Finite Group Databasebeta

← #360464

Subgroups — #360464

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1930C212yesmaximal
2#20800C211yesmaximal, characteristic
3#20647C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5#20441C211yesmaximal, characteristic
6C2×C4×#101C211yesmaximal, characteristic
7#11290C212yesmaximal
8#11145C212yesmaximal
9#10896C212yesmaximal
10#10857C211yesmaximal, characteristic
11#10853C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#428C2212yes—
14C2×#428C2212yes—
15C2×#423C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C2×#419C2212yes—
18C22×#103C2212yes—
19C22×#101C2211yescharacteristic
20C22×#101C2211yescharacteristic
21C2×#405C2212yes—
22C2×C4×M16C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#2285C2211yescharacteristic
25C2×#384C2211yescharacteristic
26#1930—28no—
27#1847—22no—
28#1845—21no—
29#1844—21no—
30#1805—22no—
31#1803—21no—
32#1803—21no—
33#1803—22no—
34#1803—21no—
35#1803—21no—
36#1771—22no—
37#1770—21no—
38C8×C4⋊C4—21no—
39C4×#101C418yescomplemented
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#343C2211yescharacteristic
43C2×#343C2211yescharacteristic
44C2×#341C2212yes—
45C2×#341C2212yes—
46#1736C2212yes—
47C2×#340C2212yes—
48#1729C2212yes—
49C23×Q8C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C23×C8C2312yes—
56C2×C4×Q8—22no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8C2×C414yes—
64C2×#117C2312yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#428—24no—
73#428—24no—
74#428—42no—
75#428—42no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#106—22no—
79C2×#106—22no—
80#423—22no—
81#423—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8D411yescharacteristic
87C2×C4⋊C8D411yescharacteristic
88C2×C4⋊C8D412yes—
89#419—24no—
90#419—42no—
91C2×#103—24no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—42no—
97C2×#101—22no—
98C2×#101—22no—
99C2×#101C2×C414yes—
100C2×#101C2×C414yes—
101C2×#101D414yes—
102C2×#101Q814yes—
103#405—24no—
104#405—42no—
105C4×M16—22no—
106C4×M16—22no—
107C4×M16C2×C414yes—
108C4×M16C2×C414yes—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97—22no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—22no—
115#384—21no—
116#384—21no—
117#384—21no—
118#384—21no—
119#384D411yescharacteristic
120#384D411yescharacteristic
121#382D412yes—
122#380—21no—
123#380—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126#343—24no—
127#343—42no—
128#343C2×C414yes—
129#343C2×C414yes—
130#341—24no—
131#341—24no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#336—22no—
137#336—22no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8C22×C412yes—
150C22×Q8C2×D412yes—
151C22×Q8C2×Q812yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4#4111yescharacteristic
156C23×C4#4111yescharacteristic
157C23×C4C2411yescharacteristic
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—21no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16C22×C412yes—
179C2×M16C22×C412yes—
180C2×M16C22×C412yes—
181C2×M16C22×C412yes—
182C2×M16C2×D412yes—
183C2×M16C2×D412yes—
184C2×M16C2×Q812yes—
185C2×M16C2×Q812yes—
186C2×M16#4112yes—
187C2×M16#4112yes—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—22no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C4×Q8—24no—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×Q8—24no—
205C4×Q8—42no—
206C4×Q8—42no—
207C4×Q8—42no—
208C4×Q8—42no—
209#117—22no—
210#117—22no—
211#117—22no—
212#117—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4—82no—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C22⋊C4—22no—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42—42no—
247C2×C42C22×C412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42#4112yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255#107—44no—
256#107—44no—
257#106—44no—
258#106—44no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—24no—
262C4⋊C8—42no—
263C4⋊C8—42no—
264#103—44no—
265#103—44no—
266#103—44no—
267#103—44no—
268#101—28no—
269#101C4⋊C4116yescomplemented
270#97—24no—
271#97—24no—
272#97—44no—
273C4×C8—24no—
274C4×C8—24no—
275C4×C8—44no—
276#95—22no—
277#95—22no—
278C24C2×#4111yescharacteristic
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8C2×C4⋊C414yes—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
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—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4#11712yes—
331C22×C4#11712yes—
332C22×C4#12111yescharacteristic
333C22×C4#12111yescharacteristic
334C22×C4#12111yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4#12612yes—
337C22×C4#12612yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×Q811yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4H(5,𝔽2)11yescharacteristic
349C22×C4H(5,𝔽2)11yescharacteristic
350C22×C42−1+411yescharacteristic
351C22×C42−1+411yescharacteristic
352M16—22no—
353M16—22no—
354M16—22no—
355M16—22no—
356M16—22no—
357M16—22no—
358M16—22no—
359M16—22no—
360M16—22no—
361M16—22no—
362M16—22no—
363M16—41no—
364M16—41no—
365M16—42no—
366M16—42no—
367M16—42no—
368M16—42no—
369M16C2×C4⋊C414yes—
370M16C2×C4⋊C414yes—
371M16#11714yes—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C42—42no—
422C42—42no—
423C42—44no—
424C42—44no—
425C42C4×D412yes—
426C42C4×D412yes—
427C42C4×D412yes—
428C42C4×D412yes—
429C42#12312yes—
430C42#12312yes—
431C42#12412yes—
432C42#12412yes—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23C2×#11711yescharacteristic
438C23C2×#12111yescharacteristic
439C23C2×#12611yescharacteristic
440C23#53211yescharacteristic
441C23#53311yescharacteristic
442C23#54011yescharacteristic
443C23#54111yescharacteristic
444Q8—24no—
445Q8—24no—
446Q8—42no—
447Q8—42no—
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—162no—
484C2×C4—162no—
485C2×C4#38812yes—
486C2×C4#38812yes—
487C2×C4#38812yes—
488C2×C4#38812yes—
489C2×C4#47112yes—
490C2×C4#47312yes—
491C2×C4C22×C4⋊C411yescharacteristic
492C2×C4C2×#11711yescharacteristic
493C2×C4C2×C4×D411yescharacteristic
494C2×C4C2×C4×D411yescharacteristic
495C2×C4#52011yescharacteristic
496C2×C4#52011yescharacteristic
497C2×C4C2×#12311yescharacteristic
498C2×C4C2×#12411yescharacteristic
499C2×C4#53211yescharacteristic
500C2×C4#53311yescharacteristic
501C2×C4#54211yescharacteristic
502C2×C4#54211yescharacteristic
503C2×C4D4211yescharacteristic
504C2×C4D4211yescharacteristic
505C2×C4#54611yescharacteristic
506C2×C4#54711yescharacteristic
507C2×C4#54711yescharacteristic
508C2×C4#54711yescharacteristic
509C2×C4#54811yescharacteristic
510C2×C4D4×Q811yescharacteristic
511C2×C4D4×Q811yescharacteristic
512C2×C4#55011yescharacteristic
513C2×C4#55211yescharacteristic
514C2×C4#55311yescharacteristic
515C2×C4#55411yescharacteristic
516C2×C4#55411yescharacteristic
517C2×C4#55511yescharacteristic
518C2×C4#55611yescharacteristic
519C8—24no—
520C8—24no—
521C8—24no—
522C8—42no—
523C8—42no—
524C8—42no—
525C8—82no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22#228511yescharacteristic
539C22C2×#38811yescharacteristic
540C22#231411yescharacteristic
541C22#242811yescharacteristic
542C22#242911yescharacteristic
543C22#249611yescharacteristic
544C22#249711yescharacteristic
545C22C2×#47111yescharacteristic
546C22C2×#47311yescharacteristic
547C22#306811yescharacteristic
548C22#306911yescharacteristic
549C4—22no—
550C4—22no—
551C4—22no—
552C4—44no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4—162no—
557C4#193014yescomplemented
558C4D4×C4⋊C411yescharacteristic
559C4D4×C4⋊C411yescharacteristic
560C4#235211yescharacteristic
561C4#235411yescharacteristic
562C4#242211yescharacteristic
563C4#242311yescharacteristic
564C4#249111yescharacteristic
565C4#249211yescharacteristic
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2#1387611yescharacteristic
571C2C2×#193011yescharacteristic
572C2#2107611yescharacteristic
573C2#2540611yescharacteristic
574C2#2540711yescharacteristic
575C2#2549111yescharacteristic
576C2#2549211yescharacteristic
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
39. C4×#101C4semidirect
269. #101C4⋊C4semidirect
557. C4#1930semidirect

Maximal subgroups

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

subgroupindex
C2×#19302
#208002
#206472
C2×#18032
#204412
C2×C4×#1012
#112902
#111452
#108962
#108572
#108532