Finite Group Databasebeta

← #360411

Subgroups — #360411

Order 512.

974 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20764C211yesmaximal, characteristic
2#20647C211yesmaximal, characteristic
3C2×#1803C211yesmaximal, characteristic
4#20441C211yesmaximal, characteristic
5C2×C4×#101C211yesmaximal, characteristic
6#11303C212yesmaximal
7#11164C212yesmaximal
8#10942C212yesmaximal
9#10857C211yesmaximal, characteristic
10#10853C211yesmaximal, characteristic
11#10827C212yesmaximal
12C22×C4×Q8C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×#419C2212yes—
16C22×#103C2212yes—
17C22×#101C2211yescharacteristic
18C22×#101C2211yescharacteristic
19C2×#405C2212yes—
20C2×C4×M16C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#1836—22no—
25#1834—21no—
26#1834—21no—
27#1805—22no—
28#1803—21no—
29#1803—21no—
30#1803—22no—
31#1803—21no—
32#1803—21no—
33#1771—22no—
34#1770—21no—
35C8×C4⋊C4—21no—
36C4×#101C418yescomplemented
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#343C2211yescharacteristic
40C2×#343C2211yescharacteristic
41C2×#341C2212yes—
42C2×#341C2212yes—
43#1736C2212yes—
44#1736C2212yes—
45#1736C2212yes—
46C2×#340C2212yes—
47#1729C2212yes—
48C23×Q8C2311yescharacteristic
49C22×M16C2312yes—
50C22×M16C2311yescharacteristic
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C23×C8C2312yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8C2×C414yes—
63C2×#117C2312yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71#423—22no—
72#423—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8D411yescharacteristic
78C2×C4⋊C8D411yescharacteristic
79C2×C4⋊C8D412yes—
80#419—24no—
81#419—42no—
82C2×#103—24no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—42no—
88C2×#101—22no—
89C2×#101—22no—
90C2×#101—22no—
91C2×#101—22no—
92C2×#101C2×C414yes—
93C2×#101C2×C414yes—
94#405—24no—
95#405—42no—
96C4×M16—22no—
97C4×M16—22no—
98C4×M16C2×C414yes—
99C4×M16C2×C414yes—
100C2×#97—21no—
101C2×#97—21no—
102C2×#97—22no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—22no—
106#384—21no—
107#384—21no—
108#384—21no—
109#384—21no—
110#384D411yescharacteristic
111#384D411yescharacteristic
112#382D412yes—
113#380—21no—
114#380—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117#343—24no—
118#343—42no—
119#343C2×C414yes—
120#343C2×C414yes—
121#341—42no—
122#341—42no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#336—22no—
132#336—22no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8C22×C412yes—
145C22×Q8#4112yes—
146C22×Q8#4112yes—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4#4111yescharacteristic
151C23×C4#4111yescharacteristic
152C23×C4C2411yescharacteristic
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16C22×C412yes—
178C2×M16C22×C412yes—
179C2×M16C22×C412yes—
180C2×M16C22×C412yes—
181C2×M16#4112yes—
182C2×M16#4112yes—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8C2×D412yes—
195C22×C8C2×D412yes—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C22⋊C4—22no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—42no—
242C2×C42C22×C412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—24no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255#103—44no—
256#103—44no—
257#103—44no—
258#103—44no—
259#101—28no—
260#101—28no—
261#97—24no—
262#97—24no—
263#97—44no—
264C4×C8—24no—
265C4×C8—24no—
266C4×C8—44no—
267#95—22no—
268#95—22no—
269C24C2×#4111yescharacteristic
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8#11714yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4#11712yes—
322C22×C4#11712yes—
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4#12111yescharacteristic
327C22×C4#12612yes—
328C22×C4#12612yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C42−1+411yescharacteristic
342C22×C42−1+411yescharacteristic
343M16—22no—
344M16—22no—
345M16—22no—
346M16—22no—
347M16—22no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—22no—
352M16—41no—
353M16—41no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
358M16—42no—
359M16—42no—
360M16—42no—
361M16—42no—
362M16#11714yes—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—81no—
405C4⋊C4—81no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C42—42no—
413C42—42no—
414C42—44no—
415C42—44no—
416C42C4×D412yes—
417C42C4×D412yes—
418C42C4×D412yes—
419C42C4×D412yes—
420C42#12312yes—
421C42#12312yes—
422C42#12412yes—
423C42#12412yes—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23C2×#11711yescharacteristic
429C23C2×#12111yescharacteristic
430C23C2×#12611yescharacteristic
431C23#52911yescharacteristic
432C23#52911yescharacteristic
433C23#54211yescharacteristic
434C23#54211yescharacteristic
435Q8—24no—
436Q8—24no—
437Q8—42no—
438Q8—42no—
439Q8—42no—
440Q8—42no—
441Q8—42no—
442Q8—42no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—162no—
475C2×C4—162no—
476C2×C4#38812yes—
477C2×C4#38812yes—
478C2×C4#38812yes—
479C2×C4#38812yes—
480C2×C4#47112yes—
481C2×C4#47312yes—
482C2×C4C2×#11711yescharacteristic
483C2×C4C2×#11711yescharacteristic
484C2×C4C2×C4×D411yescharacteristic
485C2×C4C2×C4×D411yescharacteristic
486C2×C4#52011yescharacteristic
487C2×C4#52011yescharacteristic
488C2×C4C2×#12311yescharacteristic
489C2×C4C2×#12411yescharacteristic
490C2×C4#52911yescharacteristic
491C2×C4#52911yescharacteristic
492C2×C4#54011yescharacteristic
493C2×C4#54111yescharacteristic
494C2×C4#54611yescharacteristic
495C2×C4#54611yescharacteristic
496C2×C4#54611yescharacteristic
497C2×C4#54711yescharacteristic
498C2×C4#54811yescharacteristic
499C2×C4#54811yescharacteristic
500C2×C4#54811yescharacteristic
501C2×C4#55011yescharacteristic
502C2×C4#55211yescharacteristic
503C2×C4#55211yescharacteristic
504C2×C4#55211yescharacteristic
505C2×C4#55311yescharacteristic
506C2×C4#55311yescharacteristic
507C2×C4#55311yescharacteristic
508C2×C4#55511yescharacteristic
509C2×C4#55611yescharacteristic
510C8—24no—
511C8—42no—
512C8—42no—
513C8—42no—
514C8—42no—
515C8—42no—
516C8—82no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22#228311yescharacteristic
530C22C2×#38811yescharacteristic
531C22#231411yescharacteristic
532C22#242611yescharacteristic
533C22#242711yescharacteristic
534C22#249411yescharacteristic
535C22#249511yescharacteristic
536C22C2×#47111yescharacteristic
537C22C2×#47311yescharacteristic
538C22#306811yescharacteristic
539C22#306911yescharacteristic
540C4—22no—
541C4—22no—
542C4—22no—
543C4—44no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—162no—
548C4#193014yes—
549C4#235211yescharacteristic
550C4#235211yescharacteristic
551C4#235211yescharacteristic
552C4#235411yescharacteristic
553C4#242011yescharacteristic
554C4#242111yescharacteristic
555C4#248911yescharacteristic
556C4#249011yescharacteristic
557C2—21no—
558C2—21no—
559C2—21no—
560C2—21no—
561C2#1387511yescharacteristic
562C2C2×#193011yescharacteristic
563C2#2107611yescharacteristic
564C2#2531411yescharacteristic
565C2#2531511yescharacteristic
566C2#2536011yescharacteristic
567C2#2536111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
36. C4×#101C4semidirect

Maximal subgroups

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

subgroupindex
#207642
#206472
C2×#18032
#204412
C2×C4×#1012
#113032
#111642
#109422
#108572
#108532
#108272