Finite Group Databasebeta

← #6379166

Subgroups — #6379166

Order 512.

4624 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#578C211yesmaximal, characteristic
2C2×#2951C211yesmaximal, characteristic, complemented
3C2×C4×#135C211yesmaximal, characteristic, complemented
4#34259C212yesmaximal, complemented
5C22⋊C4×Q16C214yesmaximal, complemented
6#22448C214yesmaximal, complemented
7#22444C218yesmaximal, complemented
8#21232C211yesmaximal, characteristic, complemented
9C2×#1969C211yesmaximal, characteristic, complemented
10#20931C212yesmaximal, complemented
11#20927C212yesmaximal, complemented
12#20663C211yesmaximal, characteristic, complemented
13#20603C212yesmaximal, complemented
14#20383C211yesmaximal, characteristic, complemented
15C22×2−1+4C2212yes—
16C2×#578—28no—
17C2×#578—28no—
18C2×#578C4116yescomplemented
19C22×#137C2214yes—
20C22×#137C2212yes—
21C22×#135C2212yes—
22C22×#135C2211yescharacteristic
23C23×Q16C2212yes—
24C23×Q16C2211yescharacteristic
25C22×#131C2211yescharacteristic
26C2×#520C2212yes—
27C2×C4×#41C2212yes—
28#2951—28no—
29C2×#440C2214yes—
30C4×#135—28no—
31C2×C4×Q16C2212yes—
32C2×C4×Q16C2212yes—
33C2×C4×SD16C2214yes—
34C2×C4×D8C2212yes—
35C2×C4×D8C2212yes—
36C2×#427C2211yescharacteristic
37C22×#107C2211yescharacteristic
38C2×#417C2212yes—
39C2×#419C2212yes—
40C2×#416C2212yes—
41C22×#103C2212yes—
42C2×#408C2211yescharacteristic
43C2×#405C2211yescharacteristic
44C22×C4×C8C2211yescharacteristic
45Q8×C22⋊C4C2218yes—
46#2339—28no—
47#2339C2218yes—
48#2311—22no—
49#2311C2212yes—
50#2310—22no—
51#2310C2212yes—
52#2297—22no—
53#2297C2211yescharacteristic
54C2×#391—22no—
55C2×#391C2212yes—
56#1971C2214yes—
57#1969C2214yes—
58#1969C2214yes—
59#1969C2214yes—
60#1878C2218yes—
61#1878C2218yes—
62#1877C2218yes—
63#1877C2218yes—
64#1809C2214yes—
65#1808C2214yes—
66#1788C2218yes—
67#1787C2218yes—
68#1753C2214yes—
69C8×C22⋊C4C2214yes—
70#1736C2212yes—
71#1729C2211yescharacteristic
72#1728C2211yescharacteristic
73C2×2−1+4—28no—
74C2×2−1+4—28no—
75C2×2−1+4—28no—
76C2×2−1+4—44no—
77C2×2−1+4—44no—
78C2×2−1+4C2×C4116yescomplemented
79C22×#41—22no—
80C22×#41—24no—
81C22×#41C2314yes—
82C22×#41C2312yes—
83C22×#41C2311yescharacteristic
84C23×Q8—22no—
85C23×Q8C2312yes—
86C23×Q8C2314yes—
87#578—2128no—
88C2×#137—28no—
89C2×#137—28no—
90C2×#137—216no—
91C2×#137—216no—
92C2×#137C2×C4116yescomplemented
93C2×#137C2×C4132yescomplemented
94C2×#135—28no—
95C2×#135—28no—
96C2×#135C2×C418yes—
97C2×#135C2×C4116yescomplemented
98C2×#135D418yes—
99C2×#135D418yes—
100C22×Q16—28no—
101C22×Q16—28no—
102C22×Q16—24no—
103C22×Q16—24no—
104C22×Q16C2×C4116yescomplemented
105C22×Q16C2×C418yescomplemented
106C22×Q16C2314yes—
107C22×Q16C2312yes—
108C22×Q16C2312yes—
109C22×Q16C2311yescharacteristic
110C22×SD16C2314yes—
111C22×SD16C2312yes—
112C22×D8C2311yescharacteristic
113C2×#131—24no—
114C2×#131—24no—
115C2×#131C2×C418yescomplemented
116C22×M16C2311yescharacteristic
117C22×M16C2312yes—
118C22×M16C2311yescharacteristic
119C23×C8C2311yescharacteristic
120C23×C8C2312yes—
121C23×C8C2311yescharacteristic
122#520—28no—
123#520—44no—
124C4×#41—28no—
125C4×#41—44no—
126C2×C4×Q8C2314yes—
127C2×C4×Q8C2314yes—
128C2×C4×D4—24no—
129C2×C4×D4—24no—
130C2×C4×D4C2314yes—
131C2×C4×D4C2314yes—
132C2×#117—22no—
133C2×#117—22no—
134C2×#117C2312yes—
135C2×#117C2311yescharacteristic
136C22×C4⋊C4—22no—
137C22×C4⋊C4C2312yes—
138C22×C42—22no—
139C22×C42C2311yescharacteristic
140#440—216no—
141C4×Q16—28no—
142C4×Q16—28no—
143C4×SD16—216no—
144C4×D8—28no—
145C4×D8—28no—
146#427—24no—
147C2×#107—24no—
148C2×#107C2312yes—
149C2×#107C2312yes—
150C2×#107C2312yes—
151C2×#106C2312yes—
152#419—44no—
153#417—28no—
154#416—28no—
155C2×#103—44no—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2318yes—
159C2×#103C2318yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162#408—24no—
163C2×C22⋊C8C2314yes—
164#405—24no—
165C2×#97C2312yes—
166C2×C4×C8—24no—
167C2×C4×C8C2312yes—
168C2×C4×C8C2312yes—
169C2×C4×C8C2312yes—
170#391—22no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391—22no—
175#391—28no—
176#391—28no—
177#391D411yescharacteristic
178#391D411yescharacteristic
179#391C2318yes—
180#391C2318yes—
181#390—22no—
182#390D411yescharacteristic
183#390D411yescharacteristic
184#386—24no—
185#386—28no—
186#386—28no—
187#386D414yes—
188#381—24no—
189#381—28no—
190#380—28no—
191#380C2318yes—
192C4×C22⋊C4—28no—
193C4×C22⋊C4C2314yes—
194C2×#95—24no—
195C2×#95—28no—
196#340C2318yes—
197#336C2314yes—
198#336C2314yes—
1992−1+4—264no—
2002−1+4—432no—
2012−1+4—432no—
2022−1+4—432no—
203C2×#41—216no—
204C2×#41—232no—
205C2×#41—22no—
206C2×#41—22no—
207C2×#41—24no—
208C2×#41—28no—
209C2×#41—24no—
210C2×#41—24no—
211C2×#41—22no—
212C2×#41—22no—
213C2×#41—28no—
214C2×#41—28no—
215C2×#41—44no—
216C2×#41—44no—
217C2×#41—48no—
218C2×#41—48no—
219C2×#41—416no—
220C2×#41—416no—
221C2×#41C22×C4116yes—
222C2×#41C22×C418yes—
223C2×#41C22×C414yes—
224C2×#41C2×D418yes—
225C2×#41C2×D418yes—
226C2×#41#4114yes—
227C22×Q8—216no—
228C22×Q8—24no—
229C22×Q8—22no—
230C22×Q8—28no—
231C22×Q8—24no—
232C22×Q8—28no—
233C22×Q8—28no—
234C22×Q8—24no—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—44no—
239C22×Q8—44no—
240C22×Q8—48no—
241C22×Q8—48no—
242C22×Q8C22×C418yes—
243C22×Q8C22×C4116yes—
244C22×Q8C2×D411yescharacteristic
245C22×Q8C2414yes—
246C22×Q8C2412yes—
247C22×D4—22no—
248C22×D4—24no—
249C22×D4C2×D412yes—
250C22×D4C2×D411yescharacteristic
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—24no—
255C23×C4—24no—
256C23×C4—22no—
257C23×C4—24no—
258C23×C4—24no—
259C23×C4C2×D412yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262#137—264no—
263#137—2128no—
264#135—264no—
265#135C22⋊C4164yescomplemented
266C2×Q16—28no—
267C2×Q16—28no—
268C2×Q16—264no—
269C2×Q16—232no—
270C2×Q16—24no—
271C2×Q16—24no—
272C2×Q16—24no—
273C2×Q16—24no—
274C2×Q16C22×C418yes—
275C2×Q16C22×C4116yes—
276C2×Q16C22×C418yes—
277C2×Q16C22×C414yes—
278C2×Q16C2×D414yes—
279C2×Q16C2×D414yes—
280C2×SD16—28no—
281C2×SD16—28no—
282C2×SD16C22×C4116yes—
283C2×SD16C22×C418yes—
284C2×SD16C2×D418yes—
285C2×SD16C2×D418yes—
286C2×D8C22×C414yes—
287C2×D8C2×D414yes—
288C2×D8C2×D414yes—
289#131—232no—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16—24no—
296C2×M16—24no—
297C2×M16C22×C414yes—
298C2×M16C22×C418yes—
299C22×C8—24no—
300C22×C8—24no—
301C22×C8—22no—
302C22×C8—22no—
303C22×C8—22no—
304C22×C8C22×C418yes—
305C22×C8C22×C414yes—
306C22×C8C2×D414yes—
307C22×C8C2×D414yes—
308C22×C8C2412yes—
309C22×C8C2412yes—
310C22×C8C2412yes—
311C22×C8C2411yescharacteristic
312C22×C8C2411yescharacteristic
313C4×Q8—28no—
314C4×Q8—28no—
315C4×Q8—44no—
316C4×Q8—44no—
317C4×D4—28no—
318C4×D4—28no—
319C4×D4—44no—
320C4×D4—44no—
321C4×D4—416no—
322C4×D4—416no—
323#117—22no—
324#117—22no—
325#117—24no—
326#117—24no—
327#117—48no—
328#117—48no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—48no—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2414yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—28no—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D414yes—
345C2×C22⋊C4C2×D414yes—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—48no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2412yes—
356#107—24no—
357#107—24no—
358#107—24no—
359#106—24no—
360#103—28no—
361#103—28no—
362#103—48no—
363#103—48no—
364#102—28no—
365#102—28no—
366C22⋊C8—28no—
367#97—24no—
368C4×C8—24no—
369C4×C8—24no—
370C4×C8—24no—
371#95—28no—
372#95—28no—
373#95—28no—
374#95—28no—
375#95—28no—
376#95C2×D414yes—
377#95C2×D414yes—
378C24—22no—
379C24C22×D412yes—
380C24C22×D411yescharacteristic
381#41—28no—
382#41—216no—
383#41—28no—
384#41—216no—
385#41—216no—
386#41—28no—
387#41—28no—
388#41—232no—
389#41—416no—
390#41—416no—
391#41—416no—
392#41—464no—
393#41—4128no—
394#41C2×C22⋊C4132yes—
395C2×Q8—216no—
396C2×Q8—28no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—232no—
404C2×Q8—216no—
405C2×Q8—216no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—416no—
413C2×Q8—416no—
414C2×Q8—416no—
415C2×Q8—464no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—48no—
419C2×Q8—48no—
420C2×Q8C4×D414yes—
421C2×Q8C23×C418yes—
422C2×Q8C23×C414yes—
423C2×Q8C22×D414yes—
424C2×Q8C22×D414yes—
425C2×D4—28no—
426C2×D4—28no—
427C2×D4—28no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—216no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—48no—
437C2×D4—48no—
438C2×D4C4×D418yes—
439C2×D4C4×D414yes—
440C2×D4C23×C414yes—
441C2×D4C22×D414yes—
442C2×D4C22×D414yes—
443C22×C4—22no—
444C22×C4—28no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—216no—
452C22×C4—28no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—216no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—42no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—84no—
471C22×C4—88no—
472C22×C4—84no—
473C22×C4C23×C414yes—
474C22×C4C23×C412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4114yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4C2511yescharacteristic
488Q16—232no—
489Q16—216no—
490Q16—216no—
491Q16C2×C22⋊C4116yes—
492SD16—232no—
493SD16C2×C22⋊C4132yes—
494D8C2×C22⋊C4116yes—
495M16—28no—
496M16—216no—
497M16—44no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—216no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—44no—
508C2×C8C2×C22⋊C4116yes—
509C2×C8C23×C414yes—
510C2×C8C23×C412yes—
511C2×C8C23×C412yes—
512C2×C8C22×D412yes—
513C2×C8C22×D412yes—
514C2×C8C22×D412yes—
515C2×C8C22×D412yes—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C22⋊C4—24no—
526C22⋊C4—28no—
527C22⋊C4—44no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—44no—
531C22⋊C4—88no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—48no—
539C42—48no—
540C42—44no—
541C23—22no—
542C23—22no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—22no—
548C23—22no—
549C23—24no—
550C23—44no—
551C23—44no—
552C23C2×C4×D414yes—
553C23C2×C4×D412yes—
554C23D4212yes—
555C23D4212yes—
556C23C23×D411yescharacteristic
557Q8—28no—
558Q8—24no—
559Q8—24no—
560Q8—24no—
561Q8—28no—
562Q8—28no—
563Q8—432no—
564Q8—416no—
565Q8—44no—
566Q8—44no—
567Q8C22×C22⋊C418yes—
568D4—232no—
569D4—216no—
570D4—28no—
571D4—24no—
572D4—24no—
573D4—24no—
574D4—432no—
575D4C22×C22⋊C418yes—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—232no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—48no—
597C2×C4—432no—
598C2×C4—432no—
599C2×C4—416no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—816no—
608C2×C4—168no—
609C2×C4—164no—
610C2×C4C22×C22⋊C414yes—
611C2×C4C2×C4×D414yes—
612C2×C4C2×C4×D412yes—
613C2×C4#51814yes—
614C2×C4D4212yes—
615C2×C4D4212yes—
616C2×C4#54614yes—
617C2×C4#54614yes—
618C2×C4#57814yescomplemented
619C2×C4C24×C411yescharacteristic
620C2×C4C23×D411yescharacteristic
621C2×C4C23×D411yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C8—24no—
626C8—42no—
627C8—42no—
628C8C22×C22⋊C414yes—
629C8C22×C22⋊C414yes—
630C22—24no—
631C22—24no—
632C22—22no—
633C22—22no—
634C22—28no—
635C22—24no—
636C22—28no—
637C22—48no—
638C22—48no—
639C22—44no—
640C22—44no—
641C22—42no—
642C22—42no—
643C22—44no—
644C22D4×C22⋊C414yes—
645C22C22×C4×D411yescharacteristic
646C22C2×D4211yescharacteristic
647C22C2×D4211yescharacteristic
648C22C2×#57812yes—
649C22#358412yes—
650C4—28no—
651C4—24no—
652C4—24no—
653C4—48no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—84no—
658C4—84no—
659C4D4×C22⋊C414yes—
660C4C23×C22⋊C411yescharacteristic
661C4C2×#51811yescharacteristic
662C4C2×#54611yescharacteristic
663C4C2×#54611yescharacteristic
664C2—24no—
665C2—44no—
666C2—48no—
667C2C2×D4×C22⋊C411yescharacteristic
668C2C4×#57812yes—
669C2#6105712yes—
670C2#6109011yescharacteristic
671C2#6109111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC22×#578C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2951C2semidirect
3. C2×C4×#135C2semidirect
4. #34259C2semidirect
5. C22⋊C4×Q16C2semidirect
6. #22448C2semidirect
7. #22444C2semidirect
8. #21232C2semidirect
9. C2×#1969C2semidirect
10. #20931C2semidirect
11. #20927C2semidirect
12. #20663C2semidirect
13. #20603C2semidirect
14. #20383C2semidirect
18. C2×#578C4semidirect
78. C2×2−1+4C2×C4semidirect
92. C2×#137C2×C4semidirect
93. C2×#137C2×C4semidirect
97. C2×#135C2×C4semidirect
104. C22×Q16C2×C4semidirect
105. C22×Q16C2×C4semidirect
115. C2×#131C2×C4semidirect
265. #135C22⋊C4semidirect
618. C2×C4#578semidirect

Maximal subgroups

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

subgroupindex
C22×#5782
C2×#29512
C2×C4×#1352
#342592
C22⋊C4×Q162
#224482
#224442
#212322
C2×#19692
#209312
#209272
#206632
#206032
#203832