Finite Group Databasebeta

← #120863

Subgroups — #120863

Order 512.

792 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13966C211yesmaximal, characteristic
2#10854C211yesmaximal, characteristic
3#8023C211yesmaximal, characteristic
4C2×#1294C211yesmaximal, characteristic
5C2×#1294C211yesmaximal, characteristic
6C2×#1289C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C22×C22⋊C8C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C2×#380C2211yescharacteristic
11C2×#380C2211yescharacteristic
12C2×#380C2211yescharacteristic
13C2×C4×C22⋊C4C2211yescharacteristic
14C22×#95C2211yescharacteristic
15C2×#342—21no—
16C2×#342—21no—
17#1442—22no—
18#1434—21no—
19#1434—21no—
20C22⋊#95—21no—
21C22⋊#95—21no—
22#1294—22no—
23#1294—22no—
24#1294—21no—
25#1294—21no—
26#1289—22no—
27#1282C414yescomplemented
28C24×C4—21no—
29C24×C4C2311yescharacteristic
30C23×C8—21no—
31C23×C8—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4Q812yes—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4D411yescharacteristic
39C22×C22⋊C4D411yescharacteristic
40C22×C42—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8C2×C412yes—
52C2×C22⋊C8C2×C412yes—
53#380—21no—
54#380—21no—
55#380—21no—
56#380—21no—
57#380—21no—
58#380—21no—
59#380—21no—
60#380—21no—
61#380—21no—
62#380—21no—
63#380—21no—
64#380D412yes—
65C4×C22⋊C4—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4C2×C412yes—
68C4×C22⋊C4Q812yes—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95D411yescharacteristic
82C2×#95D411yescharacteristic
83C2×#95D412yes—
84#342—42no—
85#342—42no—
86C25C2×D411yescharacteristic
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—22no—
93C23×C4—44no—
94C23×C4—41no—
95C23×C4—41no—
96C23×C4—41no—
97C23×C4—41no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—41no—
101C23×C4—82no—
102C23×C4—82no—
103C23×C4C22×C411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×Q811yescharacteristic
108C23×C4C2×Q811yescharacteristic
109C23×C4#4111yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C23×C4#4111yescharacteristic
114C23×C4#4111yescharacteristic
115C23×C4#4111yescharacteristic
116C23×C4#4111yescharacteristic
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8—41no—
122C22×C8—41no—
123C22×C8—82no—
124C22×C8—82no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—82no—
132C2×C4⋊C4—82no—
133C2×C4⋊C4—82no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4C4⋊C412yes—
163C2×C22⋊C4C4⋊C412yes—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—82no—
167C22⋊C8—22no—
168C22⋊C8—22no—
169C22⋊C8—22no—
170C22⋊C8—22no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—21no—
179#95—21no—
180#95—22no—
181#95—22no—
182#95—21no—
183#95—21no—
184#95—44no—
185#95—44no—
186#95—41no—
187#95—41no—
188#95—41no—
189#95—41no—
190#95—41no—
191#95—82no—
192#95—82no—
193C24—42no—
194C24—42no—
195C24C2×C4⋊C411yescharacteristic
196C24#12011yescharacteristic
197C24#12111yescharacteristic
198C24C22⋊Q811yescharacteristic
199C24#12311yescharacteristic
200C24#12411yescharacteristic
201C24#12411yescharacteristic
202C24#13511yescharacteristic
203C24#13511yescharacteristic
204C24#13511yescharacteristic
205C24#13511yescharacteristic
206C24AGL(1,ℤ/8)11yescharacteristic
207C24AGL(1,ℤ/8)11yescharacteristic
208C24#13711yescharacteristic
209C24#13711yescharacteristic
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—84no—
258C22×C4—84no—
259C22×C4—84no—
260C22×C4—84no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4#11711yescharacteristic
274C22×C4#11711yescharacteristic
275C22×C4C4×D411yescharacteristic
276C22×C4C4×D411yescharacteristic
277C22×C4C4×D411yescharacteristic
278C22×C4C4×Q811yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4C22⋊Q811yescharacteristic
282C22×C4C22⋊Q811yescharacteristic
283C22×C4C22⋊Q811yescharacteristic
284C22×C4C22⋊Q811yescharacteristic
285C22×C4#12311yescharacteristic
286C22×C4#12311yescharacteristic
287C22×C4#12311yescharacteristic
288C22×C4#12311yescharacteristic
289C22×C4#12411yescharacteristic
290C22×C4#12411yescharacteristic
291C22×C4#12511yescharacteristic
292C22×C4#12511yescharacteristic
293C22×C4#12511yescharacteristic
294C22×C4#12511yescharacteristic
295C22×C4#12511yescharacteristic
296C22×C4#12611yescharacteristic
297C22×C4#12611yescharacteristic
298C22×C4#12611yescharacteristic
299C22×C4#12611yescharacteristic
300C22×C4C4⋊Q811yescharacteristic
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—82no—
304C2×C8—82no—
305C2×C8—82no—
306C2×C8—82no—
307C2×C8—82no—
308C2×C8—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—81no—
326C22⋊C4—81no—
327C22⋊C4—81no—
328C22⋊C4—81no—
329C22⋊C4—81no—
330C22⋊C4—81no—
331C22⋊C4—81no—
332C22⋊C4—81no—
333C22⋊C4—81no—
334C22⋊C4—81no—
335C42—82no—
336C42—82no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—41no—
354C23—41no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23#38211yescharacteristic
360C23#38511yescharacteristic
361C23#38911yescharacteristic
362C23#39311yescharacteristic
363C23#39411yescharacteristic
364C23#39711yescharacteristic
365C23#40011yescharacteristic
366C23#42711yescharacteristic
367C23#42711yescharacteristic
368C23#42811yescharacteristic
369C23#42811yescharacteristic
370C23#44911yescharacteristic
371C23#44911yescharacteristic
372C23#45211yescharacteristic
373C23#45211yescharacteristic
374C23#46311yescharacteristic
375C23#46311yescharacteristic
376C23#46411yescharacteristic
377C23#46411yescharacteristic
378C23#47811yescharacteristic
379C23#47811yescharacteristic
380C23#47911yescharacteristic
381C23#47911yescharacteristic
382C23#48211yescharacteristic
383C23#48211yescharacteristic
384C23#48511yescharacteristic
385C23#48511yescharacteristic
386C23#48811yescharacteristic
387C23#48811yescharacteristic
388C23#48811yescharacteristic
389C23#48811yescharacteristic
390C23#48911yescharacteristic
391C23#48911yescharacteristic
392C23#49011yescharacteristic
393C23#49111yescharacteristic
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—82no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—162no—
460C2×C4—162no—
461C2×C4—162no—
462C2×C4—162no—
463C2×C4#38711yescharacteristic
464C2×C4#38711yescharacteristic
465C2×C4#38811yescharacteristic
466C2×C4#38811yescharacteristic
467C2×C4#39811yescharacteristic
468C2×C4#39811yescharacteristic
469C2×C4#40011yescharacteristic
470C2×C4#40011yescharacteristic
471C8—82no—
472C8—82no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22#144211yescharacteristic
514C22#183511yescharacteristic
515C22#183611yescharacteristic
516C22#185411yescharacteristic
517C22#185411yescharacteristic
518C22#194911yescharacteristic
519C22#194911yescharacteristic
520C22#202511yescharacteristic
521C22#202611yescharacteristic
522C22#203112yes—
523C22#203212yes—
524C22#204211yescharacteristic
525C22#204211yescharacteristic
526C22#207311yescharacteristic
527C22#207411yescharacteristic
528C22#210111yescharacteristic
529C22#210111yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—82no—
539C4—82no—
540C4—84no—
541C4—161no—
542C4—161no—
543C4—162no—
544C4—162no—
545C2—21no—
546C2—21no—
547C2—21no—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—21no—
557C2#1037612yes—
558C2#1037712yes—
559C2#1126311yescharacteristic
560C2#1126811yescharacteristic
561C2#1127911yescharacteristic
subgroupG/H
center, socleC23#393
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1282C4semidirect

Maximal subgroups

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

subgroupindex
#139662
#108542
#80232
C2×#12942
C2×#12892
C2×#12822