Finite Group Databasebeta

← #366802

Subgroups — #366802

Order 512.

1544 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#481C211yesmaximal, characteristic
2C2×#1900C211yesmaximal, characteristic
3#20912C211yesmaximal, characteristic
4C2×#1855C211yesmaximal, characteristic, complemented
5C2×#1821C211yesmaximal, characteristic, complemented
6#20379C211yesmaximal, characteristic, complemented
7C2×#1736C211yesmaximal, characteristic, complemented
8#11366C212yesmaximal, complemented
9#11127C212yesmaximal, complemented
10#10943C212yesmaximal, complemented
11#10869C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#481—24no—
17C2×#481—21no—
18C2×#481—21no—
19C2×#481—22no—
20C2×#481C418yescomplemented
21C22×#108C2212yes—
22C22×#106C2212yes—
23C2×#418C2211yescharacteristic
24C2×#418C2211yescharacteristic
25C22×#102C2212yes—
26C22×#100C2212yes—
27C2×#407C2211yescharacteristic
28C2×#407C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C2×#405C2212yes—
31#1900—24no—
32#1875—22no—
33#1874—22no—
34#1855—24no—
35#1821—24no—
36#1750—22no—
37#1749—21no—
38C4×C22⋊C8—21no—
39C2×#343C2212yes—
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#341C2212yes—
43#1736—22no—
44#1736C2211yescharacteristic
45#1736C2212yes—
46#1736C2211yescharacteristic
47#1736C2212yes—
48C2×#340—22no—
49#1729C2212yes—
50C23×D4—21no—
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C22×M16—21no—
55C22×M16—24no—
56C22×M16—22no—
57C22×M16—22no—
58C22×M16—42no—
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2312yes—
62C22×M16C2312yes—
63C23×C8—21no—
64C23×C8C2312yes—
65C2×#121—22no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121—22no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—22no—
72C2×#121C2×C414yes—
73C2×#117—21no—
74C2×#117—21no—
75C2×#117—24no—
76C2×#117—42no—
77C2×#117D412yes—
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4C2×C414yes—
89C22×C4⋊C4D412yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C42—21no—
95#481—28no—
96#481—28no—
97#481—216no—
98C2×#108—24no—
99C2×#108—24no—
100C2×#108—42no—
101C2×#106—22no—
102C2×#106—22no—
103C2×#106—24no—
104C2×#106C2×C418yescomplemented
105#418—24no—
106#418—42no—
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—24no—
110C2×#102—22no—
111C2×#102—22no—
112C2×#102C2×C418yescomplemented
113C2×#100—24no—
114C2×#100—24no—
115C2×#100—42no—
116#407—22no—
117#407—22no—
118#407—22no—
119#407D414yes—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8C2×C414yes—
125C2×C22⋊C8D414yes—
126C2×C22⋊C8D411yescharacteristic
127C2×C22⋊C8D411yescharacteristic
128C2×C22⋊C8D412yes—
129#405—24no—
130#405—42no—
131C2×#97—22no—
132C2×C4×C8—22no—
133#343—24no—
134#343—24no—
135#343—24no—
136#341—24no—
137#340—22no—
138#340—22no—
139#340—24no—
140#340—24no—
141#336—22no—
142#336—22no—
143C25—21no—
144C25#4111yescharacteristic
145C22×D4—24no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—41no—
156C22×D4—41no—
157C22×D4—42no—
158C22×D4—84no—
159C22×D4C22×C412yes—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—41no—
169C23×C4—41no—
170C23×C4—42no—
171C23×C4—84no—
172C23×C4C22×C412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4#4111yescharacteristic
180C23×C4#4112yes—
181C23×C4C2411yescharacteristic
182C2×M16—24no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—48no—
202C2×M16—48no—
203C2×M16—42no—
204C2×M16—42no—
205C2×M16C2×D414yes—
206C2×M16#4114yes—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—24no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—42no—
217C22×C8—42no—
218C22×C8C22×C414yes—
219C22×C8C2×D412yes—
220C22×C8C2×D412yes—
221C22×C8C2×D414yes—
222#121—24no—
223#121—24no—
224#121—24no—
225#121—24no—
226#121—42no—
227#121—42no—
228#121—42no—
229#121—42no—
230#117—22no—
231#117—22no—
232#117—22no—
233#117—24no—
234#117—44no—
235#117—44no—
236#117—42no—
237#117#4114yes—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4C22×C414yes—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—42no—
275C2×C42—21no—
276C2×C42—22no—
277C2×C42—21no—
278C2×C42—44no—
279#108—48no—
280#106—28no—
281#106—28no—
282#102—28no—
283#102—28no—
284#102—44no—
285#102—44no—
286#100—48no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291C22⋊C8C22⋊C414yes—
292C22⋊C8C22⋊C414yes—
293C22⋊C8C22⋊C418yes—
294#97—44no—
295C4×C8—44no—
296C24—22no—
297C24—22no—
298C24—24no—
299C24—22no—
300C24—42no—
301C24—41no—
302C24—41no—
303C24—42no—
304C24—42no—
305C24—82no—
306C24—82no—
307C24C4×D412yes—
308C24#12111yescharacteristic
309C24#12111yescharacteristic
310C24#12412yes—
311C24C2×#4111yescharacteristic
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—42no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—88no—
333C2×D4—84no—
334C2×D4—84no—
335C2×D4—88no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—44no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—82no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4—88no—
374C22×C4—84no—
375C22×C4—82no—
376C22×C4—88no—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C4×D412yes—
380C22×C4C4×D412yes—
381C22×C4C4×D412yes—
382C22×C4#12112yes—
383C22×C4#12112yes—
384C22×C4#12111yescharacteristic
385C22×C4#12111yescharacteristic
386C22×C4#12412yes—
387C22×C4#12712yes—
388C22×C4#12712yes—
389C22×C4C23×C411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4H(5,𝔽2)12yes—
398C22×C42−1+412yes—
399M16—24no—
400M16—24no—
401M16—24no—
402M16—24no—
403M16—48no—
404M16—42no—
405M16—42no—
406M16—42no—
407M16—42no—
408M16—42no—
409M16—42no—
410M16—42no—
411M16—42no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—48no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8C2×C22⋊C414yes—
434C2×C8C2×C22⋊C414yes—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—41no—
445C4⋊C4—42no—
446C4⋊C4—41no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4#12314yes—
450C22⋊C4—24no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C42—24no—
456C42—44no—
457C42—42no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—21no—
465C23—21no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—42no—
472C23—42no—
473C23—48no—
474C23—48no—
475C23—42no—
476C23—44no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—81no—
482C23—81no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—164no—
487C23—164no—
488C23#39012yes—
489C23#39012yes—
490C23#47114yes—
491C23C2×C4×D411yescharacteristic
492C23C2×#12111yescharacteristic
493C23C2×#12411yescharacteristic
494C23#55112yes—
495C23#55212yes—
496D4—42no—
497D4—44no—
498D4—42no—
499D4—88no—
500D4—88no—
501D4—82no—
502D4—81no—
503D4—81no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—88no—
531C2×C4—88no—
532C2×C4—84no—
533C2×C4—88no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—82no—
539C2×C4#39012yes—
540C2×C4#39012yes—
541C2×C4C22×C22⋊C411yescharacteristic
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C4×#4112yes—
544C2×C4#52012yes—
545C2×C4C2×#12111yescharacteristic
546C2×C4C2×#12311yescharacteristic
547C2×C4#52511yescharacteristic
548C2×C4#52912yes—
549C2×C4C2×#12711yescharacteristic
550C2×C4#53411yescharacteristic
551C2×C4#53611yescharacteristic
552C2×C4#53912yes—
553C2×C4#54612yes—
554C2×C4#54612yes—
555C2×C4#54612yes—
556C2×C4#54712yes—
557C2×C4#54712yes—
558C2×C4#54712yes—
559C8—24no—
560C8—24no—
561C8—44no—
562C8—44no—
563C8—81no—
564C8—81no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—24no—
569C22—24no—
570C22—22no—
571C22—21no—
572C22—21no—
573C22—24no—
574C22—24no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—48no—
579C22—48no—
580C22—42no—
581C22—42no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—164no—
586C22—164no—
587C22—162no—
588C22—162no—
589C22#197714yes—
590C22C2×#39011yescharacteristic
591C22#235912yes—
592C22#241211yescharacteristic
593C22#241311yescharacteristic
594C22#248812yes—
595C22C2×#47112yes—
596C22#306812yes—
597C4—22no—
598C4—22no—
599C4—44no—
600C4—84no—
601C4—82no—
602C4—82no—
603C4C4×#12311yescharacteristic
604C4#232211yescharacteristic
605C4#233912yes—
606C4#242512yes—
607C4#250111yescharacteristic
608C4#253911yescharacteristic
609C2—22no—
610C2—22no—
611C2—24no—
612C2—42no—
613C2—162no—
614C2#1399611yescharacteristic
615C2C2×#197711yescharacteristic
616C2#2124911yescharacteristic
617C2#2457312yes—
618C2#2462312yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1855C2semidirect
5. C2×#1821C2semidirect
6. #20379C2semidirect
7. C2×#1736C2semidirect
8. #11366C2semidirect
9. #11127C2semidirect
10. #10943C2semidirect
11. #10869C2semidirect
20. C2×#481C4semidirect
104. C2×#106C2×C4semidirect
112. C2×#102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4812
C2×#19002
#209122
C2×#18552
C2×#18212
#203792
C2×#17362
#113662
#111272
#109432
#108692