Finite Group Databasebeta

← #188968

Subgroups — #188968

Order 512.

991 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60233C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic, complemented
3#21941C211yesmaximal, characteristic, complemented
4#17582C211yesmaximal, characteristic
5C2×#1513C211yesmaximal, characteristic
6#8629C212yesmaximal, complemented
7#8628C212yesmaximal, complemented
8#8627C212yesmaximal, complemented
9#8626C212yesmaximal, complemented
10#8107C211yesmaximal, characteristic, complemented
11#8104C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13#2977—22no—
14#2976—22no—
15#2974—21no—
16#2974—21no—
17#2973—21no—
18#2972—21no—
19#2933C412yescomplemented
20#2933C412yescomplemented
21#2929C2212yes—
22#2929C2212yes—
23#2924C411yescharacteristic, complemented
24#2924C411yescharacteristic, complemented
25#2880C411yescharacteristic, complemented
26#2880C411yescharacteristic, complemented
27C2×#407C2211yescharacteristic
28C2×#407C2211yescharacteristic
29C2×#407C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2280C2211yescharacteristic
32C2×#380C2211yescharacteristic
33#1855—21no—
34#1855—21no—
35#1855—22no—
36#1854—22no—
37#1853—21no—
38#1852—21no—
39#1810C2212yescomplemented
40#1810C2212yescomplemented
41#1808C2212yescomplemented
42#1807C2212yescomplemented
43#1516—21no—
44#1516—21no—
45#1513—21no—
46#1513—21no—
47#1513C2212yes—
48#1513C2212yes—
49#1513C2211yescharacteristic
50#1513C2211yescharacteristic
51#1513C2211yescharacteristic
52#1513C2211yescharacteristic
53#1508—21no—
54#1507—21no—
55C2×#328—21no—
56C2×#327—21no—
57#1305C2212yescomplemented
58#1305C2212yescomplemented
59#1305C2212yescomplemented
60#1305C2212yescomplemented
61C22×#41C2×C411yescharacteristic
62C24×C4C2311yescharacteristic
63C2×#131C2×C412yescomplemented
64C2×#131C2×C412yescomplemented
65C22×M16—21no—
66C22×M16—21no—
67C22×M16C2×C412yes—
68C23×C8C2312yes—
69#525—21no—
70#525—21no—
71#525—21no—
72#525—21no—
73#525C2×C411yescharacteristic
74#525C2×C411yescharacteristic
75#525D411yescharacteristic
76#525D411yescharacteristic
77#525Q811yescharacteristic
78#525Q811yescharacteristic
79#525C2311yescharacteristic
80#525C2311yescharacteristic
81C2×#123—21no—
82C2×C22⋊Q8D411yescharacteristic
83C2×C22⋊Q8Q811yescharacteristic
84C2×#121D411yescharacteristic
85C2×#121Q811yescharacteristic
86C2×#120—21no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×#117—21no—
90C2×#117C2×C411yescharacteristic
91C22×C4⋊C4—21no—
92C22×C22⋊C4—21no—
93#436—22no—
94#436—22no—
95#435—22no—
96#435—22no—
97#435—22no—
98#435—22no—
99#435—22no—
100#435—22no—
101#435—22no—
102#435—22no—
103C8×D4—22no—
104C8×D4—22no—
105C8×D4—22no—
106C8×D4—22no—
107C8×D4—22no—
108C8×D4—22no—
109#433—22no—
110#433—22no—
111#432—21no—
112#432—21no—
113#431—21no—
114#431—21no—
115C2×#108—22no—
116C2×#108—22no—
117C2×#107—22no—
118C2×#106—22no—
119#424C2×C412yescomplemented
120#424C2×C412yescomplemented
121#408—22no—
122#408—22no—
123#408C2×C412yescomplemented
124#408C2×C412yescomplemented
125#407—21no—
126#407—21no—
127#407—21no—
128#407—21no—
129#407—21no—
130#407—21no—
131#407—21no—
132#407—21no—
133#407—21no—
134#407—21no—
135#407C2×C411yescharacteristic
136#407C2×C411yescharacteristic
137#407C2311yescharacteristic
138#407C2312yes—
139#407C2311yescharacteristic
140#407C2312yes—
141C2×C22⋊C8—21no—
142C2×C22⋊C8—21no—
143C2×C22⋊C8—21no—
144C2×C22⋊C8—21no—
145C2×C22⋊C8—21no—
146C2×C22⋊C8—21no—
147C2×C22⋊C8—21no—
148C2×C22⋊C8—21no—
149C2×C22⋊C8C2×C411yescharacteristic
150C2×C22⋊C8C2×C411yescharacteristic
151C2×C22⋊C8C2×C412yes—
152C2×C22⋊C8D411yescharacteristic
153C2×C22⋊C8D411yescharacteristic
154C2×C22⋊C8D412yes—
155C2×C22⋊C8D411yescharacteristic
156C2×C22⋊C8D411yescharacteristic
157C2×C22⋊C8D412yes—
158#405C2×C412yescomplemented
159#405C2×C412yescomplemented
160#381—21no—
161#380—21no—
162#380—21no—
163#380—21no—
164#380—21no—
165#380C2311yescharacteristic
166#380C2312yes—
167#380C2311yescharacteristic
168#379—21no—
169C4×C22⋊C4—21no—
170C4×C22⋊C4—21no—
171C2×#95—21no—
172#341—22no—
173#341—22no—
174#340—22no—
175#340—22no—
176#328—22no—
177#328—22no—
178#328—21no—
179#328—21no—
180#328—21no—
181#328—21no—
182#327—22no—
183#327—22no—
184#327—21no—
185#327—21no—
186#327—21no—
187#327—21no—
188C25#4111yescharacteristic
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—82no—
192C2×#41C4⋊C411yescharacteristic
193C2×#41C4⋊C411yescharacteristic
194C2×#41C22×C411yescharacteristic
195C2×#41C22×C411yescharacteristic
196C22×Q8C4⋊C411yescharacteristic
197C22×D4—21no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4C4⋊C411yescharacteristic
201C23×C4—21no—
202C23×C4—41no—
203C23×C4—82no—
204C23×C4C22×C411yescharacteristic
205C23×C4C22×C411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×Q811yescharacteristic
213C23×C4C2×Q811yescharacteristic
214C23×C4#4111yescharacteristic
215C23×C4#4111yescharacteristic
216C23×C4#4111yescharacteristic
217C23×C4C2411yescharacteristic
218#131—82no—
219#131—82no—
220C2×M16—22no—
221C2×M16—22no—
222C2×M16—22no—
223C2×M16—22no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—41no—
227C2×M16—41no—
228C2×M16—82no—
229C2×M16—82no—
230C2×M16—82no—
231C2×M16C22×C412yes—
232C2×M16C22×C412yes—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—41no—
237C22×C8—41no—
238C22×C8—82no—
239C22×C8C22×C412yes—
240C22×C8C22×C412yes—
241C22×C8C2×D412yes—
242C22×C8C2×D412yes—
243C22×C8C2×D412yes—
244C22×C8C2×D412yes—
245#123—21no—
246#123—21no—
247#123—21no—
248#123—21no—
249#123—21no—
250#123—21no—
251#123—21no—
252#123—21no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8C4⋊C411yescharacteristic
258C22⋊Q8C4⋊C411yescharacteristic
259C22⋊Q8C4⋊C411yescharacteristic
260C22⋊Q8C4⋊C411yescharacteristic
261C22⋊Q8C2×D411yescharacteristic
262C22⋊Q8C2×D411yescharacteristic
263C22⋊Q8C2×Q811yescharacteristic
264C22⋊Q8C2×Q811yescharacteristic
265#121—21no—
266#121—21no—
267#121—21no—
268#121—21no—
269#121C4⋊C411yescharacteristic
270#121C4⋊C411yescharacteristic
271#121C4⋊C411yescharacteristic
272#121C4⋊C411yescharacteristic
273#121C2×D411yescharacteristic
274#121C2×D411yescharacteristic
275#121C2×Q811yescharacteristic
276#121C2×Q811yescharacteristic
277#120—21no—
278#120—21no—
279#120—21no—
280#120—21no—
281#120—21no—
282#120—21no—
283#120—21no—
284#120—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4—21no—
294C4×D4—21no—
295C4×D4—21no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4—21no—
300C4×D4—21no—
301#117—21no—
302#117—21no—
303#117—82no—
304#117C4⋊C411yescharacteristic
305#117C4⋊C411yescharacteristic
306#117C22×C411yescharacteristic
307#117C22×C411yescharacteristic
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—41no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4C4⋊C411yescharacteristic
316C2×C4⋊C4C4⋊C411yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—41no—
326C2×C22⋊C4—41no—
327C2×C22⋊C4—41no—
328C2×C22⋊C4—82no—
329C2×C42—21no—
330C2×C42—41no—
331#108—82no—
332#108—82no—
333#107—82no—
334#106—82no—
335C4⋊C8—22no—
336C4⋊C8—22no—
337C4⋊C8—22no—
338C4⋊C8—22no—
339C22⋊C8—21no—
340C22⋊C8—21no—
341C22⋊C8—21no—
342C22⋊C8—21no—
343C22⋊C8—22no—
344C22⋊C8—22no—
345C22⋊C8—22no—
346C22⋊C8—22no—
347C22⋊C8—22no—
348C22⋊C8—22no—
349C22⋊C8—22no—
350C22⋊C8—21no—
351C22⋊C8—21no—
352C22⋊C8—22no—
353C22⋊C8—21no—
354C22⋊C8—21no—
355C22⋊C8—41no—
356C22⋊C8—41no—
357C22⋊C8—41no—
358C22⋊C8—41no—
359C22⋊C8C22⋊C411yescharacteristic
360C22⋊C8C22⋊C411yescharacteristic
361C22⋊C8C22⋊C411yescharacteristic
362C22⋊C8C22⋊C411yescharacteristic
363C22⋊C8C22⋊C412yes—
364C22⋊C8C22⋊C412yes—
365C22⋊C8C22⋊C412yes—
366C22⋊C8C22⋊C412yes—
367C22⋊C8C22⋊C412yes—
368C22⋊C8C22⋊C412yes—
369#97—22no—
370#97—22no—
371C4×C8—22no—
372C4×C8—22no—
373#95—22no—
374#95—21no—
375#95—21no—
376#95—41no—
377#95—41no—
378C24—21no—
379C24—41no—
380C24—42no—
381C24—41no—
382C24—81no—
383C24#11711yescharacteristic
384C24#11711yescharacteristic
385C24#12111yescharacteristic
386C24#12111yescharacteristic
387C24C22⋊Q811yescharacteristic
388C24C22⋊Q811yescharacteristic
389C24C2×#4111yescharacteristic
390#41—42no—
391#41—42no—
392#41—82no—
393#41—82no—
394#41—81no—
395#41—81no—
396C2×Q8—21no—
397C2×Q8—81no—
398C2×Q8C2×C4⋊C411yescharacteristic
399C2×Q8C2×C4⋊C411yescharacteristic
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—44no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D4—81no—
417C2×D4—82no—
418C2×D4C2×C4⋊C411yescharacteristic
419C2×D4C2×C4⋊C411yescharacteristic
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—82no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C4⋊C411yescharacteristic
453C22×C4C2×C4⋊C411yescharacteristic
454C22×C4C2×C4⋊C411yescharacteristic
455C22×C4C2×C4⋊C411yescharacteristic
456C22×C4#11711yescharacteristic
457C22×C4#11711yescharacteristic
458C22×C4#12111yescharacteristic
459C22×C4#12111yescharacteristic
460C22×C4#12111yescharacteristic
461C22×C4#12111yescharacteristic
462C22×C4#12111yescharacteristic
463C22×C4#12111yescharacteristic
464C22×C4C22⋊Q811yescharacteristic
465C22×C4C22⋊Q811yescharacteristic
466C22×C4C22⋊Q811yescharacteristic
467C22×C4C22⋊Q811yescharacteristic
468C22×C4C22⋊Q811yescharacteristic
469C22×C4C22⋊Q811yescharacteristic
470C22×C4C23×C411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×Q811yescharacteristic
475C22×C4C2×#4111yescharacteristic
476M16—42no—
477M16—42no—
478M16—42no—
479M16—42no—
480M16—42no—
481M16—42no—
482M16—82no—
483M16—82no—
484M16—82no—
485M16—82no—
486M16—82no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—42no—
490C2×C8—42no—
491C2×C8—42no—
492C2×C8—42no—
493C2×C8—42no—
494C2×C8—42no—
495C2×C8—42no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—82no—
499C2×C8—82no—
500C2×C8—82no—
501C2×C8—82no—
502C2×C8—82no—
503C2×C8C2×C22⋊C412yes—
504C2×C8C2×C22⋊C412yes—
505C2×C8C2×C22⋊C412yes—
506C2×C8C2×C22⋊C412yes—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—82no—
522C4⋊C4C2×C4⋊C411yescharacteristic
523C4⋊C4C2×C4⋊C411yescharacteristic
524C4⋊C4C2×C4⋊C411yescharacteristic
525C4⋊C4C2×C4⋊C411yescharacteristic
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—81no—
550C22⋊C4—81no—
551C22⋊C4—82no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—82no—
557C23—22no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—22no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—42no—
584C23—81no—
585C23—81no—
586C23—84no—
587C23—84no—
588C23#38011yescharacteristic
589C23#38011yescharacteristic
590C23#38011yescharacteristic
591C23#38011yescharacteristic
592C23C2×#11711yescharacteristic
593C23C2×#12111yescharacteristic
594C23C2×C22⋊Q811yescharacteristic
595Q8—41no—
596Q8—41no—
597Q8—81no—
598D4—41no—
599D4—41no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—42no—
605D4—42no—
606D4—41no—
607D4—41no—
608D4—41no—
609D4—41no—
610D4—81no—
611D4—81no—
612D4—81no—
613D4—81no—
614D4—81no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4—81no—
657C2×C4—81no—
658C2×C4—81no—
659C2×C4—81no—
660C2×C4—82no—
661C2×C4—82no—
662C2×C4—82no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4—164no—
666C2×C4#38011yescharacteristic
667C2×C4#38011yescharacteristic
668C2×C4#38011yescharacteristic
669C2×C4#38011yescharacteristic
670C2×C4C22×C22⋊C411yescharacteristic
671C2×C4C22×C4⋊C411yescharacteristic
672C2×C4C2×#12111yescharacteristic
673C2×C4C2×C22⋊Q811yescharacteristic
674C8—22no—
675C8—22no—
676C8—22no—
677C8—22no—
678C8—42no—
679C8—42no—
680C8—82no—
681C8—81no—
682C8—81no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—21no—
688C22—21no—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—21no—
694C22—21no—
695C22—21no—
696C22—21no—
697C22—21no—
698C22—21no—
699C22—21no—
700C22—21no—
701C22—21no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—81no—
713C22—81no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—162no—
719C22#181412yes—
720C22C2×#38011yescharacteristic
721C22#306811yescharacteristic
722C22#306911yescharacteristic
723C4—21no—
724C4—21no—
725C4—21no—
726C4—21no—
727C4—21no—
728C4—21no—
729C4—21no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—161no—
737C4—161no—
738C4—162no—
739C4#288312yes—
740C2—21no—
741C2—21no—
742C2—22no—
743C2—41no—
744C2—41no—
745C2—41no—
746C2—41no—
747C2—41no—
748C2—81no—
749C2—81no—
750C2#2063311yescharacteristic
751C2C2×#181411yescharacteristic
752C2#2067911yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21948C2semidirect
3. #21941C2semidirect
6. #8629C2semidirect
7. #8628C2semidirect
8. #8627C2semidirect
9. #8626C2semidirect
10. #8107C2semidirect
11. #8104C2semidirect
19. #2933C4semidirect
20. #2933C4semidirect
23. #2924C4semidirect
24. #2924C4semidirect
25. #2880C4semidirect
26. #2880C4semidirect
39. #1810C22semidirect
40. #1810C22semidirect
41. #1808C22semidirect
42. #1807C22semidirect
57. #1305C22semidirect
58. #1305C22semidirect
59. #1305C22semidirect
60. #1305C22semidirect
63. C2×#131C2×C4semidirect
64. C2×#131C2×C4semidirect
119. #424C2×C4semidirect
120. #424C2×C4semidirect
123. #408C2×C4semidirect
124. #408C2×C4semidirect
158. #405C2×C4semidirect
159. #405C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602332
#219482
#219412
#175822
C2×#15132
#86292
#86282
#86272
#86262
#81072
#81042