Finite Group Databasebeta

← #198723

Subgroups — #198723

Order 512.

1088 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
1#60943C211yesmaximal, characteristic
2#34215C211yesmaximal, characteristic
3#21332C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic
5C2×#1518C211yesmaximal, characteristic
6#9276C218yesmaximal
7#8026C211yesmaximal, characteristic
8#7924C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#533C2211yescharacteristic
11#3249—22no—
12#3230—21no—
13#3230C412yescomplemented
14#3209—21no—
15#3209C412yescomplemented
16#3189C2214yes—
17#3170—22no—
18#3104—22no—
19#3104C414yescomplemented
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2368—22no—
23#2368C2214yes—
24#2355—22no—
25#2355—22no—
26#2331—21no—
27#2331—21no—
28#2317—21no—
29C4×#125—22no—
30#2285C2211yescharacteristic
31C2×#382C2211yescharacteristic
32#2274—21no—
33C4×#117C2211yescharacteristic
34#1996—22no—
35#1996C2214yes—
36#1740—21no—
37#1740—21no—
38#1520C2214yes—
39#1518—22no—
40#1518—22no—
41#1518—22no—
42#1503—22no—
43C2×#328C2214yes—
44#1294C2214yes—
45#1282C2214yes—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533C2×C412yes—
59#533C2×C412yes—
60#533D412yes—
61#533D412yes—
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8D411yescharacteristic
65C2×C4⋊Q8D411yescharacteristic
66C2×#125—21no—
67C2×#125—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8C2312yes—
71C2×C22⋊Q8C2312yes—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8C2×C412yes—
75C2×C4×Q8D412yes—
76C2×#117—21no—
77C2×#117—21no—
78C2×#117—21no—
79C2×#117D411yescharacteristic
80C2×#117D411yescharacteristic
81C22×C4⋊C4—21no—
82C22×C42C2311yescharacteristic
83#500—22no—
84#494—22no—
85#487—22no—
86#487—22no—
87#484—24no—
88#484C2×C418yes—
89#477—24no—
90#477—24no—
91#467—28no—
92#462—24no—
93#462—24no—
94#451—24no—
95#451C2×C418yes—
96C4×Q16—28no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431C2×C412yes—
104#431C2×C412yes—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C22⋊C8C2312yes—
108C2×C22⋊C8C2312yes—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111#391—24no—
112#391—24no—
113#389—22no—
114#389—22no—
115#387—24no—
116#387—24no—
117#387—24no—
118#387—24no—
119#385—24no—
120#384—22no—
121#384—21no—
122#382—22no—
123#382—21no—
124#382—21no—
125#382—21no—
126#382D412yes—
127#382D412yes—
128#380—21no—
129#380C2312yes—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4—21no—
140C4×C22⋊C4—24no—
141C4×C22⋊C4D412yes—
142C4×C22⋊C4C2312yes—
143#376—21no—
144C2×#95—21no—
145C2×#95C2312yes—
146C43—21no—
147#342—24no—
148#342—24no—
149#328—28no—
150#328—28no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8C22×C412yes—
154C22×Q8C2×D412yes—
155C23×C4—22no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C2×Q16—28no—
160C22×C8—22no—
161C22×C8—22no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8C22⋊C412yes—
175C4⋊Q8C22⋊C412yes—
176C4⋊Q8C22⋊C412yes—
177C4⋊Q8C22⋊C412yes—
178#125—22no—
179#125—22no—
180#125—22no—
181#125—22no—
182#125—41no—
183#125—41no—
184#125—41no—
185#125—41no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198C22⋊Q8—41no—
199C22⋊Q8—41no—
200C22⋊Q8C22×C414yes—
201C22⋊Q8C2×D414yes—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8C22⋊C414yes—
217C4×Q8C22⋊C414yes—
218#117—22no—
219#117—22no—
220#117—22no—
221#117—22no—
222#117—22no—
223#117—22no—
224#117—22no—
225#117—41no—
226#117—41no—
227#117—41no—
228#117—41no—
229#117—42no—
230#117—42no—
231#117Q1614yescomplemented
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4C2×D412yes—
259C2×C22⋊C4C2×D412yes—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—21no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—41no—
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42#4111yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42#4111yescharacteristic
281#107—28no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—41no—
285C4⋊C8—41no—
286#103—28no—
287#103—28no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—22no—
291C22⋊C8C22×C414yes—
292C4×C8—22no—
293C4×C8—22no—
294C4×C8—41no—
295C4×C8—41no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—41no—
307#95C2×D414yes—
308C24C22×D411yescharacteristic
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—82no—
317C2×Q8—82no—
318C2×Q8—82no—
319C2×Q8—82no—
320C2×Q8C2×C22⋊C414yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—81no—
332C22×C4—81no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4C4×D411yescharacteristic
336C22×C4C4×D411yescharacteristic
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4#12011yescharacteristic
340C22×C4#12011yescharacteristic
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12111yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4C2×Q1612yes—
348C22×C4#13512yes—
349C22×C4#13712yes—
350C22×C4#13712yes—
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357Q16—88no—
358C2×C8—24no—
359C2×C8—42no—
360C2×C8—82no—
361C2×C8—82no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—84no—
390C4⋊C4—84no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4C2×C22⋊C414yes—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4C2×Q1614yes—
405C42—24no—
406C42—24no—
407C42—24no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—82no—
412C42—82no—
413C42—84no—
414C42—84no—
415C42—82no—
416C42C2×C22⋊C411yescharacteristic
417C42C2×C22⋊C411yescharacteristic
418C42C2×C22⋊C411yescharacteristic
419C42C2×C22⋊C411yescharacteristic
420C42C4×D411yescharacteristic
421C42C4×D411yescharacteristic
422C42C4×D411yescharacteristic
423C42C4×D411yescharacteristic
424C42#12111yescharacteristic
425C42#12111yescharacteristic
426C42#12111yescharacteristic
427C42#12111yescharacteristic
428C42#12311yescharacteristic
429C42#12311yescharacteristic
430C42#12311yescharacteristic
431C42#12311yescharacteristic
432C23—41no—
433C23—41no—
434C23C2×C4×D411yescharacteristic
435C23C2×#12011yescharacteristic
436C23C2×#12111yescharacteristic
437C23C22×Q1611yescharacteristic
438C23C2×#13511yescharacteristic
439C23C2×#13711yescharacteristic
440C23C2×#13711yescharacteristic
441Q8—44no—
442Q8—44no—
443Q8—82no—
444Q8—82no—
445Q8—82no—
446Q8—82no—
447Q8—82no—
448Q8—82no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
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—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—162no—
484C2×C4—162no—
485C2×C4#38611yescharacteristic
486C2×C4#38611yescharacteristic
487C2×C4#38611yescharacteristic
488C2×C4#38611yescharacteristic
489C2×C4#38611yescharacteristic
490C2×C4#38611yescharacteristic
491C2×C4#38611yescharacteristic
492C2×C4#38611yescharacteristic
493C2×C4C4×Q1612yes—
494C2×C4C4×Q1612yes—
495C2×C4#44112yes—
496C2×C4#44112yes—
497C2×C4#45112yes—
498C2×C4#45112yes—
499C2×C4#45212yes—
500C2×C4#45212yes—
501C2×C4#46212yes—
502C2×C4#46212yes—
503C2×C4#46412yes—
504C2×C4#46412yes—
505C2×C4C22×C22⋊C411yescharacteristic
506C2×C4C2×C4×D411yescharacteristic
507C2×C4C2×#12111yescharacteristic
508C2×C4C2×#12311yescharacteristic
509C8—44no—
510C8—82no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22C2×#38611yescharacteristic
522C22C2×C4×Q1611yescharacteristic
523C22C2×#44111yescharacteristic
524C22C2×#45111yescharacteristic
525C22C2×#45211yescharacteristic
526C22C2×#46211yescharacteristic
527C22C2×#46411yescharacteristic
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—44no—
537C4—44no—
538C4—44no—
539C4—82no—
540C4—84no—
541C4—82no—
542C4—82no—
543C4—162no—
544C4#187812yes—
545C4#187812yes—
546C4#187812yes—
547C4#187812yes—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—21no—
554C2C2×#187811yescharacteristic
555C2#2100011yescharacteristic
556C2#2101411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3230C4semidirect
15. #3209C4semidirect
19. #3104C4semidirect
231. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#609432
#342152
#213322
#175752
C2×#15182
#92762
#80262
#79242