Finite Group Databasebeta

← #379903

Subgroups — #379903

Order 512.

2154 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#480C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic, complemented
3C2×#1874C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic, complemented
5#14698C211yesmaximal, characteristic
6#11219C214yesmaximal, complemented
7#11066C214yesmaximal, complemented
8#10730C211yesmaximal, characteristic, complemented
9#10712C211yesmaximal, characteristic, complemented
10C22×#121C2211yescharacteristic
11C23×C4⋊C4—21no—
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C2×#480—24no—
16C2×#480—24no—
17C2×#480C418yescomplemented
18C22×#107C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C2×#390C2212yes—
24C2×#389—22no—
25C2×#389—24no—
26C2×#389—24no—
27C2×#389C2214yes—
28C2×#386—21no—
29C2×#386—21no—
30C2×#380C2211yescharacteristic
31C22×#95—21no—
32#1874—24no—
33#1823—24no—
34C2×#340C2214yes—
35C2×#340C2212yes—
36C2×#340C2214yes—
37C2×#336C2214yes—
38#1437—22no—
39C4⋊#95C2212yes—
40C23×D4—21no—
41C23×D4C2311yescharacteristic
42C24×C4—21no—
43C24×C4—21no—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#121—22no—
50C2×#121—22no—
51C2×#121—22no—
52C2×#121—22no—
53C2×#121—22no—
54C2×#121C2×C414yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—28no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—44no—
64C22×C4⋊C4—44no—
65C22×C4⋊C4—44no—
66C22×C4⋊C4C2×C414yes—
67C22×C4⋊C4D414yes—
68C22×C4⋊C4D414yes—
69C22×C4⋊C4D414yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—44no—
81C22×C42D412yes—
82C22×C42C2312yes—
83#480—232no—
84C2×#107—24no—
85C2×#107—24no—
86C2×#107C2×C418yescomplemented
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—44no—
91C2×#102C2×C418yescomplemented
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8C2×C414yes—
98#390—44no—
99#389—216no—
100#389—28no—
101#389—28no—
102#386—44no—
103#386—44no—
104#380—22no—
105#380—22no—
106C2×#95—28no—
107C2×#95—24no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111#340—28no—
112#340—24no—
113#340—44no—
114#336—44no—
115C25—21no—
116C25C2×D411yescharacteristic
117C22×D4—24no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—42no—
126C22×D4—42no—
127C22×D4—42no—
128C22×D4—42no—
129C22×D4—42no—
130C22×D4—42no—
131C22×D4C22×C412yes—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4D814yes—
160C23×C4C22×C412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4#4112yes—
168C23×C4#4112yes—
169C23×C4C2411yescharacteristic
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—24no—
173C22×C8—24no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—24no—
178C22×C8—24no—
179C22×C8C22×C414yes—
180#121—28no—
181#121—28no—
182#121—44no—
183#121—44no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—216no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—432no—
211C2×C4⋊C4—48no—
212C2×C4⋊C4—48no—
213C2×C4⋊C4C22⋊C4116yescomplemented
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C414yes—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D414yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—48no—
239C2×C42—48no—
240#107—216no—
241#102—216no—
242#102—48no—
243C22⋊C8—28no—
244C22⋊C8—44no—
245#95—216no—
246#95—44no—
247#95—42no—
248#95—44no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—41no—
255C24—41no—
256C24—42no—
257C24—42no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—42no—
263C24—42no—
264C24C4×D412yes—
265C24#12012yes—
266C24#12112yes—
267C24C2×D812yes—
268C24#13512yes—
269C24AGL(1,ℤ/8)12yes—
270C24AGL(1,ℤ/8)12yes—
271C24C22×D411yescharacteristic
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—28no—
275C2×D4—48no—
276C2×D4—48no—
277C2×D4—48no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—84no—
288C2×D4—84no—
289C2×D4—84no—
290C2×D4—84no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—28no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—28no—
299C22×C4—28no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—216no—
318C22×C4—28no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—48no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4—48no—
335C22×C4—44no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—44no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—44no—
351C22×C4—48no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4C2×C22⋊C414yes—
362C22×C4C4×D414yes—
363C22×C4C4×D412yes—
364C22×C4#12012yes—
365C22×C4#12114yes—
366C22×C4#12112yes—
367C22×C4#12314yes—
368C22×C4C2×D814yes—
369C22×C4#13714yes—
370C22×C4C23×C411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4H(5,𝔽2)14yes—
377C22×C42−1+414yes—
378C2×C8—24no—
379C2×C8—28no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—44no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—44no—
387C2×C8—44no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—216no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—48no—
397C4⋊C4—432no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4C2×C22⋊C418yes—
404C22⋊C4—48no—
405C22⋊C4—88no—
406C22⋊C4—88no—
407C22⋊C4—84no—
408C42—48no—
409C42—44no—
410C42—44no—
411C23—24no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23—24no—
427C23—44no—
428C23—42no—
429C23—42no—
430C23—44no—
431C23—42no—
432C23—81no—
433C23—81no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23—81no—
447C23—81no—
448C23—81no—
449C23—82no—
450C23#38614yes—
451C23C4×D814yes—
452C23#44214yes—
453C23#44714yes—
454C23#44914yes—
455C23#45914yes—
456C23#46314yes—
457C23C2×C4×D411yescharacteristic
458C23C2×#12011yescharacteristic
459C23C2×#12111yescharacteristic
460C23#53412yes—
461C23#53612yes—
462C23C22×D811yescharacteristic
463C23C2×#13511yescharacteristic
464C23C2×AGL(1,ℤ/8)11yescharacteristic
465C23C2×AGL(1,ℤ/8)11yescharacteristic
466C23C2×#13712yes—
467C23#57512yes—
468D4—44no—
469D4—44no—
470D4—816no—
471D4—82no—
472D4—82no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—28no—
488C2×C4—416no—
489C2×C4—416no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—432no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—816no—
512C2×C4—84no—
513C2×C4—84no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#38614yes—
526C2×C4C22×C22⋊C411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4#52014yes—
529C2×C4C2×#12111yescharacteristic
530C2×C4C2×#12311yescharacteristic
531C2×C4#53412yes—
532C2×C4#53512yes—
533C2×C4#53612yes—
534C2×C4#53612yes—
535C2×C4#53714yes—
536C2×C4#53914yes—
537C2×C4#53914yes—
538C8—44no—
539C8—82no—
540C22—28no—
541C22—28no—
542C22—28no—
543C22—28no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—84no—
551C22—84no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—162no—
556C22—162no—
557C22—161no—
558C22—161no—
559C22—161no—
560C22—161no—
561C22#187918yes—
562C22C2×#38611yescharacteristic
563C22#232212yes—
564C22#261311yescharacteristic
565C22#261411yescharacteristic
566C22#261812yes—
567C22C2×C4×D811yescharacteristic
568C22C2×#44211yescharacteristic
569C22C4×#13712yes—
570C22C2×#44711yescharacteristic
571C22C2×#44911yescharacteristic
572C22#300411yescharacteristic
573C22#300511yescharacteristic
574C22C2×#45911yescharacteristic
575C22C2×#46311yescharacteristic
576C22#303712yes—
577C22#318612yes—
578C22#319012yes—
579C22#319312yes—
580C22#319412yes—
581C22#322612yes—
582C22#323212yes—
583C22#323412yes—
584C22#323812yes—
585C4—24no—
586C4—44no—
587C4—44no—
588C4—48no—
589C4—84no—
590C4—82no—
591C4—162no—
592C4#231112yes—
593C4#232212yes—
594C4#261812yes—
595C4#262312yes—
596C2—28no—
597C2—162no—
598C2#1425611yescharacteristic
599C2C2×#187911yescharacteristic
600C2#2092611yescharacteristic
601C2#2207512yes—
602C2#2208112yes—
603C2#2959211yescharacteristic
604C2#2959511yescharacteristic
605C2#2963811yescharacteristic
606C2#2964011yescharacteristic
607C2#2975412yes—
608C2#2976012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#389C2semidirect
4. C2×#1823C2semidirect
6. #11219C2semidirect
7. #11066C2semidirect
8. #10730C2semidirect
9. #10712C2semidirect
17. C2×#480C4semidirect
86. C2×#107C2×C4semidirect
91. C2×#102C2×C4semidirect
213. C2×C4⋊C4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4802
C22×#3892
C2×#18742
C2×#18232
#146982
#112192
#110662
#107302
#107122