Finite Group Databasebeta

← #6446162

Subgroups — #6446162

Order 512.

2008 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2C2×#2960C211yesmaximal, characteristic, complemented
3#35173C211yesmaximal, characteristic, complemented
4#35172C211yesmaximal, characteristic
5#25478C212yesmaximal, complemented
6#25477C212yesmaximal, complemented
7#25475C214yesmaximal, complemented
8#25473C214yesmaximal, complemented
9#25470C212yesmaximal, complemented
10#25469C212yesmaximal, complemented
11#23671C212yesmaximal, complemented
12#23614C212yesmaximal
13#21318C211yesmaximal, characteristic, complemented
14#21170C211yesmaximal, characteristic, complemented
15#21160C212yesmaximal, complemented
16#21089C212yesmaximal
17C2×C4×#102C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#3077—28no—
22C2×#478C2214yes—
23C2×#476C2212yes—
24C2×#474C2212yes—
25#2960—28no—
26C2×#446C2212yescomplemented
27C2×#446C2212yescomplemented
28C2×C8×Q8C2212yescomplemented
29C2×C8×Q8C2212yescomplemented
30C2×#428C2212yes—
31C2×#428C2212yes—
32C2×#424C2212yes—
33C2×#423C2211yescharacteristic
34C2×#418C2212yes—
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C2×#405C2212yes—
38C22×C4×C8C2211yescharacteristic
39#2857—22no—
40#2848—22no—
41#2659—22no—
42#2626—22no—
43#2584—22no—
44#2584C2214yescomplemented
45#2580C2214yes—
46#2567—21no—
47#2567C2212yescomplemented
48#2566—22no—
49#2563—21no—
50#2546—21no—
51#2546C2212yescomplemented
52#2542—21no—
53#2538C2212yes—
54#2538C2212yes—
55#2317C2212yes—
56#2316C2212yes—
57#2316C2212yes—
58#2309C2212yes—
59#2285C2211yescharacteristic
60C2×C4×C4⋊C4C2211yescharacteristic
61#2088C2214yescomplemented
62#2088C2214yescomplemented
63#2087C2214yescomplemented
64#2086C2214yescomplemented
65#2067C2214yes—
66#2067C2214yes—
67#2065C2214yes—
68#2065C2214yes—
69#1991C2212yescomplemented
70#1991C2212yescomplemented
71#1990C2211yescharacteristic, complemented
72#1990C2211yescharacteristic, complemented
73#1990C2211yescharacteristic, complemented
74#1990C2211yescharacteristic, complemented
75#1949C2214yescomplemented
76#1948C2212yescomplemented
77#1947C2212yescomplemented
78#1944C2214yescomplemented
79#1943C2214yescomplemented
80#1942C2214yescomplemented
81#1941C2214yescomplemented
82#1936C2214yes—
83#1936C2214yes—
84#1934C2214yes—
85#1934C2214yes—
86C4×#102C2212yes—
87C4×#102C2212yes—
88C4×#102C2212yes—
89C4×#102C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92#1735C2212yes—
93C2×#340C2211yescharacteristic
94C2×#340C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×M16C2312yes—
98C23×C8C2311yescharacteristic
99C23×C8C2311yescharacteristic
100#533—24no—
101#533—24no—
102#533—24no—
103#533—24no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8D411yescharacteristic
106C2×C4⋊Q8D411yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×#125D412yes—
110C2×#125C2312yes—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8D412yes—
116C2×C22⋊Q8D412yes—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—24no—
124C2×C4×D4—24no—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×#117C2312yes—
128C2×#117C2312yes—
129C2×#117C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—21no—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136#478—216no—
137#476—28no—
138#474—28no—
139#446—28no—
140#446—28no—
141C8×Q8—28no—
142C8×Q8—28no—
143#428—28no—
144#428—28no—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149#424—28no—
150#423D418yes—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155#418—28no—
156C2×#102—24no—
157C2×#102Q818yescomplemented
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164#405—28no—
165C2×#97C2314yes—
166C2×C4×C8—24no—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2314yes—
172#399—24no—
173#398—24no—
174#398—24no—
175#396—24no—
176#396—24no—
177#395—24no—
178#395—24no—
179#393—24no—
180#391—22no—
181#391—22no—
182#390—22no—
183#390—22no—
184#390C2314yes—
185#390C2314yes—
186#389—22no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—24no—
191#389—24no—
192#389C2314yes—
193#389C2314yes—
194#389C2314yes—
195#389C2314yes—
196#389C2314yes—
197#389C2314yes—
198#385—24no—
199#385—24no—
200#385—24no—
201#384—22no—
202#384—22no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#382—22no—
206#382C2312yes—
207#380—21no—
208#380—21no—
209#380—22no—
210#380—22no—
211#380—24no—
212#380—24no—
213#379—24no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C2×#95—21no—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235C25—21no—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8C2×D412yes—
239C22×Q8C2×D412yes—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4C2×Q814yes—
247C22×D4C2412yes—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—44no—
254C23×C4—82no—
255C23×C4—82no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C2×M16—24no—
260C2×M16—24no—
261C2×M16C2×D418yes—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8C2×Q814yes—
268C22×C8C2411yescharacteristic
269C22×C8C2411yescharacteristic
270C22×C8C2414yes—
271C22×C8C2411yescharacteristic
272C22×C8C2411yescharacteristic
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4⋊Q8—84no—
280#125—28no—
281#125—24no—
282#125—24no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287C22⋊Q8—84no—
288C22⋊Q8—84no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—44no—
300C4×D4—44no—
301#117—24no—
302#117—24no—
303#117—24no—
304#117—24no—
305#117—24no—
306#117—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D414yes—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×Q814yes—
335C2×C4⋊C4#4114yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4C2×D414yes—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—82no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D414yes—
361C2×C42#4114yes—
362C2×C42C2414yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365#107—28no—
366#107—28no—
367#106—28no—
368#106—28no—
369C4⋊C8—28no—
370C4⋊C8—28no—
371C4⋊C8C2×D418yes—
372C4⋊C8C2×D418yes—
373#102—28no—
374#102—28no—
375#102—24no—
376#102—24no—
377#102C2×Q818yes—
378#102C2×Q818yes—
379#97—28no—
380C4×C8—28no—
381C4×C8—22no—
382C4×C8—22no—
383C4×C8—22no—
384C4×C8—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395C24—22no—
396C24—24no—
397C24—42no—
398C24—42no—
399C24—81no—
400C24—81no—
401C24—81no—
402C24—81no—
403C24C22×D411yescharacteristic
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—81no—
415C2×Q8—81no—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4C22×Q814yes—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—816no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4#12114yes—
457C22×C4#12114yes—
458C22×C4C22×D414yes—
459C22×C4C22×D414yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×Q812yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C42−1+412yes—
470C22×C42−1+412yes—
471C22×C4C2511yescharacteristic
472M16—216no—
473C2×C8—28no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—24no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—24no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—44no—
487C2×C8C22×D418yes—
488C2×C8C22×Q812yes—
489C2×C8C22×Q812yes—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4C22×Q814yes—
526C4⋊C4C2×#4114yes—
527C22⋊C4—28no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C42—24no—
537C42—24no—
538C42—44no—
539C42—44no—
540C42—44no—
541C42—82no—
542C42#12114yes—
543C42#12114yes—
544C42C22×D412yes—
545C42C22×D412yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—24no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—48no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23C2×#12112yes—
567C23#54812yes—
568C23D4×Q812yes—
569C23C23×D411yescharacteristic
570Q8—48no—
571Q8—48no—
572Q8—84no—
573Q8—84no—
574D4—24no—
575D4—24no—
576D4—48no—
577D4—42no—
578D4—42no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—44no—
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—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—88no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—88no—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C2×#12112yes—
622C2×C4C2×#12112yes—
623C2×C4C2×#12112yes—
624C2×C4#53314yes—
625C2×C4#54114yes—
626C2×C4#54714yes—
627C2×C4#54812yes—
628C2×C4D4×Q812yes—
629C2×C4D4×Q814yes—
630C2×C4#57612yes—
631C2×C4#57712yes—
632C2×C4C23×D411yescharacteristic
633C2×C4C23×Q811yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C8—28no—
639C8—22no—
640C8—22no—
641C8—41no—
642C8—41no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—816no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22#242914yes—
658C22C22×#12111yescharacteristic
659C22C2×#54811yescharacteristic
660C22C2×D4×Q811yescharacteristic
661C22C2×#57611yescharacteristic
662C22C2×#57711yescharacteristic
663C22#358411yescharacteristic
664C22#358411yescharacteristic
665C4—22no—
666C4—22no—
667C4—22no—
668C4—44no—
669C4—44no—
670C4—44no—
671C4—44no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4#242914yes—
678C4#306214yes—
679C4C2×#53311yescharacteristic
680C4C2×#54111yescharacteristic
681C4C2×#54711yescharacteristic
682C4C2×D4×Q811yescharacteristic
683C2—24no—
684C2—84no—
685C2C2×#242911yescharacteristic
686C2C2×#306211yescharacteristic
687C2#6045311yescharacteristic
688C2#6132511yescharacteristic
689C2#6132711yescharacteristic
690C2#6137011yescharacteristic
691C2#6137111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2960C2semidirect
3. #35173C2semidirect
5. #25478C2semidirect
6. #25477C2semidirect
7. #25475C2semidirect
8. #25473C2semidirect
9. #25470C2semidirect
10. #25469C2semidirect
11. #23671C2semidirect
13. #21318C2semidirect
14. #21170C2semidirect
15. #21160C2semidirect
26. C2×#446C22semidirect
27. C2×#446C22semidirect
28. C2×C8×Q8C22semidirect
29. C2×C8×Q8C22semidirect
44. #2584C22semidirect
47. #2567C22semidirect
51. #2546C22semidirect
61. #2088C22semidirect
62. #2088C22semidirect
63. #2087C22semidirect
64. #2086C22semidirect
69. #1991C22semidirect
70. #1991C22semidirect
71. #1990C22semidirect
72. #1990C22semidirect
73. #1990C22semidirect
74. #1990C22semidirect
75. #1949C22semidirect
76. #1948C22semidirect
77. #1947C22semidirect
78. #1944C22semidirect
79. #1943C22semidirect
80. #1942C22semidirect
81. #1941C22semidirect
157. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30772
C2×#29602
#351732
#351722
#254782
#254772
#254752
#254732
#254702
#254692
#236712
#236142
#213182
#211702
#211602
#210892
C2×C4×#1022