Finite Group Databasebeta

← #6460648

Subgroups — #6460648

Order 512.

1822 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35558C211yesmaximal, characteristic, complemented
2#35515C211yesmaximal, characteristic, complemented
3#29096C214yesmaximal, complemented
4#29095C214yesmaximal, complemented
5#29072C214yesmaximal, complemented
6#29071C214yesmaximal, complemented
7#23667C211yesmaximal, characteristic, complemented
8#23628C211yesmaximal, characteristic, complemented
9#23540C211yesmaximal, characteristic
10#23454C211yesmaximal, characteristic, complemented
11#21318C211yesmaximal, characteristic, complemented
12C2×#1970C211yesmaximal, characteristic
13#21091C211yesmaximal, characteristic, complemented
14#21091C211yesmaximal, characteristic, complemented
15#21089C211yesmaximal, characteristic, complemented
16#21037C211yesmaximal, characteristic, complemented
17#20934C211yesmaximal, characteristic
18#20435C211yesmaximal, characteristic, complemented
19#20416C211yesmaximal, characteristic, complemented
20C23×D8C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#518C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C2×#424C2211yescharacteristic
25C2×#418C2211yescharacteristic
26C2×#418C2211yescharacteristic
27C2×#418C2211yescharacteristic
28C2×#418C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C2×#405C2211yescharacteristic
32C2×#405C2211yescharacteristic
33C22×C4×C8C2211yescharacteristic
34#2580—22no—
35#2538—22no—
36#2513—22no—
37#2513C2214yescomplemented
38#2512—22no—
39#2512C2214yescomplemented
40#2480—24no—
41#2454C2214yescomplemented
42#2452C2214yescomplemented
43#2437—24no—
44#2317C2211yescharacteristic
45#2317C2211yescharacteristic
46#2316C2211yescharacteristic
47#2309C2211yescharacteristic
48#2309C2211yescharacteristic
49#2309C2211yescharacteristic
50#2296C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#390C2211yescharacteristic
53#2285C2211yescharacteristic
54#2274C2211yescharacteristic
55#2087C2214yescomplemented
56#2086C2214yescomplemented
57#2073C2214yescomplemented
58#2071C2214yescomplemented
59#2054C2214yes—
60#2054C2214yes—
61#2025C2214yescomplemented
62#2023C2214yescomplemented
63#1991C2212yescomplemented
64#1991C2212yescomplemented
65#1990C2212yescomplemented
66#1990C2212yescomplemented
67#1970C2212yes—
68#1970C2212yes—
69#1970C2212yes—
70#1970C2212yes—
71#1936C2214yescomplemented
72#1936C2214yescomplemented
73#1936C2214yescomplemented
74#1934C2214yescomplemented
75#1934C2214yescomplemented
76#1934C2214yescomplemented
77#1921C2214yescomplemented
78#1918C2214yescomplemented
79#1879C2214yes—
80#1879C2214yes—
81#1771C2212yescomplemented
82#1771C2212yescomplemented
83C8×C4⋊C4C2212yescomplemented
84C8×C4⋊C4C2212yescomplemented
85#1762C2214yescomplemented
86C4×#102C2214yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C23×D4C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×D8—24no—
100C22×D8—24no—
101C22×D8—42no—
102C22×D8C2312yes—
103C22×D8C2312yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#533—24no—
109#533—42no—
110C2×C4⋊Q8C2312yes—
111C2×#126—24no—
112C2×#125C2312yes—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—24no—
115#518—24no—
116#518—42no—
117C2×C4×Q8—22no—
118C2×C4×D4—22no—
119C2×C4×D4C2314yes—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—22no—
129C22×C22⋊C4—22no—
130C22×C22⋊C4—22no—
131C22×C42C2311yescharacteristic
132C2×#107—24no—
133C2×#107C2312yes—
134C2×#107C2312yes—
135#424—24no—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138#418—24no—
139#418—24no—
140#418—24no—
141#418—42no—
142C2×#102—24no—
143C2×#102—24no—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152#405—24no—
153#405—42no—
154C2×#97C2312yes—
155C2×#97C2312yes—
156C2×C4×C8—42no—
157C2×C4×C8C2312yes—
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2312yes—
163#399—24no—
164#397—24no—
165#397—24no—
166#397—24no—
167#396—24no—
168#396—24no—
169#396—24no—
170#393—24no—
171#390—21no—
172#390—21no—
173#390—22no—
174#390—21no—
175#390—21no—
176#390—22no—
177#390—22no—
178#390C2314yes—
179#390C2314yes—
180#390C2314yes—
181#390C2312yes—
182#390C2312yes—
183#390C2312yes—
184#390C2312yes—
185#389—22no—
186#389D412yes—
187#389D412yes—
188#389C2314yes—
189#389C2314yes—
190#389C2314yes—
191#388—24no—
192#388—24no—
193#388—24no—
194#385—24no—
195#385—22no—
196#385—22no—
197#385—24no—
198#385D414yes—
199#384—21no—
200#384—21no—
201#384C2312yes—
202#382—21no—
203#382—21no—
204#382C2312yes—
205#380—22no—
206#380—22no—
207#380—24no—
208#379—22no—
209#379—22no—
210#379—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217#340C2312yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233C25—21no—
234C25—21no—
235C22×Q8—22no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4#4114yes—
250C22×D4C2412yes—
251C22×D4C2412yes—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—24no—
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C2×D8—24no—
258C2×D8—24no—
259C2×D8—24no—
260C2×D8—24no—
261C2×D8—42no—
262C2×D8—42no—
263C2×D8—416no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8#4114yes—
275C22×C8#4114yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2411yescharacteristic
279C22×C8C2411yescharacteristic
280C22×C8C2411yescharacteristic
281C22×C8C2411yescharacteristic
282C4⋊Q8—24no—
283C4⋊Q8—42no—
284#126—88no—
285#125—24no—
286#125—42no—
287C22⋊Q8—48no—
288C22⋊Q8—88no—
289C4×Q8—48no—
290C4×D4—28no—
291C4×D4—44no—
292C4×D4—48no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117#4114yes—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—48no—
346C2×C22⋊C4—84no—
347C2×C22⋊C4—84no—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—24no—
357C2×C42C2×D412yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—24no—
363#107—24no—
364C4⋊C8—24no—
365C4⋊C8—24no—
366#102—28no—
367#102—28no—
368#102—24no—
369#102—24no—
370#102—24no—
371#102—24no—
372#102—28no—
373#102—44no—
374#97—24no—
375#97—42no—
376C4×C8—24no—
377C4×C8—41no—
378C4×C8—41no—
379C4×C8—41no—
380C4×C8—41no—
381C4×C8—42no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95C2×D414yes—
389#95C2×D414yes—
390C24—22no—
391C24—24no—
392C24—22no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—82no—
399C24—82no—
400C24—82no—
401C24—82no—
402C24C22×D411yescharacteristic
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—82no—
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—28no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4C2×#4114yes—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—48no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C42−1+412yes—
460C22×C4C2511yescharacteristic
461D8—48no—
462D8—48no—
463M16—44no—
464M16—44no—
465C2×C8—28no—
466C2×C8—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8C2×#4112yes—
482C2×C8C2×#4112yes—
483C2×C8C2×#4112yes—
484C2×C8C2×#4112yes—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—28no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—82no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4C2×#4114yes—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—48no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—21no—
535C42—22no—
536C42—21no—
537C42—21no—
538C42—22no—
539C42—42no—
540C42—42no—
541C42—42no—
542C42—84no—
543C23—22no—
544C23—22no—
545C23—24no—
546C23—44no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—48no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—44no—
556C23—88no—
557C23—88no—
558C23—84no—
559C23—162no—
560C23—162no—
561C23—162no—
562C23—162no—
563C23—164no—
564C23—164no—
565C23#53212yes—
566C23#54612yes—
567C23#54812yes—
568C23C23×D411yescharacteristic
569Q8—88no—
570D4—24no—
571D4—24no—
572D4—48no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—84no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—48no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—168no—
610C2×C4—162no—
611C2×C4—162no—
612C2×C4#52914yes—
613C2×C4#53212yes—
614C2×C4#53914yes—
615C2×C4#54612yes—
616C2×C4#54812yes—
617C2×C4#55114yes—
618C2×C4#55614yes—
619C2×C4#57614yes—
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C2×C4C2×2−1+411yescharacteristic
626C8—22no—
627C8—22no—
628C8—22no—
629C8—22no—
630C8—42no—
631C8—42no—
632C22—22no—
633C22—22no—
634C22—24no—
635C22—24no—
636C22—44no—
637C22—48no—
638C22—82no—
639C22—82no—
640C22—84no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—84no—
645C22—84no—
646C22—168no—
647C22—168no—
648C22#256014yes—
649C22C2×#53211yescharacteristic
650C22C2×#54611yescharacteristic
651C22C2×#54811yescharacteristic
652C22C2×#57612yes—
653C22#358412yes—
654C4—24no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—44no—
659C4—44no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4#256014yes—
666C4C2×#52911yescharacteristic
667C4C2×#53911yescharacteristic
668C4C2×#55111yescharacteristic
669C4C2×#55611yescharacteristic
670C2—24no—
671C2—84no—
672C2—162no—
673C2C2×#256011yescharacteristic
674C2#6071311yescharacteristic
675C2#6071411yescharacteristic
676C2#6117112yes—
677C2#6132512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35558C2semidirect
2. #35515C2semidirect
3. #29096C2semidirect
4. #29095C2semidirect
5. #29072C2semidirect
6. #29071C2semidirect
7. #23667C2semidirect
8. #23628C2semidirect
10. #23454C2semidirect
11. #21318C2semidirect
13. #21091C2semidirect
14. #21091C2semidirect
15. #21089C2semidirect
16. #21037C2semidirect
18. #20435C2semidirect
19. #20416C2semidirect
37. #2513C22semidirect
39. #2512C22semidirect
41. #2454C22semidirect
42. #2452C22semidirect
55. #2087C22semidirect
56. #2086C22semidirect
57. #2073C22semidirect
58. #2071C22semidirect
61. #2025C22semidirect
62. #2023C22semidirect
63. #1991C22semidirect
64. #1991C22semidirect
65. #1990C22semidirect
66. #1990C22semidirect
71. #1936C22semidirect
72. #1936C22semidirect
73. #1936C22semidirect
74. #1934C22semidirect
75. #1934C22semidirect
76. #1934C22semidirect
77. #1921C22semidirect
78. #1918C22semidirect
81. #1771C22semidirect
82. #1771C22semidirect
83. C8×C4⋊C4C22semidirect
84. C8×C4⋊C4C22semidirect
85. #1762C22semidirect
86. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#355582
#355152
#290962
#290952
#290722
#290712
#236672
#236282
#235402
#234542
#213182
C2×#19702
#210912
#210892
#210372
#209342
#204352
#204162