Finite Group Databasebeta

← #366493

Subgroups — #366493

Order 512.

1026 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21170C211yesmaximal, characteristic
2C2×#1930C212yesmaximal
3#20647C211yesmaximal, characteristic
4C2×#1801C211yesmaximal, characteristic
5#20437C211yesmaximal, characteristic
6C2×C4×#103C211yesmaximal, characteristic
7#11290C212yesmaximal
8#11164C212yesmaximal
9#10847C211yesmaximal, characteristic
10#10844C211yesmaximal, characteristic
11#10827C212yesmaximal
12C22×C4×Q8C2211yescharacteristic
13C2×#428C2212yes—
14C2×#423C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×#423C2211yescharacteristic
17C2×#419C2212yes—
18C22×#103C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#101C2212yes—
21C2×#405C2212yes—
22C22×C4×C8C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1949—22no—
26#1948—21no—
27#1947—21no—
28#1930—28no—
29#1805—22no—
30#1803—21no—
31#1803—21no—
32#1801C418yescomplemented
33#1770—21no—
34#1770—21no—
35C8×C4⋊C4—21no—
36C8×C4⋊C4—21no—
37C4×#103—21no—
38C4×#103—21no—
39C4×#103—21no—
40C4×#103—21no—
41C2×#343C2212yes—
42C2×#343C2212yes—
43C2×#343C2212yes—
44C2×#341C2212yes—
45C2×#341C2212yes—
46#1736C2212yes—
47#1736C2212yes—
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50#1729C2212yes—
51C23×Q8C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2312yes—
54C22×M16C2312yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C2×C4×Q8—22no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8C2×C414yes—
65C2×#117C2312yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#428—24no—
74#428—42no—
75C2×#107—22no—
76C2×#106—22no—
77#423—22no—
78#423—22no—
79#423—22no—
80#423—22no—
81#423C2×C414yes—
82#423C2×C414yes—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89#419—24no—
90#419—42no—
91C2×#103—22no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103D414yes—
101C2×#101—24no—
102C2×#101—24no—
103C2×#101C2×C418yes—
104#405—24no—
105#405—24no—
106C2×#97—22no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8D411yescharacteristic
111C2×C4×C8D411yescharacteristic
112C2×C4×C8D411yescharacteristic
113C2×C4×C8D411yescharacteristic
114#384D411yescharacteristic
115#384D411yescharacteristic
116#382D412yes—
117#380—21no—
118#380—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C2×#95—21no—
126#343—24no—
127#343—24no—
128#343C2×C418yes—
129#341—24no—
130#341—42no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#336—22no—
140#336—22no—
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×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8C22×C412yes—
153C22×Q8C2×D412yes—
154C22×Q8#4112yes—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4#4111yescharacteristic
160C23×C4#4111yescharacteristic
161C23×C4C2411yescharacteristic
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16C22×C414yes—
179C2×M16C22×C414yes—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D412yes—
198C22×C8#4112yes—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×Q8—24no—
202C4×Q8—24no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×Q8—42no—
207#117—22no—
208#117—22no—
209#117—22no—
210#117—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4#4112yes—
239C2×C22⋊C4—22no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—41no—
243C2×C42—41no—
244C2×C42—42no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42—82no—
248C2×C42C22×C412yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42#4112yes—
254C2×C42#4112yes—
255C2×C42#4112yes—
256#107—44no—
257#106—44no—
258C4⋊C8—24no—
259C4⋊C8—24no—
260C4⋊C8—24no—
261C4⋊C8—24no—
262C4⋊C8—42no—
263C4⋊C8—42no—
264#103—24no—
265#103—24no—
266#103—24no—
267#103—24no—
268#103—44no—
269#103—44no—
270#101—216no—
271#97—44no—
272C4×C8—22no—
273C4×C8—22no—
274C4×C8—22no—
275C4×C8—22no—
276C4×C8—44no—
277#95—22no—
278#95—22no—
279#95—41no—
280#95—41no—
281C24C2×#4111yescharacteristic
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—22no—
295C2×Q8—22no—
296C2×Q8—22no—
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×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8C4×D414yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—81no—
335C22×C4—81no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4#11712yes—
339C22×C4#11712yes—
340C22×C4#12111yescharacteristic
341C22×C4#12111yescharacteristic
342C22×C4#12111yescharacteristic
343C22×C4#12111yescharacteristic
344C22×C4#12612yes—
345C22×C4#12612yes—
346C22×C4C23×C411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4H(5,𝔽2)11yescharacteristic
357C22×C42−1+411yescharacteristic
358C22×C42−1+411yescharacteristic
359C22×C42−1+411yescharacteristic
360M16—24no—
361M16—24no—
362M16—24no—
363M16—24no—
364M16—42no—
365M16—42no—
366M16—42no—
367M16—42no—
368M16—42no—
369M16—42no—
370M16—42no—
371M16—42no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8#12112yes—
400C2×C8#12112yes—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C42—42no—
426C42—42no—
427C42—44no—
428C42—44no—
429C42—82no—
430C42—82no—
431C42C4×D412yes—
432C42C4×D412yes—
433C42C4×D412yes—
434C42C4×D412yes—
435C42#12312yes—
436C42#12312yes—
437C42#12412yes—
438C42#12412yes—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23C2×#11711yescharacteristic
444C23C2×#12111yescharacteristic
445C23C2×#12611yescharacteristic
446C23#54711yescharacteristic
447C23#55211yescharacteristic
448C23#55211yescharacteristic
449C23#55511yescharacteristic
450Q8—24no—
451Q8—24no—
452Q8—42no—
453Q8—42no—
454Q8—42no—
455Q8—42no—
456Q8—42no—
457Q8—42no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
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—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4#38812yes—
497C2×C4#38812yes—
498C2×C4#38812yes—
499C2×C4#38812yes—
500C2×C4#47112yes—
501C2×C4#47312yes—
502C2×C4C2×C4×D411yescharacteristic
503C2×C4C2×C4×D411yescharacteristic
504C2×C4C2×C4×D411yescharacteristic
505C2×C4C4×#4111yescharacteristic
506C2×C4#51911yescharacteristic
507C2×C4#52011yescharacteristic
508C2×C4C2×#12111yescharacteristic
509C2×C4C2×#12311yescharacteristic
510C2×C4#52511yescharacteristic
511C2×C4C2×#12411yescharacteristic
512C2×C4#52911yescharacteristic
513C2×C4#53211yescharacteristic
514C2×C4#53611yescharacteristic
515C2×C4#53711yescharacteristic
516C2×C4#54111yescharacteristic
517C2×C4#54211yescharacteristic
518C2×C4D4211yescharacteristic
519C2×C4#54711yescharacteristic
520C2×C4#54711yescharacteristic
521C2×C4#54711yescharacteristic
522C2×C4#54811yescharacteristic
523C2×C4#54811yescharacteristic
524C2×C4#55211yescharacteristic
525C2×C4#55211yescharacteristic
526C2×C4#55211yescharacteristic
527C2×C4#55311yescharacteristic
528C2×C4#55511yescharacteristic
529C2×C4#55611yescharacteristic
530C8—22no—
531C8—22no—
532C8—44no—
533C8—41no—
534C8—41no—
535C8—44no—
536C8—82no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22C2×#38811yescharacteristic
550C22#234411yescharacteristic
551C22#234611yescharacteristic
552C22#249511yescharacteristic
553C22#249611yescharacteristic
554C22#253611yescharacteristic
555C22#253711yescharacteristic
556C22C2×#47111yescharacteristic
557C22C2×#47311yescharacteristic
558C22#306711yescharacteristic
559C22#306711yescharacteristic
560C4—22no—
561C4—22no—
562C4—22no—
563C4—44no—
564C4—82no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4#193014yescomplemented
570C4C4×#12111yescharacteristic
571C4#232311yescharacteristic
572C4#236111yescharacteristic
573C4#236211yescharacteristic
574C4#248511yescharacteristic
575C4#248611yescharacteristic
576C4#248911yescharacteristic
577C4#249111yescharacteristic
578C2—21no—
579C2—21no—
580C2—21no—
581C2—21no—
582C2#1399411yescharacteristic
583C2C2×#193011yescharacteristic
584C2#2107511yescharacteristic
585C2#2821811yescharacteristic
586C2#2821911yescharacteristic
587C2#2823911yescharacteristic
588C2#2824111yescharacteristic
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
32. #1801C4semidirect
569. C4#1930semidirect

Maximal subgroups

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

subgroupindex
#211702
C2×#19302
#206472
C2×#18012
#204372
C2×C4×#1032
#112902
#111642
#108472
#108442
#108272