Finite Group Databasebeta

← #196696

Subgroups — #196696

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic, complemented
4C2×#1560C212yesmaximal, complemented
5#9142C212yesmaximal, complemented
6#9136C212yesmaximal, complemented
7#9134C212yesmaximal, complemented
8#9128C212yesmaximal, complemented
9#8041C212yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C2×#525C412yes—
12C23×C42C2211yescharacteristic
13#2920—28no—
14C2×#432—21no—
15C2×#432C2212yes—
16C2×#431—21no—
17C2×#431C2212yes—
18C2×#423C2212yes—
19C2×#423C2212yes—
20C2×#407—21no—
21C2×#407—21no—
22C2×C4×M16—21no—
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2856C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854—21no—
28#2854—21no—
29#2850C412yes—
30#2850C412yes—
31#2847—21no—
32#2847—21no—
33C4×#121—21no—
34C4×#121—21no—
35C4×#121—21no—
36C4×#121—21no—
37#2297C412yes—
38#2285C2211yescharacteristic
39#1797—21no—
40#1797—21no—
41#1797C2212yes—
42#1797C2212yes—
43#1728—21no—
44#1560—28no—
45C2×#333C2214yescomplemented
46C2×#331C2214yescomplemented
47#1296C2214yescomplemented
48#1296C2214yescomplemented
49C22×#41C2×C412yes—
50C24×C4C2×C411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C22×M16—22no—
54C22×M16—22no—
55#532—24no—
56#532—24no—
57C2×C4⋊Q8C2311yescharacteristic
58C2×#127C2311yescharacteristic
59C2×#124C2×C412yes—
60#525—48no—
61C2×#123—22no—
62C2×C22⋊Q8C2×C412yes—
63C2×C22⋊Q8C2×C412yes—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121C2×C412yes—
71C2×#121C2×C412yes—
72C2×#120—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×#117C2×C412yes—
78C22×C42—24no—
79C22×C42—24no—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#432—28no—
93#432—28no—
94#431—28no—
95#431—28no—
96#423—24no—
97#423—24no—
98#423—24no—
99#423D418yescomplemented
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8C2314yes—
103C2×C4⋊C8C2314yes—
104#407—28no—
105#407—44no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C4×M16—28no—
111C2×#97—22no—
112C2×C4×C8—22no—
113#391C2×C412yes—
114#391C2×C412yes—
115#390—21no—
116#390—21no—
117#390—21no—
118#390—21no—
119#390C2×C412yes—
120#390C2×C412yes—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#388—22no—
126#388—22no—
127#388—22no—
128#388—22no—
129#386—22no—
130#386—22no—
131#386—22no—
132#386—22no—
133#386—22no—
134#386—22no—
135#384C2311yescharacteristic
136#384C2311yescharacteristic
137#382C2×C412yes—
138#380—21no—
139#380—21no—
140#380—21no—
141#380—21no—
142#380—21no—
143#380—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4—22no—
148#336—22no—
149#336—22no—
150#333—28no—
151#331—28no—
152C25C22⋊C411yescharacteristic
153C2×#41—28no—
154C2×#41—44no—
155C2×#41—44no—
156C22×Q8C22×C412yes—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4C22×C412yes—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C2×M16—28no—
194C2×M16—44no—
195C2×M16—44no—
196C2×M16—44no—
197C2×M16—44no—
198C2×M16—44no—
199C22×C8—24no—
200C22×C8—24no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203#127—22no—
204#127—22no—
205#124—28no—
206#123—48no—
207C22⋊Q8—44no—
208C22⋊Q8—44no—
209#121—44no—
210#121—44no—
211#121—44no—
212#121—44no—
213#121—42no—
214#121—44no—
215#121—42no—
216#121—42no—
217#121—42no—
218#121—44no—
219#120—48no—
220C4×D4—48no—
221C4×D4—48no—
222C4×D4—42no—
223C4×D4—44no—
224C4×D4—42no—
225#117—44no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D414yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42#4114yes—
281C2×C42C2411yescharacteristic
282C4⋊C8—28no—
283C4⋊C8—24no—
284C4⋊C8—24no—
285C4⋊C8—24no—
286C4⋊C8—44no—
287C4⋊C8C2×D418yes—
288C22⋊C8—28no—
289C22⋊C8—28no—
290C22⋊C8—44no—
291C22⋊C8—44no—
292#97—28no—
293C4×C8—28no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24C2×C22⋊C411yescharacteristic
309C24C2×C22⋊C411yescharacteristic
310C24C2×C22⋊C411yescharacteristic
311#41—816no—
312C2×Q8—24no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×D4—24no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—41no—
328C2×D4—42no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C22×C4—22no—
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—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—416no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4#10014yes—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C4×D414yes—
410C22×C4#12012yes—
411C22×C4#12114yes—
412C22×C4#12312yes—
413C22×C4AGL(1,ℤ/8)12yes—
414C22×C4#13712yes—
415C22×C4C23×C411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418M16—416no—
419M16—88no—
420C2×C8—28no—
421C2×C8—44no—
422C2×C8—44no—
423C2×C8—44no—
424C2×C8—44no—
425C2×C8—44no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C42—22no—
450C42—22no—
451C42—21no—
452C42—21no—
453C42—22no—
454C42—22no—
455C42—21no—
456C42—21no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—44no—
464C42C4×D414yes—
465C42#12012yes—
466C42#12114yes—
467C42#12312yes—
468C42C22×D412yes—
469C42C2×#4112yes—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23#38612yes—
503C23C2×#10012yes—
504C23#41712yes—
505C23C22×C22⋊C411yescharacteristic
506Q8—84no—
507D4—84no—
508D4—88no—
509D4—88no—
510D4—88no—
511D4—82no—
512D4—82no—
513D4—82no—
514D4—82no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—88no—
571C2×C4#38612yes—
572C2×C4#38612yes—
573C2×C4#38612yes—
574C2×C4C2×#10012yes—
575C2×C4C2×#10012yes—
576C2×C4#41312yes—
577C2×C4#41712yes—
578C2×C4C2×C4×D412yes—
579C2×C4C2×#12011yescharacteristic
580C2×C4C2×#12112yes—
581C2×C4C2×#12311yescharacteristic
582C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
583C2×C4C2×#13711yescharacteristic
584C2×C4#57512yes—
585C2×C4#57512yes—
586C2×C4#57512yes—
587C8—48no—
588C8—84no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—416no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22C2×#38611yescharacteristic
619C22C22×#10011yescharacteristic
620C22C2×#41311yescharacteristic
621C22C2×#41711yescharacteristic
622C22#289611yescharacteristic
623C22#289611yescharacteristic
624C22#289611yescharacteristic
625C4—22no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4#190014yes—
638C4#300512yes—
639C4#305512yes—
640C4#305612yes—
641C4#309312yes—
642C2—42no—
643C2—42no—
644C2—42no—
645C2—162no—
646C2#1773711yescharacteristic
647C2#1773811yescharacteristic
648C2#1775011yescharacteristic
649C2#1775711yescharacteristic
650C2#2091011yescharacteristic
651C2#2097111yescharacteristic
652C2C2×#190011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2920C2semidirect
3. #20626C2semidirect
4. C2×#1560C2semidirect
5. #9142C2semidirect
6. #9136C2semidirect
7. #9134C2semidirect
8. #9128C2semidirect
9. #8041C2semidirect
45. C2×#333C22semidirect
46. C2×#331C22semidirect
47. #1296C22semidirect
48. #1296C22semidirect
99. #423D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
#366242
#206262
C2×#15602
#91422
#91362
#91342
#91282
#80412