Finite Group Databasebeta

← #215933

Subgroups — #215933

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2#21906C211yesmaximal, characteristic
3#13947C211yesmaximal, characteristic
4#9913C212yesmaximal, complemented
5#9910C212yesmaximal, complemented
6#9889C211yesmaximal, characteristic, complemented
7#9889C211yesmaximal, characteristic, complemented
8#9888C211yesmaximal, characteristic, complemented
9#9888C211yesmaximal, characteristic, complemented
10#8715C212yesmaximal
11#8715C212yesmaximal
12#2857C2211yescharacteristic
13#2857C2211yescharacteristic
14#2851C2212yes—
15#2626—21no—
16#2626—21no—
17#2608—22no—
18#2607—22no—
19#2441C412yescomplemented
20#2441C412yescomplemented
21#2439C412yescomplemented
22#2439C412yescomplemented
23#2329—21no—
24#2329—21no—
25#2280C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1924—22no—
28#1924—22no—
29#1797C2212yes—
30#1779C2212yes—
31#1749C2211yescharacteristic
32#1749C2211yescharacteristic
33#1443—21no—
34#1440—21no—
35#1437—21no—
36#1434—21no—
37#1434C2211yescharacteristic
38#1434C2211yescharacteristic
39#1434C2211yescharacteristic
40C22⋊#95C2211yescharacteristic
41C22⋊#95C2211yescharacteristic
42C4×#95C2211yescharacteristic
43#1294C2212yes—
44#1294C2212yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C24×C4C2311yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125C2×C412yes—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×#117C2×C412yes—
58C2×#117C2×C412yes—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4D411yescharacteristic
62C22×C22⋊C4D411yescharacteristic
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×C4⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8C2312yes—
70C2×C22⋊C8C2312yes—
71C2×#97—21no—
72C2×#97—21no—
73#399—22no—
74#399—22no—
75#396—22no—
76#396—22no—
77#396—22no—
78#396—22no—
79#393—22no—
80#393—22no—
81#391—21no—
82#391—21no—
83#391—22no—
84#391—22no—
85#389—21no—
86#389—21no—
87#389—22no—
88#389—22no—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#385—22no—
100#385—22no—
101#384C2×C411yescharacteristic
102#384C2×C411yescharacteristic
103#382C2×C412yes—
104#381—22no—
105#380—21no—
106#380—21no—
107#380C2×C412yes—
108#380C2×C412yes—
109#380C2312yes—
110#380C2312yes—
111#379—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4C2×C412yes—
115C4×C22⋊C4C2×C412yes—
116C4×C22⋊C4D412yes—
117C4×C22⋊C4D412yes—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95D411yescharacteristic
128C2×#95D411yescharacteristic
129C2×#95C2311yescharacteristic
130C2×#95C2311yescharacteristic
131C2×#95C2311yescharacteristic
132C2×#95C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—21no—
136C25C2×D411yescharacteristic
137C22×Q8—21no—
138C22×Q8—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—81no—
146C23×C4—81no—
147C23×C4C22×C412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C4⋊Q8—81no—
158C4⋊Q8—81no—
159#125—82no—
160C22⋊Q8—42no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164#117—82no—
165#117—82no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—81no—
197C2×C22⋊C4—81no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22⋊C412yes—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—81no—
207C2×C42—81no—
208C2×C42—81no—
209C2×C42C22×C412yes—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C4⋊C8—82no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—82no—
221C22⋊C8—82no—
222#97—81no—
223#97—81no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—42no—
239#95—42no—
240#95—42no—
241#95—42no—
242#95—81no—
243#95—81no—
244#95—81no—
245#95—81no—
246#95—81no—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251C24—21no—
252C24—21no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24C2×C22⋊C412yes—
262C24C22×D411yescharacteristic
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—81no—
266C2×Q8—81no—
267C2×Q8—81no—
268C2×Q8—81no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
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—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C42−1+412yes—
341C22×C42−1+412yes—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—81no—
363C4⋊C4—81no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—82no—
373C22⋊C4—82no—
374C22⋊C4—82no—
375C22⋊C4—82no—
376C22⋊C4—84no—
377C22⋊C4—84no—
378C42—41no—
379C42—41no—
380C42—42no—
381C42—42no—
382C42—41no—
383C42—41no—
384C42—82no—
385C42—82no—
386C42—84no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23C22×C22⋊C411yescharacteristic
415C23#53611yescharacteristic
416C23#53611yescharacteristic
417C23#57511yescharacteristic
418C23#57511yescharacteristic
419C23#57511yescharacteristic
420C23#57511yescharacteristic
421Q8—162no—
422Q8—162no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—48no—
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—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—84no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4#51912yes—
503C2×C4#53612yes—
504C8—82no—
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—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22#231011yescharacteristic
530C22#288311yescharacteristic
531C22#288311yescharacteristic
532C22#289611yescharacteristic
533C22#289611yescharacteristic
534C22#290111yescharacteristic
535C22#290111yescharacteristic
536C22#324011yescharacteristic
537C22#324011yescharacteristic
538C22#324011yescharacteristic
539C22#324011yescharacteristic
540C4—22no—
541C4—22no—
542C4—42no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—42no—
548C4—42no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4—82no—
554C4—82no—
555C4—161no—
556C4—161no—
557C2—21no—
558C2—21no—
559C2—21no—
560C2—21no—
561C2—82no—
562C2#1762311yescharacteristic
563C2#1762511yescharacteristic
564C2#1762911yescharacteristic
565C2#1762911yescharacteristic
566C2#2182711yescharacteristic
567C2#2184211yescharacteristic
568C2#2184611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9913C2semidirect
5. #9910C2semidirect
6. #9889C2semidirect
7. #9889C2semidirect
8. #9888C2semidirect
9. #9888C2semidirect
19. #2441C4semidirect
20. #2441C4semidirect
21. #2439C4semidirect
22. #2439C4semidirect

Maximal subgroups

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

subgroupindex
#350332
#219062
#139472
#99132
#99102
#98892
#98882
#87152