Finite Group Databasebeta

← #7646436

Subgroups — #7646436

Order 512.

7200 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic, complemented
2C2×#3461C211yesmaximal, characteristic, complemented
3#42794C218yesmaximal, complemented
4#42790C218yesmaximal, complemented
5#39598C218yesmaximal, complemented
6#39126C218yesmaximal, complemented
7#37549C214yesmaximal, complemented
8#36719C218yesmaximal, complemented
9#35661C214yesmaximal, complemented
10#35279C212yesmaximal, complemented
11#35236C214yesmaximal, complemented
12#35224C212yesmaximal, complemented
13#35223C212yesmaximal, complemented
14#34935C212yesmaximal, complemented
15#34182C211yesmaximal, characteristic, complemented
16C2×#554—216no—
17C2×#554C2218yes—
18C2×#554C2218yes—
19#3461—216no—
20C2×#544C2211yescharacteristic
21C2×#542C2214yescomplemented
22C2×#541C2214yescomplemented
23C2×#539C2214yes—
24C2×#537C2211yescharacteristic
25C2×#536C2212yes—
26C2×#533C2214yescomplemented
27C22×C4⋊Q8C2211yescharacteristic
28C2×#529C2214yes—
29C2×#529C2214yescomplemented
30C22×#125C2212yes—
31C2×#525C2214yes—
32C22×C22⋊Q8C2212yes—
33C22×C22⋊Q8C2214yes—
34C2×#519C2212yes—
35C2×C4×#41C2212yes—
36C22×C4×Q8C2211yescharacteristic
37C22×C4×D4C2211yescharacteristic
38C22×C4×D4C2212yes—
39C23×C4⋊C4C2212yes—
40#2857C2218yescomplemented
41#2848C2218yes—
42#2847C2218yes—
43#2822C2218yescomplemented
44#2820C2218yescomplemented
45#2704C22132yescomplemented
46#2702C22132yescomplemented
47#2683C22132yescomplemented
48#2681C22132yescomplemented
49#2664C22116yescomplemented
50#2636C22116yes—
51#2629C22116yescomplemented
52#2628C2218yescomplemented
53#2627C2218yes—
54#2618C2218yes—
55#2584C2218yescomplemented
56#2568C2214yescomplemented
57#2567C2214yescomplemented
58#2563C2214yescomplemented
59#2562C2218yes—
60#2562C2218yescomplemented
61#2542C2214yescomplemented
62#2540C2218yes—
63#2538C2214yes—
64#2538C2218yes—
65#2538C22116yescomplemented
66#2537C2214yes—
67#2537C2218yes—
68#2536C2214yes—
69#2536C22116yescomplemented
70#2535C2214yes—
71#2534C2218yes—
72#2487C2218yes—
73#2486C2214yes—
74#2442C22116yescomplemented
75#2440C22116yescomplemented
76#2429C22116yescomplemented
77#2426C22116yescomplemented
78#2417C22116yescomplemented
79#2416C22116yescomplemented
80#2413C22116yes—
81#2411C22116yes—
82#2409C2218yes—
83#2408C2214yes—
84#2392C2218yes—
85#2351C2218yescomplemented
86Q8×C4⋊C4C2218yescomplemented
87#2324C22116yes—
88#2317C2214yes—
89#2316C2214yes—
90#2316C2212yes—
91#2311C2214yes—
92C4×#121C2218yes—
93#2285C2212yes—
94#2285C2214yes—
95#2280C2214yes—
96C22×#41C2312yes—
97C23×Q8C2311yescharacteristic
98C23×Q8C2312yes—
99C23×D4C2311yescharacteristic
100C24×C4C2312yes—
101C24×C4C2314yes—
102#554—264no—
103#554—264no—
104#544—28no—
105#542—232no—
106#541—232no—
107#539—232no—
108#537—28no—
109#536—216no—
110#533—232no—
111C2×C4⋊Q8—28no—
112C2×C4⋊Q8C2314yes—
113C2×C4⋊Q8C2314yes—
114C2×C4⋊Q8C2314yes—
115C2×C4⋊Q8C2314yes—
116#529—232no—
117#529—232no—
118C2×#126C2318yes—
119C2×#126C2318yes—
120C2×#125—216no—
121C2×#125C2314yes—
122C2×#125C2318yes—
123C2×#125C2318yes—
124C2×#125C2314yes—
125C2×#125C2318yes—
126C2×#124C2314yes—
127C2×#124C2314yes—
128#525—416no—
129C2×#123C2318yes—
130C2×#123C2318yes—
131C2×#123C2318yes—
132C2×C22⋊Q8—216no—
133C2×C22⋊Q8—232no—
134C2×C22⋊Q8C2314yes—
135C2×C22⋊Q8C23132yes—
136C2×C22⋊Q8C23116yes—
137C2×C22⋊Q8C2314yes—
138C2×C22⋊Q8C2314yes—
139C2×C22⋊Q8C2314yes—
140C2×C22⋊Q8C2318yes—
141C2×C22⋊Q8C2318yes—
142C2×C22⋊Q8C23116yes—
143C2×C22⋊Q8C23116yes—
144C2×#121C2318yes—
145C2×#121C2314yes—
146C2×#121C2314yes—
147C2×#121C2318yes—
148C2×#121C23116yes—
149C2×#120C2318yes—
150#519—48no—
151C4×#41—216no—
152C2×C4×Q8D4116yescomplemented
153C2×C4×Q8C2312yes—
154C2×C4×Q8C2312yes—
155C2×C4×Q8C2314yes—
156C2×C4×Q8C23116yes—
157C2×C4×Q8C2314yes—
158C2×C4×Q8C2314yes—
159C2×C4×D4—28no—
160C2×C4×D4—216no—
161C2×C4×D4C2318yes—
162C2×C4×D4C2314yes—
163C2×C4×D4C2312yes—
164C2×C4×D4C2318yes—
165C2×C4×D4C2314yes—
166C2×C4×D4C2314yes—
167C2×C4×D4C2314yes—
168C2×C4×D4C2314yes—
169C2×C4×D4C2312yes—
170C2×C4×D4C23116yes—
171C2×#117C2314yes—
172C2×#117C2312yes—
173C2×#117C2318yes—
174C2×#117C2314yes—
175C2×#117C2318yes—
176C22×C4⋊C4—216no—
177C22×C4⋊C4C2312yes—
178C22×C4⋊C4C2314yes—
179C22×C4⋊C4C23116yes—
180C22×C4⋊C4C2311yescharacteristic
181C22×C4⋊C4C2312yes—
182C22×C4⋊C4C2314yes—
183C22×C4⋊C4C2311yescharacteristic
184C22×C4⋊C4C2312yes—
185C22×C22⋊C4C2314yes—
186C22×C22⋊C4C2312yes—
187C22×C42C2314yes—
188C22×C42C2311yescharacteristic
189C22×C42C2312yes—
190#400C23116yes—
191#400C23116yes—
192#399C23116yes—
193#398C2318yes—
194#398C23116yes—
195#398C23132yes—
196#398C23132yes—
197#397C2318yes—
198#396C23132yes—
199#396C23132yes—
200#396C23132yes—
201#396C23116yes—
202#395C23116yes—
203#395C23116yes—
204#394C2318yes—
205#394C23116yes—
206#394C23132yes—
207#394C23116yes—
208#393C23116yes—
209#392C2318yes—
210#391C2318yes—
211#391C23116yes—
212#390C2318yes—
213#390C2318yes—
214#390C23132yes—
215#389C23116yes—
216#389C23116yes—
217#389C23132yes—
218#389C23116yes—
219#389C2318yes—
220#389C23132yes—
221#388C23132yes—
222#388C23132yes—
223#388C23116yes—
224#386C23116yes—
225#386C23132yes—
226#386C23116yes—
227#385C23116yes—
228#385C23132yes—
229#384C2318yes—
230#384C2314yes—
231#384C2318yes—
232#382C2318yes—
233#382C2318yes—
234#382C2314yes—
235#381C2318yes—
236#380C23116yes—
237#380C23116yes—
238#380C2318yes—
239#380C23116yes—
240#380C23116yes—
241#380C2318yes—
242#380C2314yes—
243#380C2314yes—
244#380C2318yes—
245#380C2318yes—
246#379C2318yes—
247C4×C4⋊C4C2318yes—
248C4×C4⋊C4C2318yes—
249C4×C4⋊C4C2314yes—
250C4×C4⋊C4C2314yes—
251C4×C22⋊C4C23116yes—
252C25C2412yes—
253C2×#41—28no—
254C2×#41—28no—
255C2×#41—28no—
256C22×Q8—28no—
257C22×Q8—24no—
258C22×Q8—24no—
259C22×Q8—84no—
260C22×Q8C2×D418yes—
261C22×Q8C2412yes—
262C22×Q8C2414yes—
263C22×Q8C2414yes—
264C22×Q8C2418yes—
265C22×D4—28no—
266C22×D4—24no—
267C22×D4C2412yes—
268C22×D4C2414yes—
269C22×D4C2412yes—
270C22×D4C2412yes—
271C22×D4C24116yes—
272C22×D4C2414yes—
273C23×C4—28no—
274C23×C4—28no—
275C23×C4—28no—
276C23×C4—216no—
277C23×C4—88no—
278C23×C4C2412yes—
279C23×C4C2418yes—
280C23×C4C2414yes—
281C23×C4C2414yes—
282C23×C4C2412yes—
283C23×C4C2411yescharacteristic
284C23×C4C2414yes—
285C23×C4C24116yes—
286C23×C4C2414yes—
287C23×C4C2411yescharacteristic
288C23×C4C2412yes—
289C23×C4C2411yescharacteristic
290C4⋊Q8—216no—
291C4⋊Q8—216no—
292C4⋊Q8—216no—
293C4⋊Q8—216no—
294#126—232no—
295#126—232no—
296#125—216no—
297#125—232no—
298#125—232no—
299#125—232no—
300#125—216no—
301#124—216no—
302#124—216no—
303#123—232no—
304#123—232no—
305#123—416no—
306C22⋊Q8—264no—
307C22⋊Q8—216no—
308C22⋊Q8—216no—
309C22⋊Q8—216no—
310C22⋊Q8—2128no—
311C22⋊Q8—216no—
312C22⋊Q8—232no—
313C22⋊Q8—416no—
314C22⋊Q8—816no—
315C22⋊Q8—816no—
316#121—232no—
317#121—216no—
318#121—216no—
319#121—416no—
320#121—816no—
321#120—416no—
322C4×Q8—28no—
323C4×Q8—28no—
324C4×Q8—216no—
325C4×Q8—432no—
326C4×Q8C2×D4132yes—
327C4×Q8C2×D4132yes—
328C4×D4—216no—
329C4×D4—216no—
330C4×D4—216no—
331C4×D4—232no—
332C4×D4—216no—
333C4×D4—232no—
334C4×D4—28no—
335C4×D4—28no—
336C4×D4—216no—
337C4×D4—432no—
338#117—28no—
339#117—216no—
340#117—216no—
341#117—232no—
342#117—416no—
343C2×C4⋊C4—216no—
344C2×C4⋊C4—28no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—28no—
347C2×C4⋊C4—216no—
348C2×C4⋊C4—264no—
349C2×C4⋊C4C2×D418yes—
350C2×C4⋊C4C2×D4116yes—
351C2×C4⋊C4C24116yes—
352C2×C4⋊C4C2418yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2414yes—
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C24116yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2414yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2414yes—
362C2×C4⋊C4C2418yes—
363C2×C4⋊C4C2418yes—
364C2×C4⋊C4C2418yes—
365C2×C4⋊C4C2414yes—
366C2×C4⋊C4C2414yes—
367C2×C4⋊C4C2418yes—
368C2×C4⋊C4C2418yes—
369C2×C4⋊C4C24116yes—
370C2×C4⋊C4C24116yes—
371C2×C4⋊C4C24116yes—
372C2×C4⋊C4C2418yes—
373C2×C4⋊C4C24132yes—
374C2×C4⋊C4C2414yes—
375C2×C4⋊C4C2418yes—
376C2×C22⋊C4—28no—
377C2×C22⋊C4—216no—
378C2×C22⋊C4C2418yes—
379C2×C22⋊C4C24116yes—
380C2×C22⋊C4C2418yes—
381C2×C22⋊C4C2414yes—
382C2×C22⋊C4C2414yes—
383C2×C22⋊C4C2414yes—
384C2×C22⋊C4C2418yes—
385C2×C22⋊C4C2418yes—
386C2×C22⋊C4C2418yes—
387C2×C22⋊C4C24116yes—
388C2×C22⋊C4C24116yes—
389C2×C22⋊C4C24116yes—
390C2×C22⋊C4C24132yes—
391C2×C22⋊C4C24116yes—
392C2×C42—216no—
393C2×C42C2×D418yes—
394C2×C42C2×D4116yes—
395C2×C42C2414yes—
396C2×C42C2414yes—
397C2×C42C2418yes—
398C2×C42C2414yes—
399C2×C42C24116yes—
400C2×C42C2414yes—
401C2×C42C2412yes—
402C2×C42C2412yes—
403C2×C42C2414yes—
404#95C24116yes—
405#95C24116yes—
406#95C2418yes—
407#95C24116yes—
408#95C2418yes—
409#95C24132yes—
410#95C24132yes—
411C24—24no—
412C24—28no—
413C24—24no—
414C24—24no—
415C24—44no—
416C24C2514yes—
417C24C2514yes—
418C24C2511yescharacteristic
419#41—432no—
420C2×Q8—232no—
421C2×Q8—24no—
422C2×Q8—28no—
423C2×Q8—216no—
424C2×Q8—28no—
425C2×Q8—24no—
426C2×Q8—28no—
427C2×Q8—24no—
428C2×Q8—48no—
429C2×Q8—416no—
430C2×Q8—816no—
431C2×Q8C22×D4116yes—
432C2×D4—232no—
433C2×D4—28no—
434C2×D4—28no—
435C2×D4—28no—
436C2×D4—28no—
437C2×D4—232no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—416no—
446C2×D4—48no—
447C2×D4—416no—
448C2×D42−1+418yes—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—216no—
459C22×C4—28no—
460C22×C4—216no—
461C22×C4—216no—
462C22×C4—216no—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—232no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—42no—
469C22×C4—416no—
470C22×C4—48no—
471C22×C4—416no—
472C22×C4—432no—
473C22×C4—832no—
474C22×C4C22×D414yes—
475C22×C4C22×D418yes—
476C22×C4C22×D414yes—
477C22×C4C22×D418yes—
478C22×C4C22×D414yes—
479C22×C4H(5,𝔽2)14yes—
480C22×C42−1+414yes—
481C22×C4C2514yes—
482C22×C4C25116yes—
483C22×C4C2514yes—
484C22×C4C2514yes—
485C22×C4C2518yes—
486C22×C4C2514yes—
487C22×C4C2511yescharacteristic
488C22×C4C2512yes—
489C22×C4C2514yes—
490C22×C4C2514yes—
491C22×C4C2512yes—
492C22×C4C2511yescharacteristic
493C4⋊C4—232no—
494C4⋊C4—216no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—232no—
500C4⋊C4—24no—
501C4⋊C4—28no—
502C4⋊C4—24no—
503C4⋊C4—28no—
504C4⋊C4—216no—
505C4⋊C4—216no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—216no—
509C4⋊C4—216no—
510C4⋊C4—216no—
511C4⋊C4—416no—
512C4⋊C4—416no—
513C4⋊C4—416no—
514C4⋊C4—48no—
515C4⋊C4—432no—
516C4⋊C4C22×D4116yes—
517C4⋊C4C22×D4132yes—
518C22⋊C4—28no—
519C22⋊C4—216no—
520C22⋊C4—232no—
521C22⋊C4—216no—
522C22⋊C4—232no—
523C22⋊C4—28no—
524C22⋊C4—28no—
525C22⋊C4—216no—
526C22⋊C4—216no—
527C22⋊C4—216no—
528C22⋊C4—416no—
529C22⋊C4—416no—
530C22⋊C4—432no—
531C22⋊C4—416no—
532C42—28no—
533C42—28no—
534C42—28no—
535C42—216no—
536C42—416no—
537C42C22×D418yes—
538C42C22×D4116yes—
539C42C22×D418yes—
540C42C22×D4116yes—
541C23—22no—
542C23—22no—
543C23—216no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—28no—
553C23—24no—
554C23—28no—
555C23—416no—
556C23—416no—
557C23—48no—
558C23—48no—
559C23C23×D412yes—
560C23C2×H(5,𝔽2)12yes—
561C23C2×2−1+418yes—
562C23C2×2−1+412yes—
563C23C2611yescharacteristic
564Q8—232no—
565Q8—48no—
566Q8—88no—
567Q8—816no—
568D4—216no—
569D4—216no—
570D4—416no—
571D4—48no—
572D4—88no—
573D4—832no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—216no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—216no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—216no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—216no—
594C2×C4—28no—
595C2×C4—216no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—416no—
599C2×C4—48no—
600C2×C4—432no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—88no—
604C2×C4—832no—
605C2×C4#537116yes—
606C2×C4C23×D414yes—
607C2×C4C23×D412yes—
608C2×C4C23×D418yes—
609C2×C4C23×D418yes—
610C2×C4C23×D414yes—
611C2×C4C23×D412yes—
612C2×C4C2×H(5,𝔽2)14yes—
613C2×C4C2×2−1+414yes—
614C2×C4C2×2−1+414yes—
615C22—22no—
616C22—22no—
617C22—24no—
618C22—24no—
619C22—28no—
620C22—28no—
621C22—28no—
622C22—432no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—48no—
631C22—48no—
632C22C2×#53714yes—
633C22C24×D411yescharacteristic
634C22C22×H(5,𝔽2)11yescharacteristic
635C22C22×2−1+412yes—
636C22C22×2−1+411yescharacteristic
637C22H(7,𝔽2)12yes—
638C4—22no—
639C4—28no—
640C4—24no—
641C4—28no—
642C4—44no—
643C4—48no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—416no—
648C4C2×#53718yes—
649C4#348614yes—
650C2—24no—
651C2—48no—
652C2—84no—
653C2C22×#53711yescharacteristic
654C2C2×#348611yescharacteristic
655C2#6285211yescharacteristic
656C2#6294412yes—
657C2#6294512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#554C2semidirect
2. C2×#3461C2semidirect
3. #42794C2semidirect
4. #42790C2semidirect
5. #39598C2semidirect
6. #39126C2semidirect
7. #37549C2semidirect
8. #36719C2semidirect
9. #35661C2semidirect
10. #35279C2semidirect
11. #35236C2semidirect
12. #35224C2semidirect
13. #35223C2semidirect
14. #34935C2semidirect
15. #34182C2semidirect
21. C2×#542C22semidirect
22. C2×#541C22semidirect
26. C2×#533C22semidirect
29. C2×#529C22semidirect
40. #2857C22semidirect
43. #2822C22semidirect
44. #2820C22semidirect
45. #2704C22semidirect
46. #2702C22semidirect
47. #2683C22semidirect
48. #2681C22semidirect
49. #2664C22semidirect
51. #2629C22semidirect
52. #2628C22semidirect
55. #2584C22semidirect
56. #2568C22semidirect
57. #2567C22semidirect
58. #2563C22semidirect
60. #2562C22semidirect
61. #2542C22semidirect
65. #2538C22semidirect
69. #2536C22semidirect
74. #2442C22semidirect
75. #2440C22semidirect
76. #2429C22semidirect
77. #2426C22semidirect
78. #2417C22semidirect
79. #2416C22semidirect
85. #2351C22semidirect
86. Q8×C4⋊C4C22semidirect
152. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#34612
#427942
#427902
#395982
#391262
#375492
#367192
#356612
#352792
#352362
#352242
#352232
#349352
#341822