Finite Group Databasebeta

← #380081

Subgroups — #380081

Order 512.

1494 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C2×#2319C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic
5#14283C211yesmaximal, characteristic, complemented
6#11220C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#11189C212yesmaximal
9#11189C212yesmaximal
10#10849C211yesmaximal, characteristic, complemented
11#10833C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#478—24no—
15C2×#478—24no—
16C2×#478C418yescomplemented
17C22×#107C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2212yes—
24#2319—24no—
25#2319—24no—
26C2×#390C2212yes—
27C2×#389C2212yes—
28C2×#388—22no—
29C2×#384C2211yescharacteristic
30C2×#381—21no—
31C2×#380—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1989—24no—
35#1802—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1441—21no—
44#1441—21no—
45#1441—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125C2×C414yes—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4C2×C414yes—
63C2×C4×D4D414yes—
64C2×C4×D4D414yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81#478—232no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#107C2×C414yes—
85C2×#106—22no—
86C2×#106—22no—
87C2×#106C2×C414yes—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8C2×C414yes—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—24no—
99C2×#102—42no—
100C2×#102—42no—
101C2×#102C2×C414yes—
102C2×#102C2×C414yes—
103#390—24no—
104#390—24no—
105#389—24no—
106#389—24no—
107#388—216no—
108#384—22no—
109#384—22no—
110#381—28no—
111#380—28no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4D414yes—
114#376—22no—
115#376—22no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95D411yescharacteristic
123C2×#95D411yescharacteristic
124C2×#95D412yes—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—42no—
129#340—42no—
130#336—42no—
131#336—42no—
132C25—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4C22×C412yes—
141C22×D4C2×D412yes—
142C22×D4C2×D412yes—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8C22×C412yes—
174C22×C8C22×C412yes—
175#125—28no—
176#125—28no—
177#125—44no—
178#125—44no—
179C4×D4—28no—
180C4×D4—28no—
181C4×D4—28no—
182C4×D4C22⋊C4116yes—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—81no—
244C2×C42—81no—
245C2×C42C22×C412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251#107—28no—
252#106—28no—
253C4⋊C8—28no—
254C4⋊C8—44no—
255#102—28no—
256#102—28no—
257#102—48no—
258#95—28no—
259#95—24no—
260#95—22no—
261#95—22no—
262#95—24no—
263#95—82no—
264#95—82no—
265#95—82no—
266#95—82no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—42no—
271C24—42no—
272C24—41no—
273C24—42no—
274C24—41no—
275C24—42no—
276C24—41no—
277C24—41no—
278C24C22×D411yescharacteristic
279C2×D4—22no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—24no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4C2×C22⋊C414yes—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—216no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—88no—
350C22×C4—88no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4C4×D412yes—
354C22×C4C4×D412yes—
355C22×C4#12012yes—
356C22×C4#12012yes—
357C22×C4#12112yes—
358C22×C4#12112yes—
359C22×C4C23×C411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4H(5,𝔽2)12yes—
366C22×C42−1+412yes—
367C22×C42−1+412yes—
368C22×C42−1+412yes—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4—88no—
410C4⋊C4C2×C22⋊C414yes—
411C22⋊C4—216no—
412C22⋊C4—88no—
413C22⋊C4—88no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42—42no—
418C42—84no—
419C42C2×C22⋊C414yes—
420C42C4×D414yes—
421C42#12114yes—
422C42#12314yes—
423C23—24no—
424C23—44no—
425C23—44no—
426C23—42no—
427C23—42no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23C2×C4×D411yescharacteristic
447C23C2×#12011yescharacteristic
448C23C2×#12111yescharacteristic
449C23#53612yes—
450C23#53712yes—
451C23#57612yes—
452C23#57611yescharacteristic
453C23#57712yes—
454C23#57711yescharacteristic
455C23#57711yescharacteristic
456C23#57811yescharacteristic
457D4—24no—
458D4—24no—
459D4—44no—
460D4—42no—
461D4—42no—
462D4—44no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—816no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4#38614yes—
520C2×C4#38614yes—
521C2×C4C22×C22⋊C411yescharacteristic
522C2×C4C2×C4×D411yescharacteristic
523C2×C4#51912yes—
524C2×C4#52012yes—
525C2×C4C2×#12111yescharacteristic
526C2×C4C2×#12311yescharacteristic
527C2×C4#53511yescharacteristic
528C2×C4#53611yescharacteristic
529C2×C4#53611yescharacteristic
530C2×C4#53612yes—
531C2×C4#53611yescharacteristic
532C2×C4#53612yes—
533C2×C4#53712yes—
534C2×C4#53712yes—
535C2×C4#54112yes—
536C2×C4#54112yes—
537C2×C4#54212yes—
538C2×C4#54212yes—
539C8—42no—
540C8—42no—
541C8—82no—
542C22—44no—
543C22—44no—
544C22—44no—
545C22—84no—
546C22—84no—
547C22—84no—
548C22—84no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22C2×#38611yescharacteristic
554C22#232312yes—
555C22#261612yes—
556C22#262011yescharacteristic
557C22#262111yescharacteristic
558C22#294612yes—
559C22#295111yescharacteristic
560C22#295211yescharacteristic
561C22#301211yescharacteristic
562C22#301212yes—
563C22#301311yescharacteristic
564C22#304111yescharacteristic
565C22#304211yescharacteristic
566C22#304511yescharacteristic
567C22#304611yescharacteristic
568C22#324112yes—
569C22#324112yes—
570C22#324312yes—
571C22#324412yes—
572C22#325712yes—
573C22#325812yes—
574C22#325912yes—
575C22#326012yes—
576C4—24no—
577C4—24no—
578C4—44no—
579C4—82no—
580C4—84no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—162no—
585C4#231011yescharacteristic
586C4#231111yescharacteristic
587C4#232312yes—
588C4#262011yescharacteristic
589C4#262111yescharacteristic
590C4#262512yes—
591C2—84no—
592C2#1425811yescharacteristic
593C2#2092911yescharacteristic
594C2#2093711yescharacteristic
595C2#2212212yes—
596C2#2212612yes—
597C2#2970012yes—
598C2#2970212yes—
599C2#2983411yescharacteristic
600C2#2983611yescharacteristic
601C2#2987211yescharacteristic
602C2#2987411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1989C2semidirect
5. #14283C2semidirect
6. #11220C2semidirect
7. #11219C2semidirect
10. #10849C2semidirect
11. #10833C2semidirect
16. C2×#478C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
C2×#23192
C2×#19892
C2×#18022
#142832
#112202
#112192
#111892
#108492
#108332