Finite Group Databasebeta

← #193650

Subgroups — #193650

Order 512.

1340 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2970C211yesmaximal, characteristic, complemented
2C2×C4×#135C211yesmaximal, characteristic
3#21380C211yesmaximal, characteristic, complemented
4C2×#1488C211yesmaximal, characteristic, complemented
5C2×#1488C211yesmaximal, characteristic, complemented
6#8952C212yesmaximal, complemented
7#8950C212yesmaximal, complemented
8#8947C212yesmaximal, complemented
9#8945C212yesmaximal, complemented
10#7875C211yesmaximal, characteristic, complemented
11#7848C211yesmaximal, characteristic, complemented
12C22×#135C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×C4×#41C2211yescharacteristic
15#2970—24no—
16#2970—24no—
17C2×#446C2212yes—
18C4×#135—24no—
19C4×#135—24no—
20C2×C4×Q16C411yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×SD16C411yescharacteristic
23C2×C4×SD16C411yescharacteristic
24C2×C4×SD16C2211yescharacteristic
25C2×C4×SD16C2211yescharacteristic
26C2×C4×D8C411yescharacteristic
27C2×C4×D8C2211yescharacteristic
28C2×#435—22no—
29C2×C8×D4C2212yes—
30C2×#432—21no—
31C2×#432C2211yescharacteristic
32C2×#432C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C2×#427C411yescharacteristic
35C2×#416C411yescharacteristic
36C2×#416C411yescharacteristic
37C2×C4×M16C2211yescharacteristic
38C22×#97—21no—
39C22×C4×C8C2211yescharacteristic
40#1984C2212yes—
41#1954C2212yes—
42#1870—22no—
43#1749—21no—
44#1749—21no—
45#1488—24no—
46#1488—24no—
47C2×Q8⋊C8C2212yes—
48C2×Q8⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50C2×D4⋊C8C2212yes—
51#1277C2212yes—
52#1277C2212yes—
53#1273C2212yes—
54#1273C2212yes—
55C22×#41C2×C411yescharacteristic
56C22×#41C2×C411yescharacteristic
57C2×#135—24no—
58C2×#135—24no—
59C2×#135—24no—
60C22×Q16C2×C411yescharacteristic
61C22×SD16C2×C411yescharacteristic
62C22×SD16C2×C411yescharacteristic
63C22×D8C2×C411yescharacteristic
64C22×M16D412yes—
65C23×C8—22no—
66C23×C8C2×C411yescharacteristic
67C23×C8C2×C411yescharacteristic
68C4×#41—22no—
69C4×#41—22no—
70C4×#41—22no—
71C4×#41—22no—
72C4×#41—22no—
73C4×#41—22no—
74C4×#41D414yes—
75C4×#41D414yes—
76C2×C4×Q8C2×C411yescharacteristic
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C2×C4×Q8C2311yescharacteristic
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2311yescharacteristic
87C2×C4×D4C2311yescharacteristic
88C2×#117—21no—
89C2×#117—21no—
90C2×#117C2×C411yescharacteristic
91C2×#117C2×C411yescharacteristic
92C22×C42—21no—
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95#446—24no—
96#446—24no—
97C4×Q16—24no—
98C4×Q16—22no—
99C4×Q16—22no—
100C4×SD16—24no—
101C4×SD16—22no—
102C4×SD16—22no—
103C4×SD16—24no—
104C4×SD16—22no—
105C4×SD16—22no—
106C4×D8—24no—
107C4×D8—22no—
108C4×D8—22no—
109#435—216no—
110C8×D4—24no—
111C8×D4—24no—
112#432—28no—
113#432—22no—
114#432—22no—
115#432—22no—
116#432D414yes—
117#431—22no—
118#431—22no—
119#427—24no—
120C2×#107C2×C411yescharacteristic
121C2×#107C2×C411yescharacteristic
122C2×#106C2×C411yescharacteristic
123C2×#106C2×C411yescharacteristic
124C2×C4⋊C8—22no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128#416—24no—
129#416—24no—
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#103C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135C2×#102C2×C411yescharacteristic
136C2×#102C2×C411yescharacteristic
137C2×#102C2×C411yescharacteristic
138C2×C22⋊C8—21no—
139C2×C22⋊C8—22no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8—22no—
143C2×C22⋊C8—22no—
144C4×M16—22no—
145C4×M16—22no—
146C2×#97—28no—
147C2×#97—22no—
148C2×#97—21no—
149C2×#97—21no—
150C2×#97D411yescharacteristic
151C2×#97D411yescharacteristic
152C2×C4×C8—22no—
153C2×C4×C8—22no—
154C2×C4×C8C2×C411yescharacteristic
155C2×C4×C8C2×C411yescharacteristic
156C2×C4×C8C2312yes—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160#336—22no—
161#336—22no—
162Q8⋊C8—24no—
163Q8⋊C8—24no—
164D4⋊C8—24no—
165D4⋊C8—24no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—22no—
169C2×#41—22no—
170C2×#41—22no—
171C2×#41—22no—
172C2×#41—22no—
173C2×#41—41no—
174C2×#41—41no—
175C2×#41—41no—
176C2×#41—41no—
177C2×#41C22⋊C414yes—
178C22×Q8C22×C411yescharacteristic
179C22×Q8C22×C411yescharacteristic
180C22×D4—21no—
181C22×D4—21no—
182C22×D4C22×C411yescharacteristic
183C22×D4C22×C411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4#4111yescharacteristic
193#135—232no—
194C2×Q16—22no—
195C2×Q16—22no—
196C2×Q16—22no—
197C2×SD16—22no—
198C2×SD16—22no—
199C2×SD16—22no—
200C2×SD16—22no—
201C2×SD16—22no—
202C2×SD16M1614yescomplemented
203C2×D8—22no—
204C2×D8—22no—
205C2×D8M1614yescomplemented
206C2×M16—24no—
207C2×M16—24no—
208C2×M16—42no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—28no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—41no—
218C22×C8—41no—
219C22×C8—44no—
220C22×C8—44no—
221C22×C8C22×C411yescharacteristic
222C22×C8C22×C411yescharacteristic
223C22×C8C22×C411yescharacteristic
224C22×C8C22×C411yescharacteristic
225C22×C8C2×D412yes—
226C22×C8C2×D412yes—
227C4×Q8—22no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8C22⋊C414yes—
237C4×Q8C2×D412yes—
238C4×Q8C2×D412yes—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—42no—
253C4×D4—42no—
254C4×D4—42no—
255C4×D4—42no—
256C4×D4C22⋊C414yes—
257C4×D4C2×D412yes—
258C4×D4C2×D412yes—
259#117—22no—
260#117—24no—
261#117—24no—
262#117—22no—
263#117—22no—
264#117—42no—
265#117—42no—
266#117C22⋊C414yes—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42C22×C411yescharacteristic
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D411yescharacteristic
296C2×C42#4112yes—
297C2×C42#4111yescharacteristic
298C2×C42C2411yescharacteristic
299#107—22no—
300#107—22no—
301#106—22no—
302#106—22no—
303C4⋊C8—28no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—42no—
308C4⋊C8C2×D414yes—
309#103—22no—
310#103—22no—
311#103—22no—
312#103—22no—
313#102—22no—
314#102—22no—
315#102—22no—
316#102—22no—
317C22⋊C8—28no—
318C22⋊C8—24no—
319C22⋊C8—44no—
320C22⋊C8—42no—
321C22⋊C8—42no—
322C22⋊C8—44no—
323#97—28no—
324#97—22no—
325#97—24no—
326#97—22no—
327#97—42no—
328C4×C8—22no—
329C4×C8—22no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—21no—
334C4×C8—22no—
335C4×C8—22no—
336C4×C8—22no—
337C4×C8—22no—
338C24—21no—
339C24—21no—
340C24C4×D411yescharacteristic
341#41—28no—
342#41—28no—
343#41—44no—
344#41—44no—
345#41—44no—
346#41—44no—
347#41—44no—
348#41—44no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8C2×C22⋊C412yes—
360C2×Q8C2×M1612yes—
361C2×D4—24no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4C2×C22⋊C412yes—
376C2×D4C2×M1612yes—
377C2×D4C2×M1612yes—
378C2×D4C2×M1612yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4C2×C22⋊C412yes—
414C22×C4C4×D412yes—
415C22×C4C4×D411yescharacteristic
416C22×C4C4×D411yescharacteristic
417C22×C4C4×D411yescharacteristic
418C22×C4#12012yes—
419C22×C4#12112yes—
420C22×C4#12112yes—
421C22×C4#12312yes—
422C22×C4C23×C411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4111yescharacteristic
425Q16—28no—
426SD16—28no—
427SD16—28no—
428D8—28no—
429M16—48no—
430C2×C8—28no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—22no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—22no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—416no—
448C2×C8—44no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8C2×M1612yes—
458C2×C8C2×M1612yes—
459C2×C8#13514yescomplemented
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—42no—
472C4⋊C4C2×C22⋊C412yes—
473C4⋊C4C2×C22⋊C412yes—
474C22⋊C4—24no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—24no—
478C22⋊C4—42no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—42no—
482C42—22no—
483C42—22no—
484C42—24no—
485C42—24no—
486C42—41no—
487C42—41no—
488C42—42no—
489C42—42no—
490C42C2×C22⋊C412yes—
491C42C4×D412yes—
492C42#12012yes—
493C42#12112yes—
494C42#12112yes—
495C42#12312yes—
496C42C22×D411yescharacteristic
497C42C22×D411yescharacteristic
498C42C22×D411yescharacteristic
499C42C2×#4111yescharacteristic
500C23—21no—
501C23—21no—
502C23—22no—
503C23—22no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23#38612yes—
521C23C2×C4×D411yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—22no—
525Q8—22no—
526Q8—41no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530D4—22no—
531D4—22no—
532D4—22no—
533D4—22no—
534D4—22no—
535D4—48no—
536D4—48no—
537D4—41no—
538D4—41no—
539D4#40714yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4#38612yes—
583C2×C4#38612yes—
584C2×C4#38612yes—
585C2×C4C22×C22⋊C411yescharacteristic
586C2×C4C2×C4×D411yescharacteristic
587C2×C4C2×#12011yescharacteristic
588C2×C4C2×#12111yescharacteristic
589C2×C4C2×#12111yescharacteristic
590C2×C4C2×#12311yescharacteristic
591C2×C4C22×M1611yescharacteristic
592C2×C4C22×M1611yescharacteristic
593C2×C4#56811yescharacteristic
594C2×C4#56811yescharacteristic
595C2×C4C2×#13512yes—
596C2×C4#57512yes—
597C8—22no—
598C8—22no—
599C8—24no—
600C8—41no—
601C8—41no—
602C8—42no—
603C8—44no—
604C8—82no—
605C8—82no—
606C22—22no—
607C22—24no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—42no—
612C22—41no—
613C22—42no—
614C22—41no—
615C22—42no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22C2×#38611yescharacteristic
628C22D4×M1611yescharacteristic
629C22D4×M1611yescharacteristic
630C22C4×#13511yescharacteristic
631C22#294311yescharacteristic
632C22#295711yescharacteristic
633C22#295711yescharacteristic
634C4—22no—
635C4—21no—
636C4—22no—
637C4—21no—
638C4—22no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—81no—
644C4—81no—
645C4C2×#40711yescharacteristic
646C4#287911yescharacteristic
647C4D4×M1611yescharacteristic
648C4D4×M1611yescharacteristic
649C4#300212yes—
650C4#303512yes—
651C2—42no—
652C2—44no—
653C2—82no—
654C2#1803511yescharacteristic
655C2#1803711yescharacteristic
656C2#1804111yescharacteristic
657C2#1804511yescharacteristic
658C2#2089811yescharacteristic
659C2#2091911yescharacteristic
660C2#2092411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×#135C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2970C2semidirect
3. #21380C2semidirect
4. C2×#1488C2semidirect
5. C2×#1488C2semidirect
6. #8952C2semidirect
7. #8950C2semidirect
8. #8947C2semidirect
9. #8945C2semidirect
10. #7875C2semidirect
11. #7848C2semidirect
202. C2×SD16M16semidirect
205. C2×D8M16semidirect
459. C2×C8#135semidirect

Maximal subgroups

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

subgroupindex
C2×#29702
C2×C4×#1352
#213802
C2×#14882
#89522
#89502
#89472
#89452
#78752
#78482