Finite Group Databasebeta

← #195627

Subgroups — #195627

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3170C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic
3#21885C211yesmaximal, characteristic
4#21312C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6#9066C212yesmaximal, complemented
7#9065C212yesmaximal, complemented
8#9060C212yesmaximal, complemented
9#9058C212yesmaximal, complemented
10#7897C211yesmaximal, characteristic
11C2×#1279C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#533C2211yescharacteristic
14#3170—24no—
15#3170C418yescomplemented
16C2×#500C2211yescharacteristic
17C2×#494C2211yescharacteristic
18C2×#487C2211yescharacteristic
19C2×#487C2211yescharacteristic
20C2×#467—22no—
21C2×C4×Q16—22no—
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2370—22no—
26#2369C2212yes—
27#2365C2212yes—
28#2355—21no—
29#2355—22no—
30Q8×C4⋊C4—21no—
31#2329—22no—
32#2317—21no—
33C4×C22⋊Q8—22no—
34#2285C2211yescharacteristic
35#2274—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1989—22no—
38#1988C2212yes—
39#1987C2212yes—
40#1987C2212yes—
41#1981C2212yes—
42#1892—22no—
43#1803—21no—
44#1803—21no—
45#1736—21no—
46#1736—21no—
47#1550—24no—
48C2×#333C2212yes—
49C2×#332C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C22×Q16—22no—
55C22×M16—21no—
56C23×C8—21no—
57C23×C8—21no—
58#533—22no—
59#533—22no—
60#533—22no—
61#533—22no—
62#533—22no—
63#533C2×C414yes—
64#533C2×C414yes—
65#533D414yes—
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×#125—21no—
71C2×#125—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8D412yes—
76C2×C4×Q8—22no—
77C2×C4×Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8D412yes—
81C2×#117—21no—
82C2×#117—21no—
83C2×#117—21no—
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4D411yescharacteristic
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#500—22no—
89#500C2×C414yescomplemented
90#494—22no—
91#494C2×C414yescomplemented
92#487—22no—
93#487—22no—
94#487C2×C414yescomplemented
95#487C2×C414yescomplemented
96#467—216no—
97C4×Q16—216no—
98C2×#107—22no—
99#423—22no—
100#423—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2312yes—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—22no—
110C2×#103—22no—
111C2×C4×C8—22no—
112C2×C4×C8—21no—
113C2×C4×C8C2×C414yes—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#391—24no—
117#391—22no—
118#389—24no—
119#389—22no—
120#389—22no—
121#389—22no—
122#387—24no—
123#387—24no—
124#387—24no—
125#385—24no—
126#384—22no—
127#384D411yescharacteristic
128#384D411yescharacteristic
129#382—22no—
130#382—21no—
131#380—21no—
132#380D412yes—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4D412yes—
143C4×C4⋊C4C2312yes—
144C4×C4⋊C4C2312yes—
145C4×C22⋊C4—24no—
146C2×#95—21no—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153#333—24no—
154#332—24no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8C2×D412yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C2×Q16—28no—
165C2×Q16—44no—
166C2×Q16—44no—
167C2×M16—42no—
168C2×M16—42no—
169C2×M16—42no—
170C22×C8—21no—
171C22×C8—24no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8—42no—
178C22×C8—81no—
179C22×C8—81no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8C22×C412yes—
191C4⋊Q8C22×C412yes—
192C4⋊Q8C22×C412yes—
193C4⋊Q8C22×C412yes—
194C4⋊Q8C2×D412yes—
195C4⋊Q8C2×D412yes—
196#125—24no—
197#125—24no—
198#125—24no—
199#125—42no—
200C22⋊Q8—24no—
201C22⋊Q8—24no—
202C22⋊Q8—24no—
203C22⋊Q8—24no—
204C22⋊Q8—24no—
205C22⋊Q8—42no—
206C22⋊Q8—42no—
207C22⋊Q8C22⋊C418yes—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—42no—
214C4×Q8—42no—
215C4×Q8—82no—
216C4×Q8—82no—
217C4×Q8C22⋊C418yes—
218#117—24no—
219#117—24no—
220#117—24no—
221#117—42no—
222#117—42no—
223#117—42no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4Q1614yescomplemented
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—41no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—41no—
262C2×C42—81no—
263C2×C42—81no—
264C2×C42C22×C412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42#4112yes—
270C2×C42#4112yes—
271C2×C42C2411yescharacteristic
272#107—28no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—42no—
278C4⋊C8—42no—
279#103—28no—
280#103—28no—
281#103—82no—
282#103—82no—
283C4×C8—24no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8C22×C412yes—
287C4×C8C22×C412yes—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296C24#12011yescharacteristic
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8C2×C22⋊C414yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—81no—
334C22×C4—81no—
335C22×C4—81no—
336C22×C4—81no—
337C22×C4C2×C22⋊C412yes—
338C22×C4C4×D412yes—
339C22×C4C4×D412yes—
340C22×C4#12011yescharacteristic
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12112yes—
344C22×C4#12112yes—
345C22×C4#12312yes—
346C22×C4C2×Q1612yes—
347C22×C4#13512yes—
348C22×C4#13712yes—
349C22×C4#13712yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353Q16—416no—
354M16—84no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—48no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—84no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—28no—
378C4⋊C4—28no—
379C4⋊C4—24no—
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—22no—
389C4⋊C4—22no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—81no—
400C4⋊C4—81no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4C2×C22⋊C414yes—
404C4⋊C4C2×Q1614yes—
405C22⋊C4—28no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C42—28no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—42no—
413C42—44no—
414C42—82no—
415C42—84no—
416C42—84no—
417C42C2×C22⋊C412yes—
418C42C4×D412yes—
419C42C4×D412yes—
420C42#12112yes—
421C42#12112yes—
422C42#12312yes—
423C42C23×C411yescharacteristic
424C42C22×D411yescharacteristic
425C42C2×#4111yescharacteristic
426C42C2×#4111yescharacteristic
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23#38612yes—
432C23#45112yes—
433C23#45212yes—
434C23C2×#12011yescharacteristic
435Q8—44no—
436Q8—44no—
437Q8—42no—
438Q8—42no—
439Q8—42no—
440Q8—42no—
441Q8—42no—
442Q8—42no—
443Q8—82no—
444Q8—82no—
445Q8—81no—
446Q8—81no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—42no—
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—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4#38612yes—
486C2×C4#38612yes—
487C2×C4#38612yes—
488C2×C4C4×Q1614yes—
489C2×C4#44114yes—
490C2×C4#45112yes—
491C2×C4#45212yes—
492C2×C4#46214yes—
493C2×C4#46414yes—
494C2×C4C22×C22⋊C411yescharacteristic
495C2×C4C2×C4×D411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×#12111yescharacteristic
498C2×C4C2×#12111yescharacteristic
499C2×C4C2×#12311yescharacteristic
500C2×C4C22×Q1611yescharacteristic
501C2×C4C2×#13511yescharacteristic
502C2×C4C2×#13711yescharacteristic
503C2×C4C2×#13711yescharacteristic
504C2×C4#57611yescharacteristic
505C2×C4#57711yescharacteristic
506C2×C4#57711yescharacteristic
507C2×C4#57811yescharacteristic
508C8—24no—
509C8—42no—
510C8—82no—
511C8—82no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22#187814yes—
519C22C2×#38611yescharacteristic
520C22C2×#45111yescharacteristic
521C22C2×#45211yescharacteristic
522C22#301211yescharacteristic
523C22#301311yescharacteristic
524C22#301811yescharacteristic
525C22#301811yescharacteristic
526C4—22no—
527C4—21no—
528C4—21no—
529C4—22no—
530C4—44no—
531C4—44no—
532C4—44no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4#187814yes—
539C4C2×C4×Q1611yescharacteristic
540C4C2×#44111yescharacteristic
541C4#295111yescharacteristic
542C4#295211yescharacteristic
543C4C2×#46211yescharacteristic
544C4C2×#46411yescharacteristic
545C4#304511yescharacteristic
546C4#304611yescharacteristic
547C2—24no—
548C2#1866511yescharacteristic
549C2#1866911yescharacteristic
550C2#1867711yescharacteristic
551C2#1868111yescharacteristic
552C2C2×#187811yescharacteristic
553C2#2093711yescharacteristic
554C2#2095211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. #9066C2semidirect
7. #9065C2semidirect
8. #9060C2semidirect
9. #9058C2semidirect
15. #3170C4semidirect
89. #500C2×C4semidirect
91. #494C2×C4semidirect
94. #487C2×C4semidirect
95. #487C2×C4semidirect
246. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#31702
#342132
#218852
#213122
C2×#15502
#90662
#90652
#90602
#90582
#78972
C2×#12792