Finite Group Databasebeta

← #222225

Subgroups — #222225

Order 512.

959 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3108C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic, complemented
3#22474C211yesmaximal, characteristic
4C2×#1592C211yesmaximal, characteristic, complemented
5C2×#1582C211yesmaximal, characteristic, complemented
6#9662C211yesmaximal, characteristic, complemented
7#9658C211yesmaximal, characteristic, complemented
8#8950C211yesmaximal, characteristic, complemented
9#8950C211yesmaximal, characteristic, complemented
10#8949C211yesmaximal, characteristic, complemented
11#8945C211yesmaximal, characteristic, complemented
12#8600C211yesmaximal, characteristic, complemented
13#8315C211yesmaximal, characteristic
14#8311C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18#3108—24no—
19#3108—24no—
20C2×#483C411yescharacteristic
21C2×#480C411yescharacteristic
22C2×#479C411yescharacteristic
23C2×#479C411yescharacteristic
24C2×#464C411yescharacteristic
25C2×#464C411yescharacteristic
26C2×#451C411yescharacteristic
27C2×#448C411yescharacteristic
28C2×#446C2211yescharacteristic
29C2×#446C2211yescharacteristic
30C2×C4×Q16C2211yescharacteristic
31C2×C4×Q16C2211yescharacteristic
32C2×C4×SD16C2211yescharacteristic
33C2×C4×SD16C2211yescharacteristic
34C2×#436C2211yescharacteristic
35C2×C8×D4C2211yescharacteristic
36C2×#431C2211yescharacteristic
37#1985—21no—
38#1985—21no—
39#1984C2211yescharacteristic
40#1984C2211yescharacteristic
41#1984C2211yescharacteristic
42#1984C2211yescharacteristic
43#1870—21no—
44#1869—21no—
45#1818—21no—
46#1797—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1749—21no—
50C4×C22⋊C8C2211yescharacteristic
51C4×C22⋊C8C2211yescharacteristic
52#1729—21no—
53#1592—24no—
54#1582—24no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×#322C2211yescharacteristic
66C2×#322C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C23×Q8C2×C411yescharacteristic
72C22×Q16C2×C411yescharacteristic
73C22×Q16C2×C411yescharacteristic
74C22×SD16C2×C411yescharacteristic
75C22×SD16C2×C411yescharacteristic
76C22×M16—21no—
77C23×C8—21no—
78#529—22no—
79#529—22no—
80#529—41no—
81#529—41no—
82C2×#126—21no—
83C2×#125C2×C411yescharacteristic
84C2×#124C2×C411yescharacteristic
85C2×#123—21no—
86C2×C22⋊Q8C2×C411yescharacteristic
87C2×#121C2×C411yescharacteristic
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—41no—
93C2×C4×Q8—41no—
94C2×C4×Q8C2×C411yescharacteristic
95C2×C4×Q8C2×C411yescharacteristic
96C2×C4×Q8C2311yescharacteristic
97C2×C4×Q8C2311yescharacteristic
98C2×C4×Q8C2311yescharacteristic
99C2×C4×D4—21no—
100C2×C4×D4C2311yescharacteristic
101C2×#117—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2×C411yescharacteristic
104C22×C42—21no—
105C22×C42C2311yescharacteristic
106#483—24no—
107#480—24no—
108#479—24no—
109#479—24no—
110#464—24no—
111#464—24no—
112#451—24no—
113#448—24no—
114#446—22no—
115#446—22no—
116#446—41no—
117#446—41no—
118C4×Q16—22no—
119C4×Q16—22no—
120C4×Q16—22no—
121C4×Q16—22no—
122C4×SD16—22no—
123C4×SD16—22no—
124C4×SD16D414yes—
125C4×SD16D414yes—
126#436—22no—
127#436—41no—
128C8×D4—22no—
129C8×D4—41no—
130#431—22no—
131#431—22no—
132C2×#107C2×C411yescharacteristic
133C2×#107C2×C411yescharacteristic
134C2×#106C2×C411yescharacteristic
135C2×#106C2×C411yescharacteristic
136C2×C4⋊C8—21no—
137C2×C4⋊C8—21no—
138C2×C4⋊C8C2×C411yescharacteristic
139C2×C4⋊C8C2×C411yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×#103C2×C411yescharacteristic
145C2×#103C2×C411yescharacteristic
146C2×#103C2×C411yescharacteristic
147C2×#103C2×C411yescharacteristic
148C2×#103C2×C411yescharacteristic
149C2×#103C2×C411yescharacteristic
150C2×#102C2×C411yescharacteristic
151C2×#102C2×C411yescharacteristic
152C2×C22⋊C8—21no—
153C2×C22⋊C8—21no—
154C2×C22⋊C8—21no—
155C2×C22⋊C8—21no—
156C2×C22⋊C8—21no—
157C2×C22⋊C8C2×C411yescharacteristic
158C2×C22⋊C8C2×C411yescharacteristic
159C2×C22⋊C8D411yescharacteristic
160C2×C22⋊C8D411yescharacteristic
161C2×#97—21no—
162C2×#97—21no—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336D411yescharacteristic
181#336D411yescharacteristic
182#335—22no—
183#334—22no—
184Q8⋊C8—22no—
185Q8⋊C8—22no—
186Q8⋊C8—41no—
187Q8⋊C8—41no—
188Q8⋊C8—41no—
189Q8⋊C8—41no—
190D4⋊C8—22no—
191D4⋊C8—22no—
192#322—22no—
193#322—22no—
194C22×Q8—21no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—41no—
198C22×Q8—41no—
199C22×Q8—41no—
200C22×Q8—41no—
201C22×Q8C22×C411yescharacteristic
202C22×Q8C22×C411yescharacteristic
203C22×Q8C22×C411yescharacteristic
204C22×D4—21no—
205C22×D4C22×C411yescharacteristic
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4C22×C411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4#4111yescharacteristic
212C2×Q16—22no—
213C2×Q16—22no—
214C2×Q16—22no—
215C2×Q16—41no—
216C2×Q16—41no—
217C2×Q16—41no—
218C2×SD16—22no—
219C2×SD16—22no—
220C2×SD16—41no—
221C2×SD16—41no—
222C2×SD16—41no—
223C2×SD16C22⋊C414yes—
224C2×M16—42no—
225C2×M16—42no—
226C2×M16—81no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—42no—
234C22×C8—42no—
235C22×C8—81no—
236C22×C8C22×C411yescharacteristic
237C22×C8C22×C411yescharacteristic
238C22×C8C22×C411yescharacteristic
239C22×C8C22×C411yescharacteristic
240C22×C8C2×D411yescharacteristic
241C22×C8C2×D411yescharacteristic
242C22×C8C2×D411yescharacteristic
243C22×C8C2×D411yescharacteristic
244#126—42no—
245#126—42no—
246#125—22no—
247#125—41no—
248#124—22no—
249#124—41no—
250#123—42no—
251#123—42no—
252C22⋊Q8—22no—
253C22⋊Q8—41no—
254#121—22no—
255#121—41no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—42no—
269C4×Q8—42no—
270C4×Q8—41no—
271C4×Q8—41no—
272C4×Q8—42no—
273C4×Q8—42no—
274C4×Q8C2×D412yes—
275C4×Q8C2×D412yes—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—42no—
279C4×D4—42no—
280C4×D4C2×D412yes—
281C4×D4C2×D412yes—
282#117—42no—
283#117—42no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—41no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42C22×C411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42#4111yescharacteristic
320C2×C42C2411yescharacteristic
321#107—22no—
322#107—22no—
323#106—22no—
324#106C22⋊C414yes—
325C4⋊C8—22no—
326C4⋊C8—22no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—42no—
336C4⋊C8—42no—
337#103—22no—
338#103—22no—
339#103—22no—
340#103—22no—
341#103—22no—
342#103C22⋊C414yes—
343#102—22no—
344#102C22⋊C414yes—
345C22⋊C8—22no—
346C22⋊C8—22no—
347C22⋊C8—22no—
348C22⋊C8—41no—
349C22⋊C8—42no—
350C22⋊C8—42no—
351C22⋊C8—42no—
352C22⋊C8—42no—
353C22⋊C8—81no—
354#97—42no—
355#97—42no—
356C4×C8—21no—
357C4×C8—21no—
358C4×C8—21no—
359C4×C8—21no—
360C4×C8—21no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8—21no—
364C4×C8—21no—
365C4×C8—21no—
366C4×C8C2×D412yes—
367C4×C8C2×D412yes—
368C24—21no—
369C24C4×D411yescharacteristic
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8C2×C22⋊C412yes—
394C2×Q8#13112yes—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—81no—
403C2×D4C2×C22⋊C412yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—81no—
439C22×C4C4×D411yescharacteristic
440C22×C4C4×D411yescharacteristic
441C22×C4C4×D411yescharacteristic
442C22×C4C23×C411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2×#4111yescharacteristic
445Q16—44no—
446Q16—44no—
447SD16—44no—
448SD16—44no—
449M16—84no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—84no—
490C2×C8C2×C22⋊C412yes—
491C2×C8C2×C22⋊C412yes—
492C2×C8#13712yes—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4C2×C22⋊C412yes—
512C4⋊C4C2×C22⋊C412yes—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C42—21no—
520C42—21no—
521C42—22no—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—42no—
528C42—42no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42C22×D411yescharacteristic
534C42C22×D411yescharacteristic
535C42H(5,𝔽2)11yescharacteristic
536C42H(5,𝔽2)11yescharacteristic
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23#44412yes—
551C23C2×C4×D411yescharacteristic
552Q8—22no—
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—48no—
557Q8—41no—
558Q8—41no—
559Q8—41no—
560Q8—41no—
561Q8—41no—
562Q8—41no—
563Q8—41no—
564Q8—41no—
565D4—22no—
566D4—22no—
567D4—41no—
568D4—41no—
569D4—84no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—48no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—84no—
618C2×C4C22×C22⋊C411yescharacteristic
619C2×C4#51811yescharacteristic
620C2×C4D4211yescharacteristic
621C2×C4D4211yescharacteristic
622C2×C4#54611yescharacteristic
623C2×C4#54611yescharacteristic
624C2×C4C2×#13111yescharacteristic
625C2×C4#56811yescharacteristic
626C2×C4#56811yescharacteristic
627C2×C4#56811yescharacteristic
628C2×C4C2×#13711yescharacteristic
629C2×C4#57511yescharacteristic
630C2×C4#57711yescharacteristic
631C2×C4#57811yescharacteristic
632C8—22no—
633C8—22no—
634C8—22no—
635C8—41no—
636C8—41no—
637C8—41no—
638C8—41no—
639C8—41no—
640C8—81no—
641C8—81no—
642C8—81no—
643C8—81no—
644C8—81no—
645C8—81no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22D4×C22⋊C411yescharacteristic
661C22#293011yescharacteristic
662C22#293311yescharacteristic
663C22C4×#13711yescharacteristic
664C22#294711yescharacteristic
665C22C2×#44411yescharacteristic
666C22#295711yescharacteristic
667C4—21no—
668C4—21no—
669C4—41no—
670C4—42no—
671C4—42no—
672C4—41no—
673C4—42no—
674C4—41no—
675C4—42no—
676C4—81no—
677C4—81no—
678C4—81no—
679C4#287911yescharacteristic
680C4#288011yescharacteristic
681C4#297611yescharacteristic
682C4#297811yescharacteristic
683C4#326611yescharacteristic
684C4#326911yescharacteristic
685C4#327511yescharacteristic
686C4#327611yescharacteristic
687C2—42no—
688C2—82no—
689C2#1793911yescharacteristic
690C2#1794711yescharacteristic
691C2#1801011yescharacteristic
692C2#1802311yescharacteristic
693C2#2240011yescharacteristic
694C2#2244411yescharacteristic
695C2#2245211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22510C2semidirect
4. C2×#1592C2semidirect
5. C2×#1582C2semidirect
6. #9662C2semidirect
7. #9658C2semidirect
8. #8950C2semidirect
9. #8950C2semidirect
10. #8949C2semidirect
11. #8945C2semidirect
12. #8600C2semidirect
14. #8311C2semidirect
15. #8149C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31082
#225102
#224742
C2×#15922
C2×#15822
#96622
#96582
#89502
#89492
#89452
#86002
#83152
#83112
#81492