D411yescharacteristic
393C2×C4⋊C4C2×D411yescharacteristic
394C2×C4⋊C4C2×D412yes—
395C2×C4⋊C4C2×D412yes—
396C2×C4⋊C4C2×D414yes—
397C2×C4⋊C4C2×D414yes—
398C2×C4⋊C4C2414yes—
399C2×C4⋊C4C2414yes—
400C2×C4⋊C4C2411yescharacteristic
401C2×C4⋊C4C2412yes—
402C2×C4⋊C4C2411yescharacteristic
403C2×C4⋊C4C2412yes—
404C2×C4⋊C4C2411yescharacteristic
405C2×C4⋊C4C2411yescharacteristic
406C2×C22⋊C4—22no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—28no—
412C2×C22⋊C4C2×D412yes—
413C2×C22⋊C4C2×D412yes—
414C2×C22⋊C4C2×D412yes—
415C2×C42—22no—
416C2×C42—22no—
417C2×C42—21no—
418C2×C42—21no—
419C2×C42—24no—
420C2×C42—24no—
421C2×C42—22no—
422C2×C42C2×D412yes—
423C2×C42C2×D412yes—
424C2×C42C2×D411yescharacteristic
425C2×C42C2×D411yescharacteristic
426C2×C42C2×D414yes—
427C2×C42C2×D414yes—
428C2×C42C2412yes—
429C2×C42C2411yescharacteristic
430C2×C42C2411yescharacteristic
431#107—216no—
432#107—28no—
433#107—28no—
434#106—216no—
435#106—28no—
436#106—28no—
437C22⋊C8C2×Q8132yescomplemented
438#97—24no—
439#97—24no—
440#97—24no—
441C4×C8—24no—
442#95—22no—
443#95—22no—
444#95—28no—
445#95—28no—
446C24C22×D412yes—
447C24C22×D411yescharacteristic
448#41—216no—
449#41—216no—
450C2×Q8—28no—
451C2×Q8—28no—
452C2×Q8—22no—
453C2×Q8—24no—
454C2×Q8—24no—
455C2×Q8—22no—
456C2×Q8—48no—
457C2×Q8—44no—
458C2×Q8—44no—
459C2×Q8—82no—
460C2×Q8—82no—
461C2×Q8—84no—
462C2×Q8—84no—
463C2×Q8—84no—
464C2×Q8—816no—
465C2×Q8C4⋊Q814yes—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4C4⋊Q818yes—
471C2×D4C4⋊Q814yes—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—28no—
482C22×C4—28no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—82no—
495C22×C4—84no—
496C22×C4—82no—
497C22×C4—84no—
498C22×C4—84no—
499C22×C4#12014yes—
500C22×C4#12014yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D412yes—
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×Q818yes—
512C22×C4C22×Q814yes—
513C22×C42−1+414yes—
514C22×C4C2511yescharacteristic
515M16—216no—
516C2×C8—216no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8C22×Q818yes—
522C2×C8C22×Q818yes—
523C4⋊C4—28no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—28no—
527C4⋊C4—28no—
528C4⋊C4—28no—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—24no—
532C4⋊C4—28no—
533C4⋊C4—24no—
534C4⋊C4—28no—
535C4⋊C4—28no—
536C4⋊C4—28no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—24no—
554C4⋊C4—24no—
555C4⋊C4—28no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—88no—
562C4⋊C4C22×D412yes—
563C4⋊C4C22×D414yes—
564C4⋊C4C22×D412yes—
565C4⋊C4C22×D412yes—
566C4⋊C4C22×D414yes—
567C4⋊C4C22×D412yes—
568C22⋊C4—28no—
569C22⋊C4—28no—
570C22⋊C4—24no—
571C22⋊C4—24no—
572C22⋊C4—24no—
573C22⋊C4—24no—
574C22⋊C4—24no—
575C22⋊C4—44no—
576C22⋊C4—88no—
577C42—28no—
578C42—28no—
579C42—24no—
580C42—24no—
581C42—24no—
582C42—24no—
583C42—22no—
584C42—22no—
585C42—44no—
586C42#12014yes—
587C42#12014yes—
588C42C22×D412yes—
589C42C22×D412yes—
590C42C22×D412yes—
591C42C22×D412yes—
592C23—22no—
593C23—22no—
594C23—24no—
595C23—24no—
596C23—22no—
597C23—22no—
598C23C2×#12012yes—
599C23C2×C4⋊Q814yes—
600C23C2×C4⋊Q812yes—
601C23#53612yes—
602C23C23×D411yescharacteristic
603Q8—28no—
604Q8—84no—
605Q8—84no—
606Q8—88no—
607Q8—88no—
608Q8—88no—
609Q8—88no—
610D4—216no—
611D4—28no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—48no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—88no—
648C2×C4—88no—
649C2×C4—88no—
650C2×C4C2×#12012yes—
651C2×C4C2×#12012yes—
652C2×C4C2×#12012yes—
653C2×C4C2×#12012yes—
654C2×C4C2×#12012yes—
655C2×C4C2×C4⋊Q814yes—
656C2×C4C2×C4⋊Q812yes—
657C2×C4#53612yes—
658C2×C4D4×Q818yes—
659C2×C4D4×Q818yes—
660C2×C4C23×D411yescharacteristic
661C2×C4C23×D411yescharacteristic
662C2×C4C23×Q812yes—
663C2×C4C2×2−1+412yes—
664C8—28no—
665C8—28no—
666C22—22no—
667C22—22no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—28no—
672C22—42no—
673C22—42no—
674C22—44no—
675C22#243414yes—
676C22C22×#12011yescharacteristic
677C22C22×C4⋊Q811yescharacteristic
678C22C2×#53611yescharacteristic
679C22#358512yes—
680C22#358612yes—
681C4—24no—
682C4—22no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—42no—
687C4—44no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—44no—
692C4—42no—
693C4—42no—
694C4—84no—
695C4—84no—
696C4#243414yes—
697C4C2×D4×Q812yes—
698C4C2×D4×Q812yes—
699C2—24no—
700C2—44no—
701C2C2×#243411yescharacteristic
702C2#6036911yescharacteristic
703C2#6037011yescharacteristic
704C2#6066812yes—
705C2#6088612yes—
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
2. C2×#3165C2semidirect
3. C22×#501C2semidirect
5. #25821C2semidirect
6. #25820C2semidirect
7. #25819C2semidirect
8. #25818C2semidirect
9. #23556C2semidirect
10. #23555C2semidirect
11. #21325C2semidirect
12. #21321C2semidirect
205. #408Q8semidirect
437. C22⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#35042
C2×#31652
C22×#5012
#352792
#258212
#258202
#258192
#258182
#235562
#235552
#213252
#213212
#208482
#208472
#203752