Finite Group Databasebeta

← #193837

Subgroups — #193837

Order 512.

1412 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60284C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#17450C212yesmaximal, complemented
4#17366C211yesmaximal, characteristic, complemented
5#8963C212yesmaximal, complemented
6#8961C212yesmaximal, complemented
7#8960C212yesmaximal, complemented
8#8958C212yesmaximal, complemented
9#7867C212yesmaximal, complemented
10C2×#532C411yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C2×C4×#41C2211yescharacteristic
13#2991—24no—
14#2982C2212yes—
15#2975—22no—
16#2975C2212yes—
17#2971—24no—
18#2920—22no—
19#2920C2212yes—
20#2920C2212yes—
21C4×C4⋊Q8C411yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×#127C411yescharacteristic
24C4×#127C2211yescharacteristic
25C4×#124C411yescharacteristic
26C4×#124C411yescharacteristic
27C4×#124C2211yescharacteristic
28C4×#124C2211yescharacteristic
29C4×#121—22no—
30C4×#121—22no—
31#2297C411yescharacteristic
32#2297C411yescharacteristic
33#2293—22no—
34#2285C411yescharacteristic
35C42×D4—22no—
36C2×C43C2211yescharacteristic
37#1489—24no—
38#1488—24no—
39#1479C2214yes—
40#1478C2214yes—
41#1465—22no—
42#1464—22no—
43#1461C2212yes—
44#1460C2212yes—
45#1276C2214yes—
46#1276C2214yes—
47C22×#41C2×C411yescharacteristic
48C22×#41C2×C411yescharacteristic
49#532—24no—
50#532—24no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#127C2×C411yescharacteristic
53C2×#124C2×C411yescharacteristic
54C2×#124C2×C411yescharacteristic
55C2×#121—22no—
56C4×#41—22no—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41—22no—
62C4×#41—22no—
63C4×#41—22no—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4D412yes—
76C2×C4×D4D412yes—
77C2×C4×D4C2311yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C2×#117—21no—
80C2×#117C2×C411yescharacteristic
81C2×#117C2×C411yescharacteristic
82C2×#117D412yes—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2×C411yescharacteristic
87C22×C42C2×C411yescharacteristic
88C22×C42D412yes—
89C22×C42C2311yescharacteristic
90C8×Q8—28no—
91#436—216no—
92#436—28no—
93#432—24no—
94#432—22no—
95#432—22no—
96#432—22no—
97#432D414yescomplemented
98#431—24no—
99#431—24no—
100#431—22no—
101#431—22no—
102#431C2314yes—
103#431C2314yes—
104#423—24no—
105#423—22no—
106#423D414yescomplemented
107#407—24no—
108#407—22no—
109#407—22no—
110C4×M16—24no—
111C4×M16—22no—
112C4×M16—22no—
113#391C2×C411yescharacteristic
114#391C2×C411yescharacteristic
115#391C2×C411yescharacteristic
116#391C2×C411yescharacteristic
117#390—22no—
118#390—22no—
119#390C2×C411yescharacteristic
120#390C2×C411yescharacteristic
121#390C2×C411yescharacteristic
122#390C2×C411yescharacteristic
123#389—22no—
124#389—22no—
125#388—22no—
126#388—22no—
127#388—22no—
128#388—22no—
129#386—22no—
130#386—22no—
131#384C2×C411yescharacteristic
132#384C2×C411yescharacteristic
133#382C2×C411yescharacteristic
134#382C2×C411yescharacteristic
135#380—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C43—21no—
146C43—21no—
147C43C2×C411yescharacteristic
148C43C2×C411yescharacteristic
149C43C2311yescharacteristic
150C43C2311yescharacteristic
151Q8⋊C8—28no—
152D4⋊C8—28no—
153#324—24no—
154#323—24no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41—22no—
161C2×#41—22no—
162C2×#41—22no—
163C2×#41—42no—
164C2×#41—42no—
165C22×Q8C22×C411yescharacteristic
166C22×Q8C22×C411yescharacteristic
167C22×D4—22no—
168C22×D4—21no—
169C22×D4C22⋊C412yes—
170C22×D4C22×C411yescharacteristic
171C22×D4C22×C411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4C22⋊C412yes—
178C23×C4C22⋊C412yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C2×M16—28no—
184C2×M16—24no—
185C2×M16—42no—
186C4⋊Q8—22no—
187C4⋊Q8M1614yescomplemented
188#127—22no—
189#127—22no—
190#124—22no—
191#124—22no—
192#124—22no—
193#124M1614yescomplemented
194#121—216no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×D4—216no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—48no—
217C4×D4—48no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222#117—22no—
223#117—22no—
224#117—24no—
225#117—24no—
226#117—22no—
227#117—22no—
228#117—42no—
229#117—42no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C22⋊C412yes—
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C2×D412yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4C22⋊C412yes—
251C2×C22⋊C4C22⋊C412yes—
252C2×C22⋊C4C2×D412yes—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—42no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—41no—
276C2×C42—41no—
277C2×C42—41no—
278C2×C42—41no—
279C2×C42C22⋊C412yes—
280C2×C42C22⋊C412yes—
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42#4112yes—
292C2×C42#4112yes—
293C2×C42C2411yescharacteristic
294C4⋊C8—28no—
295C4⋊C8—24no—
296C4⋊C8—24no—
297C4⋊C8—42no—
298C4⋊C8C2×D418yes—
299C22⋊C8—28no—
300C22⋊C8—24no—
301C22⋊C8—24no—
302C22⋊C8—24no—
303C22⋊C8—42no—
304#97—24no—
305#97—42no—
306C4×C8—28no—
307C4×C8—24no—
308C4×C8—24no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315C24—21no—
316C24C2×C22⋊C412yes—
317C24C2×C22⋊C411yescharacteristic
318#41—44no—
319#41—44no—
320#41—44no—
321#41—44no—
322#41—44no—
323#41—44no—
324#41—44no—
325#41—44no—
326C2×Q8—22no—
327C2×Q8—21no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8C2×M1612yes—
332C2×Q8C2×M1612yes—
333C2×Q8C2×M1612yes—
334C2×D4—28no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—24no—
345C2×D4—42no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4C2×M1612yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C4×D412yes—
396C22×C4C4×D412yes—
397C22×C4#12012yes—
398C22×C4#12112yes—
399C22×C4#12112yes—
400C22×C4#12312yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404M16—84no—
405M16—88no—
406M16—84no—
407C2×C8—28no—
408C2×C8—44no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—28no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C22⋊C4—28no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—28no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C42—24no—
433C42—24no—
434C42—24no—
435C42—22no—
436C42—22no—
437C42—21no—
438C42—21no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—42no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42C4×D412yes—
459C42C4×D412yes—
460C42#12011yescharacteristic
461C42#12011yescharacteristic
462C42#12112yes—
463C42#12112yes—
464C42#12311yescharacteristic
465C42#12311yescharacteristic
466C42C2×M1612yes—
467C42C2×M1612yes—
468C42C22×D412yes—
469C42C2×#4112yes—
470C23—21no—
471C23—21no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23#38612yes—
485C23C22×C22⋊C411yescharacteristic
486Q8—24no—
487Q8—24no—
488Q8—24no—
489Q8—42no—
490D4—24no—
491D4—48no—
492D4—416no—
493D4—48no—
494D4—42no—
495D4—42no—
496D4—42no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—48no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4#38611yescharacteristic
551C2×C4#38611yescharacteristic
552C2×C4#38611yescharacteristic
553C2×C4#38612yes—
554C2×C4#38611yescharacteristic
555C2×C4#40714yes—
556C2×C4C2×C4×D412yes—
557C2×C4C2×#12011yescharacteristic
558C2×C4C2×#12112yes—
559C2×C4C2×#12311yescharacteristic
560C2×C4C22×M1611yescharacteristic
561C2×C4C22×M1611yescharacteristic
562C2×C4#56812yes—
563C8—48no—
564C8—84no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—42no—
571C22—42no—
572C22—44no—
573C22—44no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—82no—
579C22—82no—
580C22—81no—
581C22—81no—
582C22C2×#38611yescharacteristic
583C22C2×#40711yescharacteristic
584C22#287911yescharacteristic
585C4—24no—
586C4—21no—
587C4—21no—
588C4—24no—
589C4—22no—
590C4—21no—
591C4—21no—
592C4—22no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4#188412yes—
602C4D4×M1612yes—
603C4D4×M1612yes—
604C4#301812yes—
605C2—21no—
606C2—21no—
607C2—42no—
608C2—82no—
609C2—82no—
610C2#2089811yescharacteristic
611C2C2×#188411yescharacteristic
612C2#2095211yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC2×#532C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60284C2semidirect
3. #17450C2semidirect
4. #17366C2semidirect
5. #8963C2semidirect
6. #8961C2semidirect
7. #8960C2semidirect
8. #8958C2semidirect
9. #7867C2semidirect
97. #432D4semidirect
106. #423D4semidirect
187. C4⋊Q8M16semidirect
193. #124M16semidirect

Maximal subgroups

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

subgroupindex
#602842
C4×#5322
#174502
#173662
#89632
#89612
#89602
#89582
#78672