Finite Group Databasebeta

← #7616403

Subgroups — #7616403

Order 512.

1229 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#45363C212yesmaximal
2#32523C212yesmaximal
3#32520C212yesmaximal
4#29292C212yesmaximal
5#28832C211yesmaximal, characteristic
6#28832C211yesmaximal, characteristic
7#28679C211yesmaximal, characteristic
8#28679C211yesmaximal, characteristic
9#27995C211yesmaximal, characteristic
10#27994C211yesmaximal, characteristic
11#27993C211yesmaximal, characteristic
12#27992C211yesmaximal, characteristic
13#26519C212yesmaximal
14#26518C212yesmaximal
15#26291C212yesmaximal
16#26290C212yesmaximal
17#23158C211yesmaximal, characteristic
18#23157C211yesmaximal, characteristic
19#22727C211yesmaximal, characteristic
20Q8×#107C211yesmaximal, characteristic
21#21725C211yesmaximal, characteristic
22#21723C211yesmaximal, characteristic
23#21715C211yesmaximal, characteristic
24C2×#500C2211yescharacteristic
25C2×#499C2211yescharacteristic
26C2×C8×Q8C2211yescharacteristic
27C2×C8×Q8C2211yescharacteristic
28#2809—22no—
29#2806—22no—
30#2802—22no—
31#2774—22no—
32#2590—22no—
33#2584C2212yes—
34#2584C2212yes—
35#2581—22no—
36#2570C2212yes—
37#2570C2212yes—
38#2509C2212yes—
39#2508C2212yes—
40#2507C2212yes—
41#2506C2212yes—
42#2462—22no—
43#2453—22no—
44#2399C2211yescharacteristic
45#2369C2212yes—
46#2365C2212yes—
47#2355C2212yes—
48Q8×C4⋊C4C2212yes—
49#2331C2212yes—
50#2330C2212yes—
51#2101C2212yes—
52#2101C2212yes—
53#2100C2212yes—
54#2099C2212yes—
55#2088C2212yes—
56#2088C2212yes—
57#2088C2212yes—
58#2088C2212yes—
59#2088C2212yes—
60#2088C2212yes—
61#2087C2212yes—
62#2086C2212yes—
63#2085C2212yes—
64#2085C2212yes—
65#2084C2212yes—
66#2084C2212yes—
67#1993C2212yes—
68#1992C2212yes—
69#1989C2212yes—
70#1989C2212yes—
71#1988C2212yes—
72#1987C2212yes—
73#1985C2211yescharacteristic
74#1985C2211yescharacteristic
75#1949C2212yes—
76#1949C2212yes—
77#1949C2212yes—
78#1949C2212yes—
79#1948C2212yes—
80#1947C2212yes—
81#1947C2212yes—
82#1947C2212yes—
83#1944C2212yes—
84#1943C2212yes—
85#1942C2211yescharacteristic
86#1942C2211yescharacteristic
87#1942C2211yescharacteristic
88#1942C2211yescharacteristic
89#1942C2211yescharacteristic
90#1942C2211yescharacteristic
91#1942C2211yescharacteristic
92#1942C2211yescharacteristic
93#1942C2211yescharacteristic
94#1942C2211yescharacteristic
95#1942C2211yescharacteristic
96#1942C2211yescharacteristic
97#1940C2211yescharacteristic
98#1940C2211yescharacteristic
99#1940C2211yescharacteristic
100#1940C2211yescharacteristic
101#1921C2212yes—
102#1921C2212yes—
103#1921C2212yes—
104#1920C2212yes—
105#1919C2212yes—
106#1918C2212yes—
107#1918C2212yes—
108#1918C2212yes—
109#1846C2211yescharacteristic
110#1846C2211yescharacteristic
111#1846C2211yescharacteristic
112#1845C2211yescharacteristic
113#1845C2211yescharacteristic
114#1845C2211yescharacteristic
115#1843C2211yescharacteristic
116#1776C2212yes—
117#1776C2212yes—
118C4×#107C2211yescharacteristic
119C4×#107C2211yescharacteristic
120C4×#107C2211yescharacteristic
121C4×#107C2211yescharacteristic
122C4×#107C2211yescharacteristic
123C4×#107C2211yescharacteristic
124C4×#107C2211yescharacteristic
125C4×#107C2211yescharacteristic
126C8×C4⋊C4C2211yescharacteristic
127C8×C4⋊C4C2211yescharacteristic
128C8×C4⋊C4C2211yescharacteristic
129C8×C4⋊C4C2211yescharacteristic
130#1726C2211yescharacteristic
131C2×C4⋊Q8C2312yes—
132C2×#125—22no—
133C2×#125C2312yes—
134C2×C4×Q8—22no—
135C2×C4×Q8—22no—
136C2×C4×Q8C2311yescharacteristic
137C2×C4×Q8C2311yescharacteristic
138#500Q818yescomplemented
139#499—24no—
140C8×Q8—24no—
141C8×Q8—24no—
142C2×#107C2311yescharacteristic
143C2×#107C2311yescharacteristic
144C2×#107C2311yescharacteristic
145C2×#107C2311yescharacteristic
146C2×#107C2311yescharacteristic
147C2×#107C2311yescharacteristic
148C2×#107C2312yes—
149C2×#107C2312yes—
150C2×#107C2312yes—
151C2×#107C2312yes—
152C2×#107C2311yescharacteristic
153C2×#107C2311yescharacteristic
154C2×#106C2312yes—
155C2×#106C2312yes—
156C2×#106C2312yes—
157C2×#106C2312yes—
158C2×C4⋊C8C2311yescharacteristic
159C2×C4⋊C8C2311yescharacteristic
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2311yescharacteristic
165C2×C4⋊C8C2311yescharacteristic
166C2×C4⋊C8C2311yescharacteristic
167C2×C4⋊C8C2311yescharacteristic
168C2×#97C2311yescharacteristic
169C2×#97C2311yescharacteristic
170C2×#97C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2311yescharacteristic
173C2×C4×C8C2311yescharacteristic
174C2×C4×C8C2311yescharacteristic
175C2×C4×C8C2311yescharacteristic
176C2×C4×C8C2311yescharacteristic
177C2×C4×C8C2311yescharacteristic
178#400—22no—
179#400—22no—
180#400—22no—
181#400—22no—
182#400—22no—
183#400—22no—
184#398—22no—
185#398—22no—
186#398—22no—
187#398—22no—
188#398—22no—
189#398—22no—
190#398—22no—
191#398—22no—
192#398D411yescharacteristic
193#398D411yescharacteristic
194#398D411yescharacteristic
195#398D411yescharacteristic
196#395—22no—
197#395—22no—
198#395D411yescharacteristic
199#395D411yescharacteristic
200#395D411yescharacteristic
201#395D411yescharacteristic
202#391—22no—
203#391—22no—
204#391—22no—
205#391—22no—
206#391—22no—
207#391—22no—
208#389—22no—
209#389—22no—
210#389—22no—
211#389—22no—
212#389—22no—
213#389—22no—
214#389—22no—
215#389—22no—
216#389—22no—
217#389—22no—
218#389—22no—
219#389—22no—
220#389C2312yes—
221#389C2312yes—
222#389C2312yes—
223#389C2312yes—
224#389C2312yes—
225#389C2312yes—
226#389C2312yes—
227#389C2312yes—
228#389C2312yes—
229#389C2312yes—
230#389C2312yes—
231#389C2312yes—
232#387—22no—
233#387—22no—
234#387—22no—
235#387—22no—
236#387—22no—
237#387—22no—
238#387—22no—
239#387—22no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387—22no—
244#387—22no—
245#387—22no—
246#387—22no—
247#387—22no—
248#384C2312yes—
249#384C2312yes—
250#384C2312yes—
251#382—22no—
252#382C2312yes—
253#382C2312yes—
254#382C2312yes—
255C4×C4⋊C4—22no—
256C4×C4⋊C4—22no—
257C4×C4⋊C4C2312yes—
258C4×C4⋊C4C2312yes—
259C4×C4⋊C4C2312yes—
260C4×C4⋊C4C2312yes—
261C4×C4⋊C4C2312yes—
262C4×C4⋊C4C2312yes—
263C4×C4⋊C4C2312yes—
264C4×C4⋊C4C2312yes—
265C4×C4⋊C4C2311yescharacteristic
266C4×C4⋊C4C2311yescharacteristic
267C4×C4⋊C4C2311yescharacteristic
268C4×C4⋊C4C2311yescharacteristic
269#376C2311yescharacteristic
270#340C2312yes—
271#340C2312yes—
272#340C2312yes—
273#340C2312yes—
274#340C2312yes—
275#340C2312yes—
276#340C2312yes—
277#340C2312yes—
278#340C2312yes—
279#340C2312yes—
280#340C2312yes—
281#340C2312yes—
282#340C2312yes—
283#340C2312yes—
284#340C2312yes—
285#340C2312yes—
286#340C2312yes—
287#340C2312yes—
288#340C2312yes—
289#340C2312yes—
290#336C2311yescharacteristic
291#336C2311yescharacteristic
292#336C2311yescharacteristic
293#336C2311yescharacteristic
294#336C2311yescharacteristic
295#336C2311yescharacteristic
296#336C2311yescharacteristic
297#336C2311yescharacteristic
298C22×Q8—22no—
299C22×Q8C2×D411yescharacteristic
300C22×Q8C2×D411yescharacteristic
301C22×C8C2411yescharacteristic
302C22×C8C2411yescharacteristic
303C22×C8C2411yescharacteristic
304C22×C8C2411yescharacteristic
305C22×C8C2411yescharacteristic
306C22×C8C2411yescharacteristic
307C22×C8C2411yescharacteristic
308C22×C8C2411yescharacteristic
309C4⋊Q8—24no—
310C4⋊Q8C2×Q818yes—
311#125—24no—
312#125—24no—
313#125—84no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—84no—
319C4×Q8—84no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378#107—22no—
379#107—22no—
380#107—22no—
381#107—22no—
382#107—22no—
383#107—22no—
384#107—42no—
385#107—42no—
386#107—42no—
387#107C2×Q818yes—
388#107C2×Q814yes—
389#107C2×Q814yes—
390#106—24no—
391#106—42no—
392#106—42no—
393#106—42no—
394C4⋊C8—22no—
395C4⋊C8—22no—
396C4⋊C8—22no—
397C4⋊C8—22no—
398C4⋊C8—22no—
399C4⋊C8—22no—
400C4⋊C8—22no—
401C4⋊C8—22no—
402C4⋊C8—41no—
403C4⋊C8—41no—
404#97—22no—
405#97—22no—
406#97—41no—
407C4×C8—22no—
408C4×C8—22no—
409C4×C8—22no—
410C4×C8—22no—
411C4×C8—22no—
412C4×C8C2×Q814yes—
413C4×C8#4114yes—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—28no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—41no—
458C2×Q8—41no—
459C2×Q8—82no—
460C2×Q8—82no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2511yescharacteristic
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8C22×Q812yes—
504C2×C8C22×Q812yes—
505C2×C8C2×#4112yes—
506C2×C8C2×#4112yes—
507C2×C82−1+412yes—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—28no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4—82no—
556C4⋊C4—82no—
557C4⋊C4C22×Q814yes—
558C4⋊C4C22×Q814yes—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—44no—
570C42—82no—
571C42—82no—
572C42—82no—
573C42—82no—
574C42—82no—
575C42—82no—
576C42—82no—
577C42C22×Q812yes—
578C42C2×#4112yes—
579C422−1+412yes—
580C422−1+412yes—
581C23C23×D411yescharacteristic
582Q8—28no—
583Q8—44no—
584Q8—88no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4#54712yes—
646C2×C4#54712yes—
647C2×C4D4×Q812yes—
648C2×C4D4×Q812yes—
649C2×C4#57612yes—
650C2×C4#57812yes—
651C2×C4C23×Q811yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×2−1+411yescharacteristic
654C2×C4C2×2−1+411yescharacteristic
655C2×C4C2×2−1+411yescharacteristic
656C2×C4#58511yescharacteristic
657C8—22no—
658C8—22no—
659C8—41no—
660C8—41no—
661C8—41no—
662C8—41no—
663C8—41no—
664C8#55714yes—
665C22C2×#54711yescharacteristic
666C22C2×D4×Q811yescharacteristic
667C22#349711yescharacteristic
668C22C2×#57611yescharacteristic
669C22C2×#57811yescharacteristic
670C22#358411yescharacteristic
671C22#358411yescharacteristic
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—44no—
687C4—44no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4C2×#55711yescharacteristic
695C4#350211yescharacteristic
696C4#350711yescharacteristic
697C4#355611yescharacteristic
698C2#5982811yescharacteristic
699C2#6125811yescharacteristic
700C2#6126111yescharacteristic
701C2#6137011yescharacteristic
702C2#6137211yescharacteristic
703C2#6237911yescharacteristic
704C2#6238011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2399 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
138. #500Q8semidirect

Maximal subgroups

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

subgroupindex
#453632
#325232
#325202
#292922
#288322
#286792
#279952
#279942
#279932
#279922
#265192
#265182
#262912
#262902
#231582
#231572
#227272
Q8×#1072
#217252
#217232
#217152