Finite Group Databasebeta

← #416280

Subgroups — #416280

Order 512.

966 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1950C211yesmaximal, characteristic
2#21145C211yesmaximal, characteristic
3#20764C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic
6C2×C4×#101C211yesmaximal, characteristic
7#11303C212yesmaximal
8#11242C212yesmaximal
9#11128C212yesmaximal
10#11062C212yesmaximal
11#10857C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#108C2212yes—
14C2×#424C2212yes—
15C2×#423C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C2×#419C2212yes—
18C22×#103C2212yes—
19C22×#101C2211yescharacteristic
20C22×#101C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×#384C2211yescharacteristic
25#1950—24no—
26#1940—22no—
27#1939—22no—
28#1836—22no—
29#1834—21no—
30#1834—21no—
31#1805—22no—
32#1803—22no—
33#1803—22no—
34#1803—21no—
35#1803—21no—
36C4×#101—24no—
37C2×#343C2211yescharacteristic
38C2×#343C2212yes—
39C2×#343C2211yescharacteristic
40C2×#343C2212yes—
41C2×#341C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44#1736C2212yes—
45C2×#340C2212yes—
46#1729C2212yes—
47C23×Q8C2311yescharacteristic
48C22×M16C2312yes—
49C22×M16C2311yescharacteristic
50C22×M16C2311yescharacteristic
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C23×C8C2312yes—
54C2×C4×Q8—22no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8Q814yes—
62C2×#117C2312yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70C2×#108—24no—
71C2×#108—42no—
72C2×#108—42no—
73#424—24no—
74#424—42no—
75#423—22no—
76#423—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8D412yes—
83C2×C4⋊C8D412yes—
84#419—24no—
85#419—42no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—42no—
91C2×#101—22no—
92C2×#101—22no—
93C2×#101—22no—
94C2×#101—22no—
95C2×#101—22no—
96C2×#101—22no—
97C4×M16—22no—
98C4×M16—22no—
99C4×M16—22no—
100C4×M16D414yes—
101C2×#97—21no—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105#384—21no—
106#384—21no—
107#384—21no—
108#384—21no—
109#384D411yescharacteristic
110#384D411yescharacteristic
111#382D412yes—
112#380—21no—
113#380—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116#343—22no—
117#343—22no—
118#343—24no—
119#343—42no—
120#341—24no—
121#340—24no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#336—24no—
129C22×Q8—21no—
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×Q8C2×Q812yes—
141C22×Q8#4112yes—
142C22×Q8#4112yes—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4#4111yescharacteristic
147C23×C4#4111yescharacteristic
148C23×C4C2411yescharacteristic
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—21no—
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—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16C2×D412yes—
176C2×M16C2×D412yes—
177C2×M16#4112yes—
178C2×M16#4112yes—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8C2×D414yes—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—42no—
199#117—22no—
200#117—22no—
201#117—22no—
202#117—22no—
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—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×Q812yes—
231C2×C22⋊C4—22no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42—42no—
237C2×C42C2×D412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×Q812yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245#108—48no—
246C4⋊C8—24no—
247C4⋊C8—24no—
248C4⋊C8—48no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251#103—44no—
252#103—44no—
253#103—48no—
254#101—28no—
255#101—28no—
256#97—24no—
257#97—42no—
258C4×C8—24no—
259C4×C8—42no—
260#95—22no—
261#95—22no—
262C24C2×#4111yescharacteristic
263C2×Q8—22no—
264C2×Q8—22no—
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—41no—
279C2×Q8—41no—
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#12514yes—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4C22⋊Q812yes—
319C22×C4C22⋊Q812yes—
320C22×C4#12612yes—
321C22×C4#12612yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×Q811yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4H(5,𝔽2)11yescharacteristic
333C22×C4H(5,𝔽2)11yescharacteristic
334C22×C42−1+411yescharacteristic
335C22×C42−1+411yescharacteristic
336M16—22no—
337M16—22no—
338M16—22no—
339M16—22no—
340M16—22no—
341M16—22no—
342M16—22no—
343M16—22no—
344M16—41no—
345M16—41no—
346M16—41no—
347M16—41no—
348M16—41no—
349M16—41no—
350M16—41no—
351M16—41no—
352M16—42no—
353M16—42no—
354M16—42no—
355M16—42no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—22no—
361C2×C8—24no—
362C2×C8—22no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—81no—
395C4⋊C4—81no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C42—42no—
403C42—42no—
404C42—44no—
405C42—44no—
406C42#12112yes—
407C42#12112yes—
408C42#12112yes—
409C42#12112yes—
410C42C22⋊Q812yes—
411C42C22⋊Q812yes—
412C42C22⋊Q812yes—
413C42C22⋊Q812yes—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23C2×#12111yescharacteristic
419C23C2×C22⋊Q811yescharacteristic
420C23C2×#12611yescharacteristic
421C23#52911yescharacteristic
422C23#52911yescharacteristic
423C23#54211yescharacteristic
424C23#54211yescharacteristic
425Q8—24no—
426Q8—24no—
427Q8—42no—
428Q8—42no—
429Q8—42no—
430Q8—42no—
431Q8—42no—
432Q8—42no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—42no—
443C2×C4—42no—
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—82no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—162no—
465C2×C4—162no—
466C2×C4#39612yes—
467C2×C4#39612yes—
468C2×C4#39612yes—
469C2×C4#39612yes—
470C2×C4#47112yes—
471C2×C4#47312yes—
472C2×C4C2×#12111yescharacteristic
473C2×C4C2×#12111yescharacteristic
474C2×C4C2×C22⋊Q811yescharacteristic
475C2×C4C2×C22⋊Q811yescharacteristic
476C2×C4#52511yescharacteristic
477C2×C4#52511yescharacteristic
478C2×C4C2×#12511yescharacteristic
479C2×C4#53311yescharacteristic
480C2×C4#53511yescharacteristic
481C2×C4#53611yescharacteristic
482C2×C4#54411yescharacteristic
483C2×C4#54411yescharacteristic
484C2×C4#54611yescharacteristic
485C2×C4#54611yescharacteristic
486C2×C4#54611yescharacteristic
487C2×C4#54711yescharacteristic
488C2×C4#54811yescharacteristic
489C2×C4#54811yescharacteristic
490C2×C4#54811yescharacteristic
491C2×C4#55011yescharacteristic
492C2×C4#55211yescharacteristic
493C2×C4#55211yescharacteristic
494C2×C4#55211yescharacteristic
495C2×C4#55311yescharacteristic
496C2×C4#55311yescharacteristic
497C2×C4#55311yescharacteristic
498C2×C4#55511yescharacteristic
499C2×C4#55611yescharacteristic
500C8—42no—
501C8—42no—
502C8—44no—
503C8—42no—
504C8—42no—
505C8—82no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22C2×#39611yescharacteristic
519C22#242611yescharacteristic
520C22#242711yescharacteristic
521C22#249411yescharacteristic
522C22#249511yescharacteristic
523C22#262811yescharacteristic
524C22#285111yescharacteristic
525C22C2×#47111yescharacteristic
526C22C2×#47311yescharacteristic
527C22#306811yescharacteristic
528C22#306911yescharacteristic
529C4—22no—
530C4—22no—
531C4—22no—
532C4—44no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4—162no—
537C4#206414yes—
538C4#241411yescharacteristic
539C4#241511yescharacteristic
540C4#243911yescharacteristic
541C4#244111yescharacteristic
542C4#244211yescharacteristic
543C4#244211yescharacteristic
544C4#253511yescharacteristic
545C4#253611yescharacteristic
546C2—21no—
547C2—21no—
548C2—21no—
549C2—21no—
550C2#1522011yescharacteristic
551C2C2×#206411yescharacteristic
552C2#2360411yescharacteristic
553C2#2531411yescharacteristic
554C2#2531511yescharacteristic
555C2#2536011yescharacteristic
556C2#2536111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#19502
#211452
#207642
#206472
C2×#18032
C2×C4×#1012
#113032
#112422
#111282
#110622
#108572