Finite Group Databasebeta

← #370087

Subgroups — #370087

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#106C211yesmaximal, characteristic
2C2×#1949C212yesmaximal, complemented
3C2×#1852C211yesmaximal, characteristic
4C2×#1850C211yesmaximal, characteristic
5#13881C211yesmaximal, characteristic
6C22⋊C4×C4⋊C4C211yesmaximal, characteristic
7#11325C212yesmaximal, complemented
8#11231C212yesmaximal, complemented
9#10908C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×#106—28no—
18C22×#106—24no—
19C22×#106C2212yes—
20C22×#106C2212yes—
21C22×C4⋊C8C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C2×#382C2212yes—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4—21no—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1949—28no—
30#1852—24no—
31#1850C418yescomplemented
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1442—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C22⋊#95—21no—
44C22⋊#95—21no—
45C22⋊#95—21no—
46C24×C4—21no—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—24no—
52C23×C8—24no—
53C23×C8C2×C414yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4D414yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—22no—
78C22×C42C2312yes—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107—24no—
82C2×#107—42no—
83C2×#106—232no—
84C2×#106—24no—
85C2×#106—24no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—24no—
89C2×#106D418yes—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8C2×C418yescomplemented
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8C2×C414yes—
96#382—24no—
97#382—42no—
98#380—22no—
99#380—22no—
100#380—22no—
101#380Q814yes—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—42no—
104C4×C22⋊C4—44no—
105C4×C22⋊C4C2×C414yes—
106C4×C22⋊C4Q814yes—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95D412yes—
117#340—24no—
118#340—24no—
119#340—22no—
120#340—22no—
121#340—24no—
122#340—42no—
123#336C2×C418yescomplemented
124C25C2×D411yescharacteristic
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—44no—
137C23×C4—44no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—82no—
144C23×C4—82no—
145C23×C4C22×C412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×Q812yes—
151C23×C4C2×Q812yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4C2411yescharacteristic
155C22×C8—216no—
156C22×C8—28no—
157C22×C8—28no—
158C22×C8—28no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—216no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8C22×C412yes—
173C22×C8C22×C412yes—
174C22×C8C22×C414yes—
175C22×C8C2×D414yes—
176C22×C8#4114yes—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×Q814yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4C4⋊C414yes—
226C2×C22⋊C4C4⋊C414yes—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—44no—
232C2×C42—44no—
233C2×C42—82no—
234C2×C42C22×C414yes—
235C2×C42C2×Q814yes—
236#107—44no—
237#107—44no—
238#106—216no—
239#106—216no—
240C4⋊C8—216no—
241C22⋊C8—28no—
242#95—24no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—42no—
248#95—82no—
249#95—82no—
250#95—82no—
251#95C4⋊C418yes—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24C2×C4⋊C412yes—
257C24#12112yes—
258C24C22⋊Q812yes—
259C24#13512yes—
260C24#13512yes—
261C24AGL(1,ℤ/8)12yes—
262C24#13712yes—
263C24C22×D411yescharacteristic
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—48no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—48no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—48no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—84no—
330C22×C4—88no—
331C22×C4—88no—
332C22×C4—88no—
333C22×C4—88no—
334C22×C4—84no—
335C22×C4—84no—
336C22×C4C2×C4⋊C414yes—
337C22×C4C2×C4⋊C414yes—
338C22×C4C2×C4⋊C412yes—
339C22×C4C4×D414yes—
340C22×C4C4×Q814yes—
341C22×C4#12112yes—
342C22×C4C22⋊Q812yes—
343C22×C4#12514yes—
344C22×C4C4⋊Q814yes—
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D412yes—
348C22×C4C22×Q811yescharacteristic
349C22×C4C22×Q811yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4112yes—
352C22×C4C2×#4111yescharacteristic
353C22×C4H(5,𝔽2)12yes—
354C22×C42−1+412yes—
355C2×C8—232no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—42no—
372C2×C8C4×D418yes—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4#12014yes—
395C22⋊C4—28no—
396C22⋊C4—28no—
397C22⋊C4—48no—
398C22⋊C4—84no—
399C42—24no—
400C42—24no—
401C42—24no—
402C42—24no—
403C42—88no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—22no—
419C23—22no—
420C23#38914yes—
421C23#42714yes—
422C23#42814yes—
423C23#46314yes—
424C23#46414yes—
425C23#47814yes—
426C23#47914yes—
427C23C22×C4⋊C411yescharacteristic
428C23C2×#12111yescharacteristic
429C23C2×C22⋊Q811yescharacteristic
430C23D4212yes—
431C23#54712yes—
432C23C2×#13511yescharacteristic
433C23C2×#13511yescharacteristic
434C23C2×AGL(1,ℤ/8)11yescharacteristic
435C23C2×#13711yescharacteristic
436C23#57712yes—
437C23#57712yes—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—28no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—816no—
477C2×C4—816no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4#38914yes—
499C2×C4C2×C4×D411yescharacteristic
500C2×C4C2×C4×D412yes—
501C2×C4C2×C4×Q811yescharacteristic
502C2×C4#52012yes—
503C2×C4C2×#12011yescharacteristic
504C2×C4#52511yescharacteristic
505C2×C4#52512yes—
506C2×C4C2×#12511yescharacteristic
507C2×C4C2×C4⋊Q811yescharacteristic
508C2×C4#53411yescharacteristic
509C2×C4#53611yescharacteristic
510C2×C4#53912yes—
511C2×C4#54612yes—
512C2×C4D4×Q812yes—
513C2×C4D4×Q812yes—
514C2×C4#55212yes—
515C2×C4#55412yes—
516C2×C4#55412yes—
517C8—28no—
518C8—42no—
519C8—42no—
520C22—24no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—28no—
525C22—28no—
526C22—24no—
527C22—24no—
528C22—28no—
529C22—28no—
530C22#194918yescomplemented
531C22C2×#38911yescharacteristic
532C22D4×C4⋊C412yes—
533C22#241011yescharacteristic
534C22#241311yescharacteristic
535C22#244912yes—
536C22C2×#42711yescharacteristic
537C22C2×#42811yescharacteristic
538C22#291212yes—
539C22C2×#46311yescharacteristic
540C22C2×#46411yescharacteristic
541C22#304211yescharacteristic
542C22#304311yescharacteristic
543C22C2×#47811yescharacteristic
544C22C2×#47911yescharacteristic
545C22#308412yes—
546C22#326512yes—
547C22#326612yes—
548C22#328212yes—
549C22#328212yes—
550C22#333712yes—
551C22#333712yes—
552C22#334412yes—
553C22#334512yes—
554C4—24no—
555C4—44no—
556C4—44no—
557C4—44no—
558C4—84no—
559C4—82no—
560C4—84no—
561C4—161no—
562C4—161no—
563C4C4×#12011yescharacteristic
564C4#232211yescharacteristic
565C4#236012yes—
566C4#243411yescharacteristic
567C4#247312yes—
568C4#258311yescharacteristic
569C2—28no—
570C2#1404311yescharacteristic
571C2C2×#194911yescharacteristic
572C2#2117411yescharacteristic
573C2#2265412yes—
574C2#2265912yes—
575C2#2435711yescharacteristic
576C2#2435911yescharacteristic
577C2#2457911yescharacteristic
578C2#2458111yescharacteristic
579C2#2634612yes—
580C2#2634812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1949C2semidirect
7. #11325C2semidirect
8. #11231C2semidirect
9. #10908C2semidirect
31. #1850C4semidirect
92. C2×C4⋊C8C2×C4semidirect
123. #336C2×C4semidirect
530. C22#1949semidirect

Maximal subgroups

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

subgroupindex
C23×#1062
C2×#19492
C2×#18522
C2×#18502
#138812
C22⋊C4×C4⋊C42
#113252
#112312
#109082
#107302
#107212