Finite Group Databasebeta

← #628186

Subgroups — #628186

Order 512.

2578 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2641C211yesmaximal, characteristic
2C2×#2330C211yesmaximal, characteristic, complemented
3#15701C214yesmaximal, complemented
4#15696C214yesmaximal, complemented
5#15381C214yesmaximal, complemented
6#15373C214yesmaximal, complemented
7#15148C212yesmaximal, complemented
8#15138C212yesmaximal, complemented
9#14713C211yesmaximal, characteristic
10#14709C211yesmaximal, characteristic, complemented
11#14277C211yesmaximal, characteristic, complemented
12#14272C211yesmaximal, characteristic
13#14141C212yesmaximal, complemented
14#14022C212yesmaximal, complemented
15#13755C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#125C2211yescharacteristic
18#2641—28no—
19C2×#399C2214yes—
20C2×#397C2214yes—
21C2×#394C2214yes—
22C2×#394C2214yes—
23#2330—28no—
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2214yes—
29C2×#390C2214yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#387C2214yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×#376C2211yescharacteristic
44C2×#376C2211yescharacteristic
45C2×C43C2211yescharacteristic
46#1444C2214yes—
47#1442C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69#1434C2212yes—
70#1434C2212yes—
71C4×#95C2211yescharacteristic
72C4×#95C2211yescharacteristic
73C23×D4C2312yes—
74C23×D4C2314yes—
75C2×#127—24no—
76C2×#127—24no—
77C2×#127—24no—
78C2×#125—24no—
79C2×#125—24no—
80C2×#125D418yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2314yes—
112C22×C42C2311yescharacteristic
113#399—48no—
114#397—48no—
115#394—216no—
116#394—216no—
117#390—28no—
118#390—28no—
119#390—28no—
120#390—28no—
121#390—48no—
122#390—48no—
123#390—44no—
124#390—44no—
125#389—216no—
126#389—28no—
127#389—44no—
128#388—48no—
129#388—48no—
130#387—216no—
131#384—24no—
132#382—24no—
133C4×C4⋊C4—28no—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—24no—
136#376—24no—
137#376—24no—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C43—24no—
160C25C2414yes—
161C22×D4—24no—
162C22×D4—28no—
163C22×D4—28no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—28no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4—44no—
175C23×C4C2411yescharacteristic
176C23×C4C2411yescharacteristic
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2411yescharacteristic
180C23×C4C2412yes—
181C23×C4C2411yescharacteristic
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189#127—416no—
190#125—232no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4C2×D418yes—
218C2×C4⋊C4C2×D4116yes—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—28no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42C2×D414yes—
281C2×C42#4114yes—
282C2×C42#4114yes—
283C2×C42#4114yes—
284C2×C42#4114yes—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—28no—
290#95—28no—
291#95—24no—
292#95—22no—
293#95—22no—
294#95—28no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—24no—
299#95—42no—
300#95—44no—
301#95—44no—
302#95—42no—
303#95—44no—
304#95—44no—
305#95—44no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24C2511yescharacteristic
322C2×D4—416no—
323C2×D4—416no—
324C2×D4—416no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—88no—
333C2×D4—88no—
334C2×D4—88no—
335C2×D4—88no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4C22×D418yes—
407C22×C4C22×D414yes—
408C22×C4C22×D412yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4114yes—
411C22×C4C2×#4114yes—
412C22×C4C2×#4114yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4H(5,𝔽2)14yes—
419C22×C4H(5,𝔽2)14yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C42−1+412yes—
422C4⋊C4—232no—
423C4⋊C4—216no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—88no—
430C4⋊C4—88no—
431C4⋊C4—88no—
432C22⋊C4—416no—
433C22⋊C4—416no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C22⋊C4—88no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C42—28no—
445C42—28no—
446C42—48no—
447C42—48no—
448C42—44no—
449C42—44no—
450C42—44no—
451C42—44no—
452C42—88no—
453C42#123116yes—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23C23×D411yescharacteristic
490C23C22×#4111yescharacteristic
491C23C22×#4111yescharacteristic
492C23C22×#4111yescharacteristic
493C23C22×#4111yescharacteristic
494C23C2×H(5,𝔽2)12yes—
495C23C2×H(5,𝔽2)11yescharacteristic
496C23C2×H(5,𝔽2)12yes—
497C23C2×2−1+411yescharacteristic
498C23#58512yes—
499C23#58512yes—
500D4—832no—
501D4—816no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4C2×#12318yes—
580C2×C4C2×#12314yes—
581C2×C4#52914yes—
582C2×C4#54014yes—
583C2×C4#54214yes—
584C2×C4#54814yes—
585C2×C4#55014yes—
586C2×C4#55614yes—
587C2×C4#55914yes—
588C22—84no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22—84no—
603C22C22×#12311yescharacteristic
604C22C2×#52911yescharacteristic
605C22C2×#54011yescharacteristic
606C22C2×#54211yescharacteristic
607C22#345511yescharacteristic
608C22#346012yes—
609C22C2×#54811yescharacteristic
610C22C2×#55011yescharacteristic
611C22C2×#55611yescharacteristic
612C22C2×#55911yescharacteristic
613C22#349412yes—
614C22#349412yes—
615C22#349412yes—
616C22#349812yes—
617C22#350012yes—
618C22#350612yes—
619C22#350612yes—
620C22#350811yescharacteristic
621C22#351211yescharacteristic
622C22#354312yes—
623C22#354712yes—
624C22#355912yes—
625C22#356212yes—
626C4—24no—
627C4—28no—
628C4—44no—
629C4—44no—
630C4—48no—
631C4—42no—
632C4—82no—
633C4—84no—
634C4—81no—
635C4—81no—
636C4—84no—
637C4—81no—
638C4—81no—
639C4#244818yes—
640C2—164no—
641C2C2×#244811yescharacteristic
642C2#3527111yescharacteristic
643C2#3529511yescharacteristic
644C2#3553112yes—
645C2#3729212yes—
646C2#3938211yescharacteristic
647C2#3938611yescharacteristic
648C2#3942612yes—
649C2#3975611yescharacteristic
650C2#3976311yescharacteristic
651C2#3980812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2330C2semidirect
3. #15701C2semidirect
4. #15696C2semidirect
5. #15381C2semidirect
6. #15373C2semidirect
7. #15148C2semidirect
8. #15138C2semidirect
10. #14709C2semidirect
11. #14277C2semidirect
13. #14141C2semidirect
14. #14022C2semidirect
15. #13755C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26412
C2×#23302
#157012
#156962
#153812
#153732
#151482
#151382
#147132
#147092
#142772
#142722
#141412
#140222
#137552