Finite Group Databasebeta

← #16800170

Subgroups — #16800170

Order 3328.

886 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719080C211yesmaximal, characteristic, complemented
2C2×C4×#6118C211yesmaximal, characteristic
3#13131009C211yesmaximal, characteristic, complemented
4#13130936C211yesmaximal, characteristic, complemented
5#13130890C211yesmaximal, characteristic, complemented
6#13130889C211yesmaximal, characteristic, complemented
7#13130782C211yesmaximal, characteristic
8#13130782C211yesmaximal, characteristic
9#13130691C211yesmaximal, characteristic
10#13130635C211yesmaximal, characteristic, complemented
11#13130635C211yesmaximal, characteristic, complemented
12C4⋊C4×#1057C211yesmaximal, characteristic, complemented
13#13130577C211yesmaximal, characteristic
14#13130577C211yesmaximal, characteristic
15#13117370C211yesmaximal, characteristic
16C22×#6118C2211yescharacteristic
17C22×#6118C2211yescharacteristic
18C2×#88150C411yescharacteristic
19C2×#88141C2211yescharacteristic
20C2×C42×D13C2211yescharacteristic
21C2×#88111C2211yescharacteristic
22C2×#88111C2211yescharacteristic
23C2×#88110C2211yescharacteristic
24C2×#88110C2211yescharacteristic
25C2×#88110C2211yescharacteristic
26C2×#88110C2211yescharacteristic
27C2×C4⋊#1057C2211yescharacteristic
28C2×C4⋊#1057C2211yescharacteristic
29C2×C4⋊#1057C2211yescharacteristic
30C2×C4⋊#1057C2211yescharacteristic
31C2×C4⋊#1057C2211yescharacteristic
32C2×C4⋊#1057C2211yescharacteristic
33C2×C4×#1057C2211yescharacteristic
34C2×C4×#1057C2211yescharacteristic
35C2×C4×#1057C2211yescharacteristic
36C2×C4×#1057C2211yescharacteristic
37#11719080—24no—
38C4×#6118—24no—
39C13×#387C411yescharacteristic
40#11718650C411yescharacteristic
41C4⋊C4×Dic13C2211yescharacteristic
42#11718594C411yescharacteristic
43#11718329C411yescharacteristic
44#11718328C411yescharacteristic
45#11718327C2211yescharacteristic
46#11718327C2211yescharacteristic
47#11718310C2211yescharacteristic
48#11718310C2211yescharacteristic
49#11718308C411yescharacteristic
50#11718307C411yescharacteristic
51#11718290C2211yescharacteristic
52#11718290C2211yescharacteristic
53#11718290C2211yescharacteristic
54#11718290C2211yescharacteristic
55#11718290C2211yescharacteristic
56#11718290C2211yescharacteristic
57#11718290C2211yescharacteristic
58#11718290C2211yescharacteristic
59#11718290C2211yescharacteristic
60#11718290C2211yescharacteristic
61C22×C4×D13C2×C411yescharacteristic
62C22×C4×D13C2×C411yescharacteristic
63C22×C4×D13C2311yescharacteristic
64C22×Dic26C2×C411yescharacteristic
65C22×#1057C2311yescharacteristic
66C22×#1057C2311yescharacteristic
67C22×#1057C2311yescharacteristic
68C22×#1057C2311yescharacteristic
69C22×#1057C2311yescharacteristic
70C22×#1057C2311yescharacteristic
71C22×#1057C2311yescharacteristic
72C22×#1057C2311yescharacteristic
73C2×#6118—22no—
74C2×#6118—22no—
75C2×#6118—22no—
76C2×#6118—22no—
77C2×#6118—22no—
78C2×#6118Q814yescomplemented
79C2×C13×C4⋊C4C2×C411yescharacteristic
80C2×C13×C4⋊C4C2×C411yescharacteristic
81C2×C42×C13C2×C411yescharacteristic
82C2×#6103C2×C411yescharacteristic
83C2×#6103C2×C411yescharacteristic
84C2×#6103C2×C411yescharacteristic
85C2×#6103C2×C411yescharacteristic
86C2×C4⋊Dic13C2×C411yescharacteristic
87C2×#6101C2×C411yescharacteristic
88C2×#6101C2×C411yescharacteristic
89C2×#6101C2×C411yescharacteristic
90C2×#6101C2×C411yescharacteristic
91C2×#6101C2×C411yescharacteristic
92C2×#6101C2×C411yescharacteristic
93C2×C4×Dic13C2×C411yescharacteristic
94C2×C4×Dic13C2311yescharacteristic
95C2×C4×Dic13C2311yescharacteristic
96C2×C4×Dic13C2311yescharacteristic
97C2×C4×Dic13C2311yescharacteristic
98C2×C4×Dic13C2311yescharacteristic
99C2×C4×Dic13C2311yescharacteristic
100#88150—24no—
101#88141—22no—
102#88141—22no—
103C42×D13—22no—
104C42×D13—22no—
105#88111—22no—
106#88111—22no—
107#88110—22no—
108#88110—22no—
109#88110—22no—
110#88110D414yes—
111C4⋊#1057—22no—
112C4⋊#1057—22no—
113C4⋊#1057—22no—
114C4⋊#1057—22no—
115C4⋊#1057—22no—
116C4⋊#1057—22no—
117C4×#1057—22no—
118C4×#1057—22no—
119C4×#1057—22no—
120C4×#1057—22no—
121C13×#95C2×C411yescharacteristic
122C13×#95C2×C411yescharacteristic
123C13×#95C2×C411yescharacteristic
124C13×#95C2×C411yescharacteristic
125#88071C2×C411yescharacteristic
126#88071C2×C411yescharacteristic
127#88071C2×C411yescharacteristic
128#88071C2×C411yescharacteristic
129#88071C2×C411yescharacteristic
130#88071C2×C411yescharacteristic
131#21600—131nomaximal, Sylow
132C23×D13C22×C411yescharacteristic
133C22×C4×C13C22×C411yescharacteristic
134C22×C4×C13C22×C411yescharacteristic
135C22×C4×C13C22×C411yescharacteristic
136C22×C4×C13C22×C411yescharacteristic
137C22×C4×C13C22×C411yescharacteristic
138C22×C4×C13C22×C411yescharacteristic
139C22×C4×C13C22×C411yescharacteristic
140C22×Dic13C22×C411yescharacteristic
141C22×Dic13C22×C411yescharacteristic
142C22×Dic13C22×C411yescharacteristic
143C22×Dic13C22×C411yescharacteristic
144C22×Dic13C22×C411yescharacteristic
145C22×Dic13C22×C411yescharacteristic
146C22×Dic13C2411yescharacteristic
147C2×C4×D13—21no—
148C2×C4×D13—21no—
149C2×C4×D13—21no—
150C2×C4×D13—21no—
151C2×C4×D13—22no—
152C2×C4×D13—22no—
153C2×C4×D13—22no—
154C2×C4×D13—22no—
155C2×C4×D13—41no—
156C2×C4×D13—41no—
157C2×C4×D13C2×Q812yes—
158C2×C4×D13#4112yes—
159C2×Dic26—22no—
160C2×Dic26—22no—
161C2×Dic26—41no—
162C2×#1057—21no—
163C2×#1057—21no—
164C2×#1057—21no—
165C2×#1057—21no—
166C2×#1057—21no—
167C2×#1057—21no—
168C2×#1057—21no—
169C2×#1057—21no—
170C2×#1057—21no—
171C2×#1057—21no—
172C2×#1057—21no—
173C2×#1057—21no—
174C2×#1057—21no—
175C2×#1057—21no—
176C2×#1057—21no—
177C2×#1057—21no—
178C2×#1057—21no—
179C2×#1057—21no—
180C2×#1057C2×D412yes—
181C2×#1057C2×D412yes—
182C2×#1057C2×Q812yes—
183C2×#1057C2×Q812yes—
184C2×#1057#4112yes—
185C2×#1057#4112yes—
186#6118—28no—
187#6118—28no—
188C13×C4⋊C4—22no—
189C13×C4⋊C4M1614yescomplemented
190C42×C13—22no—
191#6103—22no—
192#6103—22no—
193#6103—41no—
194#6103—41no—
195C4⋊Dic13—22no—
196#6101—22no—
197#6101—22no—
198#6101—22no—
199#6101—22no—
200#6101—22no—
201#6101—41no—
202C4×Dic13—22no—
203C4×Dic13—21no—
204C4×Dic13—21no—
205C4×Dic13—21no—
206C4×Dic13—21no—
207C4×Dic13—21no—
208C4×Dic13—21no—
209C4×Dic13—21no—
210C4×Dic13—21no—
211C4×Dic13—21no—
212C4×Dic13—21no—
213C4×Dic13C2×D412yes—
214C4×Dic13#4112yes—
215C2×#446—131no—
216C2×C4×M16—131no—
217C4×C22⋊Q8—131no—
218#1985—131no—
219#1940—131no—
220#1916—131no—
221#1916—131no—
222#1851—131no—
223#1851—131no—
224#1797—131no—
225#1771—131no—
226#1771—131no—
227C8×C4⋊C4—131no—
228#1753—131no—
229#1753—131no—
230C23×C13C23×C411yescharacteristic
231C22×D13—21no—
232C22×D13—21no—
233C22×D13—21no—
234C22×D13—21no—
235C22×D13—21no—
236C22×D13—21no—
237C22×D13C4×Q812yes—
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—21no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—21no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—21no—
254C2×C4×C13C4×D412yes—
255C2×C4×C13C4×Q812yes—
256C2×C4×C13C2×M1612yes—
257C2×C4×C13C2×M1612yes—
258C2×C4×C13C2×M1612yes—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—21no—
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—21no—
265C2×Dic13—21no—
266C2×Dic13—21no—
267C2×Dic13—21no—
268C2×Dic13—21no—
269C2×Dic13—21no—
270C2×Dic13—21no—
271C2×Dic13—21no—
272C2×Dic13—21no—
273C2×Dic13—21no—
274C2×Dic13—21no—
275C2×Dic13C4×D412yes—
276C2×Dic13#13112yes—
277C2×Dic13C22×D411yescharacteristic
278C2×Dic13C22×Q811yescharacteristic
279C2×Dic13C2×#4111yescharacteristic
280C2×Dic13C2×#4111yescharacteristic
281C2×Dic13C2×#4111yescharacteristic
282C2×Dic13C2×#4111yescharacteristic
283C4×D13—22no—
284C4×D13—22no—
285C4×D13—22no—
286C4×D13—22no—
287C4×D13—44no—
288C4×D13—44no—
289Dic26—44no—
290#1057—22no—
291#1057—22no—
292#1057—22no—
293#1057—41no—
294#1057—41no—
295#1057—41no—
296#1057—41no—
297#1057C22⋊Q814yescomplemented
298C22×M16—131no—
299C22×M16—131no—
300C2×C22⋊Q8—131no—
301C2×C4×Q8—131no—
302C22×C42—131no—
303#446—264no—
304C2×C4⋊C8—131no—
305C2×C4⋊C8—131no—
306C2×C4⋊C8—131no—
307C2×C4⋊C8—131no—
308C2×C4⋊C8—131no—
309C2×C4⋊C8—131no—
310C2×C22⋊C8—131no—
311C2×C22⋊C8—131no—
312C4×M16—264no—
313C2×#97—131no—
314C2×#97—131no—
315C2×#97—131no—
316C2×#97—131no—
317C2×C4×C8—131no—
318C2×C4×C8—131no—
319C2×C4×C8—131no—
320C2×C4×C8—131no—
321#391—131no—
322#389—131no—
323#387—131no—
324#387#26111yescharacteristic
325#385—131no—
326#385—131no—
327#380—131no—
328C4×C4⋊C4—131no—
329C4×C4⋊C4—131no—
330C4×C4⋊C4—131no—
331C4×C22⋊C4—131no—
332C4×C22⋊C4—131no—
333#336—131no—
334#336—131no—
335#336—131no—
336#336—131no—
337#336—131no—
338#336—131no—
339#336—131no—
340#336—131no—
341#336—131no—
342#336—131no—
343C22×C13C2×C4×D411yescharacteristic
344C22×C13C2×C4×Q811yescharacteristic
345C22×C13C4×#4111yescharacteristic
346C22×C13C22×M1611yescharacteristic
347C22×C13C2×#13111yescharacteristic
348C22×C13#56811yescharacteristic
349C22×C13#56811yescharacteristic
350C2×D13—22no—
351C2×D13—22no—
352C2×D13—22no—
353C2×D13—41no—
354C2×D13—41no—
355C2×D13—41no—
356C2×D13—41no—
357C4×C13—22no—
358C4×C13—22no—
359C4×C13—22no—
360C4×C13—22no—
361C4×C13—22no—
362C4×C13—41no—
363C4×C13—41no—
364Dic13—22no—
365Dic13—22no—
366Dic13—41no—
367Dic13—41no—
368Dic13—41no—
369Dic13—41no—
370Dic13C2×C22⋊Q811yescharacteristic
371Dic13#52511yescharacteristic
372Dic13#52911yescharacteristic
373Dic13#53311yescharacteristic
374C22×Q8—131no—
375C23×C4—131no—
376C23×C4—131no—
377C23×C4—131no—
378C2×M16—134no—
379C2×M16—262no—
380C2×M16—262no—
381C2×M16—262no—
382C2×M16—262no—
383C2×M16—262no—
384C22×C8—131no—
385C22×C8—131no—
386C22×C8—131no—
387C22×C8—131no—
388C22×C8—131no—
389C22×C8—131no—
390C22×C8—131no—
391C22×C8—131no—
392C22⋊Q8—264no—
393C4×Q8—262no—
394C4×Q8—262no—
395C2×C4⋊C4—131no—
396C2×C4⋊C4—131no—
397C2×C4⋊C4—131no—
398C2×C4⋊C4—131no—
399C2×C4⋊C4—131no—
400C2×C4⋊C4—131no—
401C2×C4⋊C4—131no—
402C2×C4⋊C4C2×#26111yescharacteristic
403C2×C4⋊C4C2×#26111yescharacteristic
404C2×C22⋊C4—131no—
405C2×C22⋊C4—131no—
406C2×C22⋊C4—131no—
407C2×C22⋊C4—131no—
408C2×C42—131no—
409C2×C42—131no—
410C2×C42—131no—
411C2×C42—131no—
412C2×C42—131no—
413C2×C42—131no—
414C2×C42—131no—
415C2×C42—262no—
416C2×C42—262no—
417C2×C42C2×#26111yescharacteristic
418C4⋊C8—262no—
419C4⋊C8—262no—
420C4⋊C8—262no—
421C4⋊C8—262no—
422C4⋊C8—262no—
423C4⋊C8—262no—
424C22⋊C8—262no—
425C22⋊C8—262no—
426#97—134no—
427#97—262no—
428#97—262no—
429#97—262no—
430C4×C8—262no—
431C4×C8—262no—
432C4×C8—262no—
433C4×C8—262no—
434#95—131no—
435#95—131no—
436#95—131no—
437#95—131no—
438#95—131no—
439#95—131no—
440#95C2×#26111yescharacteristic
441#95C2×#26111yescharacteristic
442#95C2×#26111yescharacteristic
443#95C2×#26111yescharacteristic
444C2×C13C4×C22⋊Q811yescharacteristic
445C2×C13#293311yescharacteristic
446C2×C13D4×M1611yescharacteristic
447C2×C13#296211yescharacteristic
448C2×C13Q8×M1611yescharacteristic
449C2×C13#296511yescharacteristic
450C2×C13#296711yescharacteristic
451D13—42no—
452C24—131no—
453C2×Q8—262no—
454C2×Q8—262no—
455C2×Q8—521no—
456C22×C4—131no—
457C22×C4—131no—
458C22×C4—131no—
459C22×C4—132no—
460C22×C4—132no—
461C22×C4—131no—
462C22×C4—131no—
463C22×C4—131no—
464C22×C4—131no—
465C22×C4—261no—
466C22×C4—261no—
467C22×C4—261no—
468C22×C4—261no—
469C22×C4—262no—
470C22×C4—262no—
471C22×C4—262no—
472C22×C4—262no—
473C22×C4—521no—
474C22×C4—521no—
475C22×C4C22×#26111yescharacteristic
476C22×C4C22×#26111yescharacteristic
477C22×C4C22×#26111yescharacteristic
478C22×C4C22×#26111yescharacteristic
479C22×C4C22×#26111yescharacteristic
480C22×C4C22×#26111yescharacteristic
481C22×C4C22×#26111yescharacteristic
482M16—268no—
483M16—268no—
484C2×C8—132no—
485C2×C8—132no—
486C2×C8—132no—
487C2×C8—132no—
488C2×C8—132no—
489C2×C8—132no—
490C2×C8—261no—
491C2×C8—261no—
492C2×C8—261no—
493C2×C8—261no—
494C2×C8—261no—
495C2×C8—261no—
496C2×C8—261no—
497C2×C8—261no—
498C2×C8—261no—
499C2×C8—261no—
500C2×C8—261no—
501C2×C8—261no—
502C2×C8—261no—
503C2×C8—261no—
504C2×C8—261no—
505C2×C8—261no—
506C2×C8—261no—
507C2×C8—261no—
508C4⋊C4—22no—
509C4⋊C4—262no—
510C4⋊C4—262no—
511C4⋊C4—262no—
512C4⋊C4—262no—
513C4⋊C4—262no—
514C4⋊C4—262no—
515C4⋊C4—521no—
516C4⋊C4#611814yescomplemented
517C22⋊C4—262no—
518C22⋊C4—262no—
519C22⋊C4—521no—
520C22⋊C4—521no—
521C42—22no—
522C42—132no—
523C42—132no—
524C42—261no—
525C42—261no—
526C42—261no—
527C42—261no—
528C42—261no—
529C42—261no—
530C42—261no—
531C42—261no—
532C42—262no—
533C42—261no—
534C42—261no—
535C13#2160011yescharacteristic, complemented, Sylow
536C23—132no—
537C23—261no—
538C23—261no—
539C23—261no—
540C23—261no—
541C23—261no—
542C23—261no—
543C23C23×#26111yescharacteristic
544Q8—524no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—131no—
562C2×C4—131no—
563C2×C4—131no—
564C2×C4—131no—
565C2×C4—131no—
566C2×C4—131no—
567C2×C4—132no—
568C2×C4—132no—
569C2×C4—262no—
570C2×C4—262no—
571C2×C4—262no—
572C2×C4—262no—
573C2×C4—261no—
574C2×C4—261no—
575C2×C4—261no—
576C2×C4—261no—
577C2×C4—261no—
578C2×C4—261no—
579C2×C4—261no—
580C2×C4—261no—
581C2×C4—261no—
582C2×C4—261no—
583C2×C4—261no—
584C2×C4—261no—
585C2×C4—261no—
586C2×C4—261no—
587C2×C4—261no—
588C2×C4—261no—
589C2×C4—524no—
590C2×C4—524no—
591C2×C4C2×#611812yes—
592C2×C4C2×#611812yes—
593C2×C4C2×#611812yes—
594C2×C4D4×#26112yes—
595C2×C4Q8×#26112yes—
596C8—134no—
597C8—262no—
598C8—262no—
599C8—262no—
600C8—521no—
601C8—521no—
602C8—521no—
603C8—521no—
604C22—262no—
605C22—262no—
606C22—262no—
607C22—521no—
608C22—521no—
609C22—521no—
610C22—521no—
611C22C22×#611811yescharacteristic
612C22C2×D4×#26111yescharacteristic
613C22C2×Q8×#26111yescharacteristic
614C22D13⋊#13111yescharacteristic
615C22#1171962911yescharacteristic
616C22#1171962911yescharacteristic
617C22#41×#26111yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—41no—
624C4—41no—
625C4—131no—
626C4—131no—
627C4—131no—
628C4—131no—
629C4—262no—
630C4—262no—
631C4—521no—
632C4—521no—
633C4—521no—
634C4—521no—
635C2—522no—
636C2C22⋊Q8×#26111yescharacteristic
637C2#1313157011yescharacteristic
638C2D4×#611811yescharacteristic
639C2#1313163411yescharacteristic
640C2Q8×#611811yescharacteristic
641C2#1313164311yescharacteristic
642C2#1313165911yescharacteristic
subgroupG/H
center, FrattiniC23C23×#261
commutator, socleC23×C13C23×C4
FittingC13×#387C4
prime radicalC23×D13C22×C4
cosocleC22×Dic13C24
hypercenter#387#261
hypocenterC13#21600
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. C2×#11719080C2semidirect
3. #13131009C2semidirect
4. #13130936C2semidirect
5. #13130890C2semidirect
6. #13130889C2semidirect
10. #13130635C2semidirect
11. #13130635C2semidirect
12. C4⋊C4×#1057C2semidirect
78. C2×#6118Q8semidirect
189. C13×C4⋊C4M16semidirect
297. #1057C22⋊Q8semidirect
516. C4⋊C4#6118semidirect
535. C13#21600semidirect

Maximal subgroups

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

subgroupindex
C2×#117190802
C2×C4×#61182
#131310092
#131309362
#131308902
#131308892
#131307822
#131306912
#131306352
C4⋊C4×#10572
#131305772
#131173702
#2160013