Finite Group Databasebeta

← #415210

Subgroups — #415210

Order 512.

1028 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1990C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic
3#14161C211yesmaximal, characteristic
4#13911C211yesmaximal, characteristic
5#11358C212yesmaximal
6#11315C212yesmaximal
7#11315C212yesmaximal
8#11268C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10834C211yesmaximal, characteristic
11C4×#340C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×#382C2212yes—
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1990—24no—
21#1987—24no—
22C2×#340C2211yescharacteristic
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1446—22no—
35#1442—22no—
36C4⋊#95—21no—
37C4⋊#95—21no—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×Q8—21no—
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8D414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C42—21no—
65C22×C42—21no—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4×C8—22no—
72C2×C4×C8—22no—
73C2×C4×C8—22no—
74#384—22no—
75#384—22no—
76#382—24no—
77#382—42no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95D412yes—
95#340—22no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—22no—
101#340—22no—
102#340—24no—
103#340—42no—
104#340Q814yescomplemented
105#336—24no—
106#336—42no—
107C22×Q8—24no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—81no—
111C22×Q8—81no—
112C22×Q8—81no—
113C22×Q8—81no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2411yescharacteristic
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8C2×Q812yes—
145C22×C8#4112yes—
146C4⋊Q8—28no—
147C4⋊Q8—28no—
148C4⋊Q8—44no—
149C4⋊Q8—44no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
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—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4C2×Q812yes—
198C2×C4⋊C4C2×Q812yes—
199C2×C4⋊C4#4112yes—
200C2×C4⋊C4#4112yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42C2×D412yes—
214C2×C42#4112yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42#4112yes—
218C2×C42#4112yes—
219C4⋊C8—44no—
220C4×C8—44no—
221#95—24no—
222#95—44no—
223#95—44no—
224#95—42no—
225#95—42no—
226#95—82no—
227#95—82no—
228#95—82no—
229#95—82no—
230#95—82no—
231#95—82no—
232C24C22×D411yescharacteristic
233C2×Q8—48no—
234C2×Q8—84no—
235C2×Q8—84no—
236C2×Q8—84no—
237C2×Q8—84no—
238C2×Q8—84no—
239C2×Q8—84no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4#12112yes—
301C22×C4#12112yes—
302C22×C4#12312yes—
303C22×C4#12312yes—
304C22×C4#12412yes—
305C22×C4#12412yes—
306C22×C4#13512yes—
307C22×C4#13512yes—
308C22×C4#13712yes—
309C22×C4#13712yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C22×Q811yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4H(5,𝔽2)11yescharacteristic
321C22×C42−1+411yescharacteristic
322C22×C42−1+411yescharacteristic
323C22×C42−1+411yescharacteristic
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—48no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—84no—
379C42—84no—
380C42#12314yes—
381C42#12314yes—
382C42#12614yes—
383C42#12614yes—
384C23C2×#12111yescharacteristic
385C23C2×#12311yescharacteristic
386C23C2×#12411yescharacteristic
387C23#54611yescharacteristic
388C23#54711yescharacteristic
389C23#54811yescharacteristic
390C23D4×Q811yescharacteristic
391C23C2×#13511yescharacteristic
392C23C2×#13511yescharacteristic
393C23C2×#13711yescharacteristic
394C23C2×#13711yescharacteristic
395C23#57611yescharacteristic
396C23#57611yescharacteristic
397C23#57711yescharacteristic
398C23#57711yescharacteristic
399Q8—816no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4#39714yes—
463C2×C4#39714yes—
464C2×C4#46414yes—
465C2×C4#46414yes—
466C2×C4#48514yes—
467C2×C4#48514yes—
468C2×C4#48814yes—
469C2×C4#49114yes—
470C2×C4C2×#12311yescharacteristic
471C2×C4C2×#12311yescharacteristic
472C2×C4C2×#12611yescharacteristic
473C2×C4C2×#12611yescharacteristic
474C2×C4#52911yescharacteristic
475C2×C4#52911yescharacteristic
476C2×C4#52911yescharacteristic
477C2×C4#53311yescharacteristic
478C2×C4#53311yescharacteristic
479C2×C4#53311yescharacteristic
480C2×C4#54111yescharacteristic
481C2×C4#54111yescharacteristic
482C2×C4#54111yescharacteristic
483C2×C4#54211yescharacteristic
484C2×C4#54211yescharacteristic
485C2×C4#54211yescharacteristic
486C2×C4#54811yescharacteristic
487C2×C4#54811yescharacteristic
488C2×C4D4×Q811yescharacteristic
489C2×C4#55011yescharacteristic
490C2×C4#55211yescharacteristic
491C2×C4#55211yescharacteristic
492C2×C4#55411yescharacteristic
493C2×C4#55611yescharacteristic
494C2×C4#55611yescharacteristic
495C2×C4#55611yescharacteristic
496C2×C4#55711yescharacteristic
497C2×C4#55911yescharacteristic
498C8—42no—
499C8—42no—
500C8—82no—
501C22C2×#39711yescharacteristic
502C22#242611yescharacteristic
503C22#242911yescharacteristic
504C22#244611yescharacteristic
505C22#244711yescharacteristic
506C22#248911yescharacteristic
507C22#249211yescharacteristic
508C22C2×#46411yescharacteristic
509C22C2×#46411yescharacteristic
510C22#304111yescharacteristic
511C22#304211yescharacteristic
512C22C2×#48511yescharacteristic
513C22C2×#48511yescharacteristic
514C22#309611yescharacteristic
515C22#309611yescharacteristic
516C22C2×#48811yescharacteristic
517C22C2×#49111yescharacteristic
518C22#313611yescharacteristic
519C22#313611yescharacteristic
520C22#330311yescharacteristic
521C22#330511yescharacteristic
522C22#333111yescharacteristic
523C22#333311yescharacteristic
524C22#334011yescharacteristic
525C22#334111yescharacteristic
526C22#335611yescharacteristic
527C22#335711yescharacteristic
528C22#336011yescharacteristic
529C22#336111yescharacteristic
530C22#336911yescharacteristic
531C22#337511yescharacteristic
532C22#338011yescharacteristic
533C22#338111yescharacteristic
534C22#338511yescharacteristic
535C22#338711yescharacteristic
536C4—24no—
537C4—84no—
538C4—82no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—162no—
545C4#207418yes—
546C4#242111yescharacteristic
547C4#242311yescharacteristic
548C4#244611yescharacteristic
549C4#244811yescharacteristic
550C4#251111yescharacteristic
551C4#251411yescharacteristic
552C4#264511yescharacteristic
553C4#286311yescharacteristic
554C2#1518611yescharacteristic
555C2C2×#207411yescharacteristic
556C2#2363211yescharacteristic
557C2#2527311yescharacteristic
558C2#2527711yescharacteristic
559C2#2545011yescharacteristic
560C2#2545411yescharacteristic
561C2#2627711yescharacteristic
562C2#2627911yescharacteristic
563C2#2629711yescharacteristic
564C2#2629911yescharacteristic
565C2#2772811yescharacteristic
566C2#2773811yescharacteristic
567C2#2781411yescharacteristic
568C2#2782411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
104. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19902
C2×#19872
#141612
#139112
#113582
#113152
#112682
#108492
#108342
C4×#3402