Finite Group Databasebeta

← #368918

Subgroups — #368918

Order 512.

1446 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic
2C2×#2370C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1987C211yesmaximal, characteristic
5#14155C211yesmaximal, characteristic
6#11220C212yesmaximal
7#11219C212yesmaximal
8#10908C212yesmaximal
9#10907C212yesmaximal
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C2×#501—24no—
15C2×#501—24no—
16C2×#501C418yescomplemented
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#97C2211yescharacteristic
21#2370—24no—
22#2370—24no—
23C2×#391—21no—
24C2×#391—21no—
25C2×#389—21no—
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#387—21no—
29C2×#387—21no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1989—24no—
37#1987—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1445—21no—
47#1445—21no—
48C4⋊#95C2212yes—
49C4⋊#95C2212yes—
50C23×Q8—21no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125C2×C414yes—
63C2×#125D414yes—
64C2×#125D414yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C42—21no—
79C22×C42—21no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#501—232no—
86C2×#107—24no—
87C2×#107—24no—
88C2×#107C2×C418yes—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106C2×C418yes—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×#97—22no—
96C2×#97—22no—
97C2×#97C2×C414yes—
98#391—44no—
99#391—44no—
100#389—28no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#387—28no—
106#387—28no—
107C4×C4⋊C4—28no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4D414yes—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—21no—
128C2×#95D411yescharacteristic
129C2×#95D411yescharacteristic
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340Q818yescomplemented
136#336—24no—
137#336—24no—
138C22×Q8—24no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—42no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8C22×C414yes—
170C22×C8C2×Q814yes—
171C22×C8#4114yes—
172C4⋊Q8—28no—
173C4⋊Q8—28no—
174C4⋊Q8—44no—
175C4⋊Q8—44no—
176#125—28no—
177#125—28no—
178#125—28no—
179#125C22⋊C4116yes—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
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—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4C22×C414yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×Q814yes—
231C2×C4⋊C4C2×Q814yes—
232C2×C42—24no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—82no—
249C2×C42—82no—
250C2×C42C22×C412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42#4112yes—
255C2×C42#4112yes—
256#107—216no—
257#106—216no—
258C4⋊C8—44no—
259#97—28no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—22no—
266#95—22no—
267#95—44no—
268#95—44no—
269#95—42no—
270#95—44no—
271#95—44no—
272C24C22×D411yescharacteristic
273C2×Q8—48no—
274C2×Q8—48no—
275C2×Q8—48no—
276C2×Q8—84no—
277C2×Q8—84no—
278C2×Q8—84no—
279C2×Q8—84no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4C4×D412yes—
341C22×C4C4×D412yes—
342C22×C4#12012yes—
343C22×C4#12012yes—
344C22×C4#12112yes—
345C22×C4#12112yes—
346C22×C4AGL(1,ℤ/8)12yes—
347C22×C4AGL(1,ℤ/8)12yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×Q812yes—
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4112yes—
355C22×C4C2×#4111yescharacteristic
356C22×C42−1+412yes—
357C22×C42−1+412yes—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8C4×Q818yes—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—216no—
381C4⋊C4—216no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—48no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—48no—
397C4⋊C4—84no—
398C4⋊C4C2×C22⋊C414yes—
399C4⋊C4C2×C22⋊C414yes—
400C4⋊C4C4⋊Q814yes—
401C42—48no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—42no—
407C42—42no—
408C42—84no—
409C42—84no—
410C42—84no—
411C42C2×C22⋊C414yes—
412C42C4×D414yes—
413C42#12114yes—
414C42#12314yes—
415C23C2×C4×D411yescharacteristic
416C23C2×#12011yescharacteristic
417C23C2×#12111yescharacteristic
418C23#54712yes—
419C23D4×Q812yes—
420C23C2×AGL(1,ℤ/8)11yescharacteristic
421C23C2×AGL(1,ℤ/8)11yescharacteristic
422C23#57511yescharacteristic
423C23#57511yescharacteristic
424C23#57712yes—
425C23#57812yes—
426Q8—816no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4#38614yes—
494C2×C4#38614yes—
495C2×C4C22×C22⋊C411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×C4×Q812yes—
498C2×C4#51912yes—
499C2×C4C2×#12111yescharacteristic
500C2×C4C2×#12311yescharacteristic
501C2×C4C2×C4⋊Q811yescharacteristic
502C2×C4#53211yescharacteristic
503C2×C4#53312yes—
504C2×C4#53611yescharacteristic
505C2×C4#53611yescharacteristic
506C2×C4#54112yes—
507C2×C4#54812yes—
508C2×C4#54812yes—
509C2×C4D4×Q812yes—
510C2×C4D4×Q812yes—
511C2×C4#55612yes—
512C2×C4#55712yes—
513C8—28no—
514C8—82no—
515C22C2×#38611yescharacteristic
516C22#236512yes—
517C22#242912yes—
518C22#243211yescharacteristic
519C22#243411yescharacteristic
520C22C4×AGL(1,ℤ/8)11yescharacteristic
521C22C4×AGL(1,ℤ/8)11yescharacteristic
522C22#294712yes—
523C22#300611yescharacteristic
524C22#300611yescharacteristic
525C22#300911yescharacteristic
526C22#301111yescharacteristic
527C22#303811yescharacteristic
528C22#304011yescharacteristic
529C22#304612yes—
530C22#334312yes—
531C22#334512yes—
532C22#334912yes—
533C22#335012yes—
534C22#338412yes—
535C22#338612yes—
536C22#338812yes—
537C22#339012yes—
538C4—24no—
539C4—24no—
540C4—42no—
541C4—42no—
542C4—44no—
543C4—44no—
544C4—44no—
545C4—81no—
546C4—82no—
547C4—81no—
548C4—84no—
549C4—84no—
550C4C4×C4⋊Q811yescharacteristic
551C4#232711yescharacteristic
552C4Q8×C22⋊C412yes—
553C4#244612yes—
554C4#254511yescharacteristic
555C4#254611yescharacteristic
556C2#1402411yescharacteristic
557C2#2092811yescharacteristic
558C2#2093111yescharacteristic
559C2#2316612yes—
560C2#2316812yes—
561C2#2546512yes—
562C2#2546812yes—
563C2#2571411yescharacteristic
564C2#2571611yescharacteristic
565C2#2584911yescharacteristic
566C2#2585111yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. C2×#501C4semidirect
135. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#23702
C2×#19892
C2×#19872
#141552
#112202
#112192
#109082
#109072
#108352
#107032