Finite Group Databasebeta

← #223229

Subgroups — #223229

Order 512.

822 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36189C211yesmaximal, characteristic
2#22797C211yesmaximal, characteristic
3#22749C211yesmaximal, characteristic
4#9907C211yesmaximal, characteristic, complemented
5#9904C211yesmaximal, characteristic, complemented
6#9136C211yesmaximal, characteristic, complemented
7#9135C211yesmaximal, characteristic, complemented
8#9128C211yesmaximal, characteristic, complemented
9#9128C211yesmaximal, characteristic, complemented
10#8726C211yesmaximal, characteristic, complemented
11#8725C211yesmaximal, characteristic, complemented
12#8590C211yesmaximal, characteristic, complemented
13#8232C211yesmaximal, characteristic, complemented
14#8046C211yesmaximal, characteristic, complemented
15#8043C211yesmaximal, characteristic, complemented
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20#2859C2211yescharacteristic
21#2857C2211yescharacteristic
22#2857C2211yescharacteristic
23#2570—21no—
24#2511C412yes—
25#2507—21no—
26#2497—21no—
27#2487—21no—
28#2442—21no—
29#2441—21no—
30#2423C412yes—
31#2417C412yes—
32#2360C412yes—
33#2352—21no—
34#2345—21no—
35C4×C4⋊Q8C2211yescharacteristic
36#2285C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1955—21no—
40#1924—21no—
41#1924—21no—
42#1923—21no—
43#1916—21no—
44#1870—21no—
45#1842C2211yescharacteristic
46#1831C2211yescharacteristic
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1771—21no—
52#1767C2211yescharacteristic
53#1767C2211yescharacteristic
54#1753—21no—
55C2×#333C2211yescharacteristic
56C2×#333C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C22×M16—21no—
72C2×C4⋊Q8C2311yescharacteristic
73C2×#125—21no—
74C2×C22⋊Q8C2×C412yes—
75C2×C22⋊Q8C2×C412yes—
76C2×C4×Q8C2×C412yes—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×#117—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4C2×C412yes—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#432—22no—
87#432—41no—
88#431—22no—
89#431—22no—
90#431—41no—
91#431—41no—
92#423—22no—
93#423—41no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117#400—22no—
118#400—22no—
119#399—22no—
120#398—22no—
121#398—22no—
122#396—22no—
123#391C2×C412yes—
124#391C2×C412yes—
125#391C2×C412yes—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—22no—
135#389C2×C412yes—
136#389C2×C412yes—
137#388—22no—
138#387—22no—
139#387—22no—
140#387—22no—
141#387—22no—
142#386—22no—
143#385—22no—
144#384—21no—
145#384—21no—
146#384C2×C412yes—
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#382—21no—
151#382—21no—
152#380—21no—
153#380—21no—
154#380—21no—
155#380—21no—
156#380—21no—
157#380—22no—
158#380C2×C412yes—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4C2×C412yes—
163C4×C4⋊C4C2×C411yescharacteristic
164C4×C4⋊C4D411yescharacteristic
165C4×C4⋊C4D411yescharacteristic
166C4×C22⋊C4—22no—
167C4×C22⋊C4C2×C411yescharacteristic
168C4×C22⋊C4C2×C411yescharacteristic
169C4×C22⋊C4D411yescharacteristic
170C4×C22⋊C4D411yescharacteristic
171C4×C22⋊C4D411yescharacteristic
172C4×C22⋊C4D411yescharacteristic
173C43C2311yescharacteristic
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#333—22no—
183#333—22no—
184#333—41no—
185#333—41no—
186C22×Q8C22×C412yes—
187C22×D4C22⋊C411yescharacteristic
188C23×C4—22no—
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C2×M16—42no—
196C2×M16—42no—
197C2×M16—81no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210#125—81no—
211#125—81no—
212C22⋊Q8—44no—
213C22⋊Q8—82no—
214C4×Q8—82no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—41no—
218C4×D4—41no—
219C4×D4—42no—
220#117—81no—
221#117—81no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—81no—
241C2×C4⋊C4—81no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C412yes—
248C2×C4⋊C4C22×C412yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4C22⋊C411yescharacteristic
253C2×C22⋊C4C22⋊C411yescharacteristic
254C2×C22⋊C4C22⋊C411yescharacteristic
255C2×C22⋊C4C22⋊C411yescharacteristic
256C2×C22⋊C4C22⋊C411yescharacteristic
257C2×C22⋊C4C22⋊C411yescharacteristic
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2411yescharacteristic
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—42no—
296C4⋊C8—42no—
297C4⋊C8—81no—
298C4⋊C8#4112yes—
299C4⋊C8#4112yes—
300C22⋊C8—42no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303C22⋊C8—42no—
304C22⋊C8—81no—
305C22⋊C8—81no—
306C22⋊C8—81no—
307#97—42no—
308#97—42no—
309#97—81no—
310C4×C8—42no—
311C4×C8—42no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95C22⋊C411yescharacteristic
323#95C22⋊C411yescharacteristic
324#95C22⋊C411yescharacteristic
325#95C22⋊C411yescharacteristic
326#95C22⋊C411yescharacteristic
327#95C22⋊C411yescharacteristic
328C24C2×C22⋊C411yescharacteristic
329C24C2×C22⋊C411yescharacteristic
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×D4—22no—
335C2×D4—41no—
336C2×D4—41no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—44no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4#13712yes—
385C22×C4#13712yes—
386C22×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389M16—84no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—44no—
438C4⋊C4—82no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—44no—
451C22⋊C4—82no—
452C22⋊C4—82no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—41no—
474C42—41no—
475C42—82no—
476C42—82no—
477C42C2×#4111yescharacteristic
478C42C2×#4111yescharacteristic
479C42H(5,𝔽2)11yescharacteristic
480C422−1+411yescharacteristic
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23#41912yes—
495C23C22×C22⋊C411yescharacteristic
496Q8—82no—
497Q8—82no—
498D4—44no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
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—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—164no—
562C2×C4#41312yes—
563C2×C4#41912yes—
564C2×C4C4×#4111yescharacteristic
565C2×C4#52011yescharacteristic
566C2×C4#54611yescharacteristic
567C2×C4#54611yescharacteristic
568C2×C4#54711yescharacteristic
569C2×C4#54711yescharacteristic
570C2×C4C2×#13711yescharacteristic
571C2×C4C2×#13711yescharacteristic
572C2×C4#57511yescharacteristic
573C2×C4#57511yescharacteristic
574C2×C4#57611yescharacteristic
575C2×C4#57711yescharacteristic
576C2×C4#57711yescharacteristic
577C2×C4#57811yescharacteristic
578C8—42no—
579C8—42no—
580C8—81no—
581C8—81no—
582C8—81no—
583C8—81no—
584C8—81no—
585C8—81no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22#233911yescharacteristic
601C22C2×#41311yescharacteristic
602C22#288811yescharacteristic
603C22C2×#41911yescharacteristic
604C22#289611yescharacteristic
605C22#289711yescharacteristic
606C22#289811yescharacteristic
607C4—21no—
608C4—21no—
609C4—22no—
610C4—22no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4#331411yescharacteristic
624C4#331511yescharacteristic
625C4#331611yescharacteristic
626C4#331711yescharacteristic
627C4#334411yescharacteristic
628C4#334611yescharacteristic
629C4#334811yescharacteristic
630C4#334911yescharacteristic
631C2—42no—
632C2—81no—
633C2#1774711yescharacteristic
634C2#1776011yescharacteristic
635C2#1776611yescharacteristic
636C2#1778111yescharacteristic
637C2#2248211yescharacteristic
638C2#2249611yescharacteristic
639C2#2249711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9907C2semidirect
5. #9904C2semidirect
6. #9136C2semidirect
7. #9135C2semidirect
8. #9128C2semidirect
9. #9128C2semidirect
10. #8726C2semidirect
11. #8725C2semidirect
12. #8590C2semidirect
13. #8232C2semidirect
14. #8046C2semidirect
15. #8043C2semidirect

Maximal subgroups

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

subgroupindex
#361892
#227972
#227492
#99072
#99042
#91362
#91352
#91282
#87262
#87252
#85902
#82322
#80462
#80432