Finite Group Databasebeta

← #365296

Subgroups — #365296

Order 512.

1026 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21171C211yesmaximal, characteristic
2C2×#1930C212yesmaximal
3C2×#1803C211yesmaximal, characteristic
4C2×#1801C211yesmaximal, characteristic
5#20431C211yesmaximal, characteristic
6C4×#419C211yesmaximal, characteristic
7#11303C212yesmaximal
8#11164C212yesmaximal
9#10896C212yesmaximal
10#10857C211yesmaximal, characteristic
11#10853C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#428C2212yes—
14C2×#423C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C2×#419C2211yescharacteristic
18C2×#419C2211yescharacteristic
19C22×#103C2212yes—
20C22×#101C2212yes—
21C2×#405C2212yes—
22C2×C4×M16C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1948—21no—
26#1948—21no—
27#1947—21no—
28#1947—21no—
29#1930—28no—
30#1803—21no—
31#1803—21no—
32#1803—22no—
33#1801C418yescomplemented
34#1768—22no—
35#1767—21no—
36C4×C4⋊C8—21no—
37#1763—21no—
38#1763—21no—
39C4×#103—21no—
40C4×#103—21no—
41C2×#343C2212yes—
42C2×#343C2212yes—
43C2×#343C2212yes—
44C2×#341C2212yes—
45C2×#341C2212yes—
46#1736C2211yescharacteristic
47#1736C2212yes—
48#1736C2211yescharacteristic
49C2×#340C2212yes—
50#1729C2212yes—
51C23×Q8C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2312yes—
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C23×C8C2312yes—
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×#117C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#428—24no—
75#428—24no—
76C2×#107—22no—
77C2×#106—22no—
78#423—22no—
79#423—22no—
80#423C2×C414yes—
81#423C2×C414yes—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8D412yes—
91#419—22no—
92#419—22no—
93#419—22no—
94#419D414yes—
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—24no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—42no—
103C2×#101—24no—
104C2×#101—24no—
105C2×#101C2×C418yes—
106#405—24no—
107#405—42no—
108C4×M16—22no—
109C4×M16—22no—
110C2×#97—22no—
111C2×#97—21no—
112C2×C4×C8—22no—
113C2×C4×C8—21no—
114#384—21no—
115#384—21no—
116#384—21no—
117#384—21no—
118C4×C4⋊C4D411yescharacteristic
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124#376—21no—
125C43—21no—
126#343—24no—
127#343—42no—
128#343C2×C418yes—
129#341—24no—
130#341—24no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
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—22no—
158C23×C4#4111yescharacteristic
159C23×C4#4111yescharacteristic
160C23×C4C2411yescharacteristic
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—21no—
176C2×M16—22no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—22no—
181C2×M16C22×C414yes—
182C2×M16C22×C414yes—
183C2×M16C2×D412yes—
184C2×M16#4112yes—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8C2×D412yes—
196C22×C8C2×D412yes—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205#117—21no—
206#117—21no—
207#117—21no—
208#117—21no—
209#117—21no—
210#117—21no—
211#117C2×D412yes—
212#117#4112yes—
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—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C42—21no—
242C2×C42—24no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—41no—
249C2×C42—41no—
250C2×C42—42no—
251C2×C42C22×C412yes—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42#4112yes—
257C2×C42#4112yes—
258C2×C42#4112yes—
259#107—44no—
260#106—44no—
261C4⋊C8—24no—
262C4⋊C8—24no—
263C4⋊C8—24no—
264C4⋊C8—24no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—24no—
268#103—24no—
269#103—24no—
270#103—24no—
271#103—24no—
272#103—44no—
273#103—44no—
274#101—216no—
275#97—44no—
276#97—42no—
277C4×C8—44no—
278C4×C8—42no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283C24C2×#4111yescharacteristic
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—22no—
298C2×Q8—22no—
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×Q8—41no—
310C2×Q8—41no—
311C2×Q8C4×D414yes—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
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—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—81no—
334C22×C4—81no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4#11712yes—
338C22×C4#11712yes—
339C22×C4#12111yescharacteristic
340C22×C4#12111yescharacteristic
341C22×C4#12111yescharacteristic
342C22×C4#12111yescharacteristic
343C22×C4#12612yes—
344C22×C4#12612yes—
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C42−1+411yescharacteristic
357C22×C42−1+411yescharacteristic
358C22×C42−1+411yescharacteristic
359M16—24no—
360M16—24no—
361M16—24no—
362M16—24no—
363M16—22no—
364M16—22no—
365M16—22no—
366M16—22no—
367M16—42no—
368M16—42no—
369M16—42no—
370M16—41no—
371M16—41no—
372M16—42no—
373M16—41no—
374M16—41no—
375C2×C8—24no—
376C2×C8—24no—
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—42no—
386C2×C8—42no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4#12112yes—
418C4⋊C4#12112yes—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C42—22no—
424C42—22no—
425C42—41no—
426C42—41no—
427C42—44no—
428C42—44no—
429C42—42no—
430C42—42no—
431C42—81no—
432C42—81no—
433C42C4×D412yes—
434C42C4×D412yes—
435C42C4×D412yes—
436C42C4×D412yes—
437C42#12312yes—
438C42#12312yes—
439C42#12412yes—
440C42#12412yes—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23C2×#11711yescharacteristic
446C23C2×#12111yescharacteristic
447C23C2×#12611yescharacteristic
448C23#54711yescharacteristic
449C23#55211yescharacteristic
450C23#55311yescharacteristic
451C23#55611yescharacteristic
452Q8—24no—
453Q8—24no—
454Q8—42no—
455Q8—42no—
456Q8—42no—
457Q8—42no—
458Q8—42no—
459Q8—42no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
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—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—162no—
498C2×C4—162no—
499C2×C4#38812yes—
500C2×C4#38812yes—
501C2×C4#38812yes—
502C2×C4#38812yes—
503C2×C4#47112yes—
504C2×C4#47312yes—
505C2×C4C2×C4×D411yescharacteristic
506C2×C4C2×C4×D411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C4×#4111yescharacteristic
509C2×C4#51911yescharacteristic
510C2×C4#52011yescharacteristic
511C2×C4C2×#12111yescharacteristic
512C2×C4C2×#12311yescharacteristic
513C2×C4#52511yescharacteristic
514C2×C4C2×#12411yescharacteristic
515C2×C4#52911yescharacteristic
516C2×C4#53211yescharacteristic
517C2×C4#53611yescharacteristic
518C2×C4#53711yescharacteristic
519C2×C4#54111yescharacteristic
520C2×C4#54211yescharacteristic
521C2×C4D4211yescharacteristic
522C2×C4#54711yescharacteristic
523C2×C4#54711yescharacteristic
524C2×C4#54711yescharacteristic
525C2×C4#54811yescharacteristic
526C2×C4#54811yescharacteristic
527C2×C4#55211yescharacteristic
528C2×C4#55211yescharacteristic
529C2×C4#55311yescharacteristic
530C2×C4#55311yescharacteristic
531C2×C4#55611yescharacteristic
532C2×C4#55611yescharacteristic
533C8—42no—
534C8—44no—
535C8—42no—
536C8—42no—
537C8—42no—
538C8—44no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22C2×#38811yescharacteristic
552C22#234411yescharacteristic
553C22#234711yescharacteristic
554C22#241511yescharacteristic
555C22#241611yescharacteristic
556C22#249411yescharacteristic
557C22#249611yescharacteristic
558C22C2×#47111yescharacteristic
559C22C2×#47311yescharacteristic
560C22#306811yescharacteristic
561C22#306911yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—44no—
566C4—42no—
567C4—42no—
568C4—82no—
569C4—81no—
570C4—81no—
571C4—162no—
572C4#193014yescomplemented
573C4C4×#12111yescharacteristic
574C4#232311yescharacteristic
575C4#236111yescharacteristic
576C4#236211yescharacteristic
577C4#242111yescharacteristic
578C4#242211yescharacteristic
579C4#254711yescharacteristic
580C4#256211yescharacteristic
581C2—21no—
582C2—21no—
583C2—21no—
584C2—21no—
585C2#1397411yescharacteristic
586C2C2×#193011yescharacteristic
587C2#2107611yescharacteristic
588C2#2472211yescharacteristic
589C2#2472311yescharacteristic
590C2#2477211yescharacteristic
591C2#2477311yescharacteristic
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
33. #1801C4semidirect
572. C4#1930semidirect

Maximal subgroups

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

subgroupindex
#211712
C2×#19302
C2×#18032
C2×#18012
#204312
C4×#4192
#113032
#111642
#108962
#108572
#108532