Finite Group Databasebeta

← #215919

Subgroups — #215919

Order 512.

918 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2#21902C211yesmaximal, characteristic
3#13948C211yesmaximal, characteristic
4#9913C212yesmaximal, complemented
5#9910C212yesmaximal, complemented
6#9885C211yesmaximal, characteristic, complemented
7#9885C211yesmaximal, characteristic, complemented
8#9884C211yesmaximal, characteristic, complemented
9#9884C211yesmaximal, 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—
25C2×#388—21no—
26#2280C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1923—22no—
29#1923—22no—
30#1797C2212yes—
31#1779C2212yes—
32C4×C22⋊C8C2211yescharacteristic
33C4×C22⋊C8C2211yescharacteristic
34#1442—21no—
35#1439—21no—
36#1436—21no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
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×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×C4⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8C2312yes—
72C2×C22⋊C8C2312yes—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75#399—22no—
76#399—22no—
77#396—22no—
78#396—22no—
79#396—22no—
80#396—22no—
81#393—22no—
82#393—22no—
83#391—21no—
84#391—21no—
85#391—22no—
86#391—22no—
87#389—21no—
88#389—21no—
89#389—22no—
90#389—22no—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#388—44no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—22no—
103#385—22no—
104#384C2×C411yescharacteristic
105#384C2×C411yescharacteristic
106#382C2×C412yes—
107#381—22no—
108#380—21no—
109#380—21no—
110#380C2×C412yes—
111#380C2×C412yes—
112#380C2312yes—
113#380C2312yes—
114#379—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4C2×C412yes—
118C4×C22⋊C4C2×C412yes—
119C4×C22⋊C4D412yes—
120C4×C22⋊C4D412yes—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95D411yescharacteristic
129C2×#95D411yescharacteristic
130C2×#95C2311yescharacteristic
131C2×#95C2311yescharacteristic
132C2×#95C2311yescharacteristic
133C2×#95C2311yescharacteristic
134#336—22no—
135#336—21no—
136#336—21no—
137C25C2×D411yescharacteristic
138C22×Q8—21no—
139C22×Q8—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—81no—
147C23×C4—81no—
148C23×C4C22×C412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—81no—
159C4⋊Q8—81no—
160#125—82no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C22⋊Q8—42no—
165#117—82no—
166#117—82no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—81no—
184C2×C4⋊C4—81no—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—81no—
204C2×C22⋊C4—81no—
205C2×C22⋊C4C22⋊C412yes—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42—81no—
214C2×C42—81no—
215C2×C42—81no—
216C2×C42C22×C412yes—
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C4⋊C8—82no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—82no—
228C22⋊C8—82no—
229C4×C8—81no—
230C4×C8—81no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—42no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95—81no—
250#95—81no—
251#95—81no—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24C2×C22⋊C412yes—
267C24C22×D411yescharacteristic
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—81no—
271C2×Q8—81no—
272C2×Q8—81no—
273C2×Q8—81no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
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—41no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4#13512yes—
340C22×C4#13512yes—
341C22×C4AGL(1,ℤ/8)12yes—
342C22×C4#13712yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C42−1+412yes—
346C22×C42−1+412yes—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—81no—
362C4⋊C4—81no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C42—41no—
386C42—41no—
387C42—42no—
388C42—42no—
389C42—41no—
390C42—41no—
391C42—82no—
392C42—82no—
393C42—84no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23C22×C22⋊C411yescharacteristic
422C23#53611yescharacteristic
423C23#53611yescharacteristic
424C23C2×#13511yescharacteristic
425C23C2×#13511yescharacteristic
426C23C2×AGL(1,ℤ/8)11yescharacteristic
427C23C2×#13711yescharacteristic
428Q8—162no—
429Q8—162no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—48no—
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—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—84no—
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—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4#41612yes—
506C2×C4#41612yes—
507C2×C4#41712yes—
508C2×C4#41712yes—
509C2×C4#51912yes—
510C2×C4#53612yes—
511C8—82no—
512C8—82no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22#231011yescharacteristic
537C22#288311yescharacteristic
538C22#288311yescharacteristic
539C22C2×#41611yescharacteristic
540C22C2×#41711yescharacteristic
541C22#290111yescharacteristic
542C22#290111yescharacteristic
543C22#322811yescharacteristic
544C22#322811yescharacteristic
545C22#323111yescharacteristic
546C22#323111yescharacteristic
547C4—22no—
548C4—22no—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—22no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—82no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4—161no—
563C4—161no—
564C2—21no—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—82no—
569C2#1761511yescharacteristic
570C2#1761711yescharacteristic
571C2#1762211yescharacteristic
572C2#1763011yescharacteristic
573C2#2182711yescharacteristic
574C2#2183511yescharacteristic
575C2#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. #9885C2semidirect
7. #9885C2semidirect
8. #9884C2semidirect
9. #9884C2semidirect
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
#219022
#139482
#99132
#99102
#98852
#98842
#87152