Finite Group Databasebeta

← #366495

Subgroups — #366495

Order 512.

1012 conjugacy classes of subgroups.

The subgroup lattice has 565 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#20438C211yesmaximal, characteristic
6C2×#1763C211yesmaximal, characteristic
7#11290C212yesmaximal
8#11164C212yesmaximal
9#10847C211yesmaximal, characteristic
10#10845C211yesmaximal, 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×#97C2211yescharacteristic
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#1771—22no—
34#1771—22no—
35#1763—22no—
36#1763—22no—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#343C2212yes—
40C2×#341C2212yes—
41C2×#341C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46#1729C2212yes—
47C23×Q8C2311yescharacteristic
48C22×M16C2312yes—
49C22×M16C2312yes—
50C22×M16C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4×Q8—22no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8C2×C414yes—
61C2×#117C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#428—24no—
70#428—42no—
71C2×#107—22no—
72C2×#106—22no—
73#423—22no—
74#423—22no—
75#423—22no—
76#423—22no—
77#423C2×C414yes—
78#423C2×C414yes—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85#419—24no—
86#419—42no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103D414yes—
95C2×#101—24no—
96C2×#101—24no—
97C2×#101C2×C418yes—
98#405—24no—
99#405—24no—
100C2×#97—22no—
101C2×#97—22no—
102C2×#97—22no—
103C2×#97D412yes—
104C2×#97D412yes—
105C2×C4×C8—22no—
106#384D411yescharacteristic
107#384D411yescharacteristic
108#382D412yes—
109#380—21no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C2×#95—21no—
118#343—24no—
119#343—24no—
120#343C2×C418yes—
121#341—24no—
122#341—42no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#336—24no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
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×Q8C22×C412yes—
142C22×Q8C2×D412yes—
143C22×Q8#4112yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4#4111yescharacteristic
149C23×C4#4111yescharacteristic
150C23×C4C2411yescharacteristic
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16C22×C414yes—
168C2×M16C22×C414yes—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C22×C8C2×D412yes—
185C22×C8#4112yes—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—24no—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×Q8—42no—
194#117—22no—
195#117—22no—
196#117—22no—
197#117—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
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—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4#4112yes—
226C2×C22⋊C4—22no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—41no—
230C2×C42—41no—
231C2×C42—42no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42—82no—
235C2×C42C22×C412yes—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4112yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243#107—44no—
244#106—44no—
245C4⋊C8—24no—
246C4⋊C8—24no—
247C4⋊C8—24no—
248C4⋊C8—24no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251#103—28no—
252#103—28no—
253#103—44no—
254#103—44no—
255#101—216no—
256#97—24no—
257#97—24no—
258#97—44no—
259C4×C8—44no—
260#95—22no—
261#95—22no—
262#95—41no—
263#95—41no—
264C24C2×#4111yescharacteristic
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
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—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8C4×D414yes—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
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—41no—
315C22×C4—81no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—81no—
320C22×C4—81no—
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×C4C22×D411yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C42−1+411yescharacteristic
341C22×C42−1+411yescharacteristic
342C22×C42−1+411yescharacteristic
343M16—24no—
344M16—24no—
345M16—24no—
346M16—24no—
347M16—42no—
348M16—42no—
349M16—42no—
350M16—42no—
351M16—42no—
352M16—42no—
353M16—42no—
354M16—42no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8#12114yes—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—81no—
397C4⋊C4—81no—
398C4⋊C4—81no—
399C4⋊C4—81no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C42—42no—
403C42—42no—
404C42—44no—
405C42—44no—
406C42—82no—
407C42—82no—
408C42C4×D412yes—
409C42C4×D412yes—
410C42C4×D412yes—
411C42C4×D412yes—
412C42#12312yes—
413C42#12312yes—
414C42#12412yes—
415C42#12412yes—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23C2×#11711yescharacteristic
421C23C2×#12111yescharacteristic
422C23C2×#12611yescharacteristic
423C23#54711yescharacteristic
424C23#55211yescharacteristic
425C23#55211yescharacteristic
426C23#55511yescharacteristic
427Q8—24no—
428Q8—24no—
429Q8—42no—
430Q8—42no—
431Q8—42no—
432Q8—42no—
433Q8—42no—
434Q8—42no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—24no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
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—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4#38812yes—
476C2×C4#38812yes—
477C2×C4#38812yes—
478C2×C4#38812yes—
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×C4×D411yescharacteristic
481C2×C4C2×C4×D411yescharacteristic
482C2×C4C4×#4111yescharacteristic
483C2×C4#51911yescharacteristic
484C2×C4#52011yescharacteristic
485C2×C4C2×#12111yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C2×C4#52511yescharacteristic
488C2×C4C2×#12411yescharacteristic
489C2×C4#52911yescharacteristic
490C2×C4#53211yescharacteristic
491C2×C4#53611yescharacteristic
492C2×C4#53711yescharacteristic
493C2×C4#54111yescharacteristic
494C2×C4#54211yescharacteristic
495C2×C4D4211yescharacteristic
496C2×C4#54711yescharacteristic
497C2×C4#54711yescharacteristic
498C2×C4#54711yescharacteristic
499C2×C4#54811yescharacteristic
500C2×C4#54811yescharacteristic
501C2×C4#55211yescharacteristic
502C2×C4#55211yescharacteristic
503C2×C4#55211yescharacteristic
504C2×C4#55311yescharacteristic
505C2×C4#55511yescharacteristic
506C2×C4#55611yescharacteristic
507C8—24no—
508C8—44no—
509C8—42no—
510C8—44no—
511C8—82no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22C2×#38811yescharacteristic
525C22#234411yescharacteristic
526C22#234611yescharacteristic
527C22#249511yescharacteristic
528C22#249611yescharacteristic
529C22#253611yescharacteristic
530C22#253711yescharacteristic
531C22#306811yescharacteristic
532C22#306911yescharacteristic
533C22#306911yescharacteristic
534C22#306911yescharacteristic
535C4—22no—
536C4—22no—
537C4—22no—
538C4—22no—
539C4—44no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4C4×#12111yescharacteristic
546C4#232311yescharacteristic
547C4#236111yescharacteristic
548C4#236211yescharacteristic
549C4#248511yescharacteristic
550C4#248611yescharacteristic
551C4#248911yescharacteristic
552C4#249111yescharacteristic
553C2—21no—
554C2—21no—
555C2—21no—
556C2—21no—
557C2#1399411yescharacteristic
558C2#2107611yescharacteristic
559C2#2107711yescharacteristic
560C2#2822111yescharacteristic
561C2#2822211yescharacteristic
562C2#2824211yescharacteristic
563C2#2824411yescharacteristic
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

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
#204382
C2×#17632
#112902
#111642
#108472
#108452
#108272