Finite Group Databasebeta

← #482119

Subgroups — #482119

Order 512.

924 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35334C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#11902C214yesmaximal, complemented
5#11897C214yesmaximal, complemented
6#11718C212yesmaximal, complemented
7#11717C212yesmaximal, complemented
8C2×#533C411yescharacteristic, complemented
9C2×C4×#41C2211yescharacteristic
10C2×#431—21no—
11C2×#410—21no—
12#2851—21no—
13#2851—21no—
14#2628C411yescharacteristic, complemented
15#2628C411yescharacteristic, complemented
16#2628C2211yescharacteristic
17#2628C2211yescharacteristic
18#2581—21no—
19#2581C412yescomplemented
20#2580—21no—
21#2580C2212yes—
22#2564—21no—
23#2564—21no—
24#2564C412yescomplemented
25#2564C2212yes—
26#2316—21no—
27#2311—21no—
28#2285—21no—
29C2×#382C411yescharacteristic, complemented
30#1799—21no—
31#1799—21no—
32#1799C2212yes—
33#1799C2212yes—
34#1740C2212yes—
35#1461—21no—
36#1460C2212yes—
37#1459—21no—
38#1459C2212yes—
39#1458—21no—
40#1458C2212yes—
41#1352C2218yes—
42#1351C2218yes—
43C22×#41D411yescharacteristic
44#533—41no—
45#533—41no—
46#533—41no—
47#533—41no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#125—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8C2×C412yes—
53C4×#41—41no—
54C4×#41—41no—
55C4×#41—41no—
56C4×#41—41no—
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C2×C4×D4C2311yescharacteristic
67C2×#117—21no—
68C2×#117—22no—
69C2×#117C2×C411yescharacteristic
70C2×#117D411yescharacteristic
71C2×#117C2311yescharacteristic
72C22×C4⋊C4—21no—
73C22×C42C2×C411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76#431—22no—
77#431—22no—
78#431—41no—
79#431—41no—
80C2×C4⋊C8—41no—
81#410—44no—
82C2×#99—22no—
83C2×#99—22no—
84C2×C22⋊C8—22no—
85C2×C4×C8—41no—
86#400—21no—
87#400—21no—
88#400C2×C412yes—
89#400C2×C412yes—
90#399—21no—
91#399—21no—
92#399C2×C412yes—
93#399C2312yes—
94#395—21no—
95#395—21no—
96#395C2×C412yes—
97#395C2×C412yes—
98#393—21no—
99#393—21no—
100#393C2×C412yes—
101#393C2312yes—
102#391—22no—
103#390—22no—
104#389—22no—
105#389—22no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#386—22no—
115#386—22no—
116#385—22no—
117#385—22no—
118#385—22no—
119#385—22no—
120#384—41no—
121#382—21no—
122#382—21no—
123#382—41no—
124#382C2×C412yes—
125#382C2×C411yescharacteristic
126#382C2×C411yescharacteristic
127#380—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380—22no—
132#380—22no—
133#380—22no—
134#380—22no—
135C4×C4⋊C4—41no—
136C4×C4⋊C4—41no—
137C2×#95—21no—
138C2×#95C2×C412yes—
139#342—22no—
140#342C2314yes—
141#324—22no—
142#324—22no—
143#323—22no—
144#323C2314yes—
145C2×#41—41no—
146C2×#41—41no—
147C2×#41—42no—
148C2×#41SD1614yes—
149C2×#41Q1614yes—
150C22×Q8—21no—
151C22×Q8C22⋊C411yescharacteristic
152C22×Q8C22×C412yes—
153C22×D4C22⋊C411yescharacteristic
154C22×D4C2×D411yescharacteristic
155C22×D4C2×D411yescharacteristic
156C23×C4—22no—
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C22×C8—42no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171#125—41no—
172#125—41no—
173#125—41no—
174#125—41no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C4×Q8—41no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199#117—41no—
200#117—41no—
201#117—41no—
202#117—41no—
203#117—41no—
204#117—41no—
205#117—42no—
206#117—42no—
207#117—44no—
208#117—44no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4C22⋊C411yescharacteristic
239C2×C22⋊C4C22⋊C411yescharacteristic
240C2×C22⋊C4C22×C411yescharacteristic
241C2×C22⋊C4C2×D411yescharacteristic
242C2×C22⋊C4C2×D411yescharacteristic
243C2×C22⋊C4C2411yescharacteristic
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—42no—
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C4⋊C8—42no—
269C4⋊C8—42no—
270#99—44no—
271#99—44no—
272C22⋊C8—24no—
273C22⋊C8—42no—
274C4×C8—42no—
275C4×C8—42no—
276#95—22no—
277#95—21no—
278#95—21no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—42no—
289#95—42no—
290#95C22×C412yes—
291#95C22×C412yes—
292C24C2×C22⋊C411yescharacteristic
293C24C2×C22⋊C411yescharacteristic
294C24C22×D411yescharacteristic
295#41—44no—
296#41—44no—
297#41—44no—
298#41—44no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—82no—
304C2×Q8—82no—
305C2×Q8#10314yes—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4#10314yes—
313C2×D4C2×SD1612yes—
314C2×D4C2×SD1612yes—
315C2×D4C2×Q1612yes—
316C2×D4C2×Q1612yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4—81no—
349C22×C4—81no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4#10314yes—
353C22×C4#10314yes—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×SD1612yes—
365C22×C4C2×Q1612yes—
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C2×C8—44no—
369C2×C8—84no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—84no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—44no—
428C42—84no—
429C42#9911yescharacteristic
430C42#9911yescharacteristic
431C42#9911yescharacteristic
432C42#9911yescharacteristic
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23C2×#10312yes—
446C23C2×#10312yes—
447C23C22×C22⋊C411yescharacteristic
448C23C22×SD1611yescharacteristic
449C23C22×Q1611yescharacteristic
450Q8—44no—
451Q8—84no—
452Q8—84no—
453D4—44no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—84no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4—164no—
514C2×C4C2×#9911yescharacteristic
515C2×C4C2×#9911yescharacteristic
516C2×C4C2×#9911yescharacteristic
517C2×C4C2×#9911yescharacteristic
518C2×C4C2×#9911yescharacteristic
519C2×C4C2×#9911yescharacteristic
520C2×C4C2×#10312yes—
521C2×C4C2×#10312yes—
522C2×C4C2×#10312yes—
523C2×C4C2×#10312yes—
524C2×C4#42112yes—
525C8—84no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—22no—
531C22—22no—
532C22—41no—
533C22—41no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—81no—
544C22—81no—
545C22C22×#9911yescharacteristic
546C22C22×#10311yescharacteristic
547C22C2×#42111yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—22no—
553C4—22no—
554C4—22no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—82no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4—164no—
566C4#152012yes—
567C4#152012yes—
568C2—21no—
569C2—21no—
570C2—41no—
571C2—42no—
572C2—41no—
573C2—42no—
574C2C2×#152011yescharacteristic
575C2#3334711yescharacteristic
576C2#3335811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11902C2semidirect
5. #11897C2semidirect
6. #11718C2semidirect
7. #11717C2semidirect
8. C2×#533C4semidirect
14. #2628C4semidirect
15. #2628C4semidirect
19. #2581C4semidirect
24. #2564C4semidirect
29. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
#353342
#206322
#173492
#119022
#118972
#117182
#117172