Finite Group Databasebeta

← #15884768

Subgroups — #15884768

Order 3328.

846 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117936C211yesmaximal, characteristic, complemented
2#13117746C211yesmaximal, characteristic
3#13115802C211yesmaximal, characteristic, complemented
4#13115606C211yesmaximal, characteristic, complemented
5#13115570C211yesmaximal, characteristic, complemented
6#13115565C211yesmaximal, characteristic, complemented
7#13115430C211yesmaximal, characteristic, complemented
8#13115423C211yesmaximal, characteristic, complemented
9#13115099C211yesmaximal, characteristic
10#13115085C211yesmaximal, characteristic
11#13114701C211yesmaximal, characteristic, complemented
12#102×Dic13C211yesmaximal, characteristic
13#102×Dic13C211yesmaximal, characteristic
14C4×#88072C211yesmaximal, characteristic
15C2×C13×#478C211yesmaximal, characteristic
16C2×C13×#125C2211yescharacteristic
17C2×C4×D4×C13C2211yescharacteristic
18C2×D4×Dic13C2211yescharacteristic
19C2×D4×Dic13C2211yescharacteristic
20C13×#478C418yescomplemented
21C2×C13×#107C2211yescharacteristic
22C2×C13×#106C2211yescharacteristic
23C2×C13×C4⋊C8C2211yescharacteristic
24C2×C13×#102C2211yescharacteristic
25C2×C13×#102C2211yescharacteristic
26C2×#88072C2211yescharacteristic
27C2×#88072C2211yescharacteristic
28#11718873—21no—
29C2×#88071—21no—
30C2×#88058C2211yescharacteristic
31C2×#88057C2211yescharacteristic
32C2×C8×Dic13C2211yescharacteristic
33C2×C8×Dic13C2211yescharacteristic
34#11718646—21no—
35#11718646C2211yescharacteristic
36#11718646C2211yescharacteristic
37#11718646C2211yescharacteristic
38#11718646C2211yescharacteristic
39#11718642—21no—
40#11718642—21no—
41C4⋊C4×Dic13—21no—
42C4⋊C4×Dic13C2211yescharacteristic
43C4⋊C4×Dic13C2211yescharacteristic
44#11718611—21no—
45#11718611—21no—
46#11718597C2211yescharacteristic
47C4×C4⋊Dic13C2211yescharacteristic
48C4×C4⋊Dic13C2211yescharacteristic
49C2×C4×#1055C2211yescharacteristic
50#11718143C2211yescharacteristic
51#11718143C2211yescharacteristic
52#11718142C2211yescharacteristic
53#11718142C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×D4×C13C2311yescharacteristic
61C23×C4×C13—21no—
62C23×Dic13—21no—
63C23×Dic13—21no—
64C22×C8×C13C2311yescharacteristic
65C22×C8×C13C2311yescharacteristic
66C13×#125—22no—
67C13×#125C2×C414yescomplemented
68C4×D4×C13—22no—
69C4×D4×C13C2×C414yes—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C22⋊C4—21no—
76C2×C42×C13C2311yescharacteristic
77C2×C22⋊Dic13—21no—
78C2×C22⋊Dic13—21no—
79D4×Dic13—22no—
80D4×Dic13—22no—
81D4×Dic13—41no—
82D4×Dic13—41no—
83C2×C4⋊Dic13—21no—
84C2×C4⋊Dic13—21no—
85C2×C4⋊Dic13—21no—
86C2×C4⋊Dic13—21no—
87C2×C4⋊Dic13D411yescharacteristic
88C2×C4⋊Dic13D411yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C2×C4×Dic13—21no—
96C2×C4×Dic13C2311yescharacteristic
97C2×C4×Dic13C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C13×#107C2×C414yescomplemented
101C13×#106C2×C414yescomplemented
102C13×C4⋊C8C2×C414yescomplemented
103C13×#102C2×C414yes—
104C13×#102C2×C414yes—
105#88072—22no—
106#88072Q814yes—
107#88071—21no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071D411yescharacteristic
117#88071D411yescharacteristic
118#88058—41no—
119#88057—41no—
120C8×Dic13—41no—
121C8×Dic13—41no—
122C4×#1055—22no—
123#22966—131nomaximal, Sylow
124C24×C13—21no—
125C2×D4×C13—21no—
126C2×D4×C13—21no—
127C2×D4×C13—21no—
128C2×D4×C13C22×C412yes—
129C2×D4×C13C2×Q812yes—
130C2×D4×C13#4112yes—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—24no—
135C22×C4×C13—81no—
136C22×C4×C13—81no—
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2411yescharacteristic
140C22×Dic13—21no—
141C22×Dic13—21no—
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—42no—
145C22×Dic13—42no—
146C22×Dic13—42no—
147C22×Dic13—42no—
148C22×Dic13—81no—
149C22×Dic13—81no—
150C22×Dic13C2×D411yescharacteristic
151C22×Dic13C2×D411yescharacteristic
152C22×Dic13C2×D411yescharacteristic
153C22×Dic13C2×D411yescharacteristic
154C2×C8×C13—21no—
155C2×C8×C13—21no—
156C2×C8×C13—21no—
157C2×C8×C13—21no—
158C2×C8×C13C22×C412yes—
159C2×C8×C13C22×C412yes—
160C13×C4⋊C4—24no—
161C13×C4⋊C4—24no—
162C13×C4⋊C4—21no—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4C22×C412yes—
166C13×C4⋊C4C22×C412yes—
167C13×C4⋊C4C22×C412yes—
168C13×C22⋊C4—24no—
169C42×C13C22×C412yes—
170C42×C13#4112yes—
171C22⋊Dic13—42no—
172C22⋊Dic13—42no—
173C4⋊Dic13—21no—
174C4⋊Dic13—22no—
175C4⋊Dic13—22no—
176C4⋊Dic13—21no—
177C4⋊Dic13—21no—
178C4⋊Dic13—21no—
179C4⋊Dic13—21no—
180C4⋊Dic13—21no—
181C4⋊Dic13—21no—
182C4⋊Dic13—42no—
183C4⋊Dic13—81no—
184C4⋊Dic13—81no—
185C4⋊Dic13—81no—
186C4⋊Dic13C2×Q812yes—
187C4×Dic13—21no—
188C4×Dic13—21no—
189C4×Dic13—21no—
190C4×Dic13—21no—
191C4×Dic13—42no—
192C4×Dic13—81no—
193C4×Dic13—81no—
194C2×#1055—21no—
195C2×#1055—21no—
196C2×#1055—21no—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055C2×Q812yes—
200C2×#478D1311yescharacteristic
201#2370—131no—
202#2349—131no—
203#1991—131no—
204#1949—131no—
205#1942—131no—
206#1941—131no—
207#1921—131no—
208#1920—131no—
209#1864—131no—
210#1862—131no—
211C8×C4⋊C4—131no—
212C4×#102—131no—
213C4×#102—131no—
214C4×#102—131no—
215C23×C13—22no—
216C23×C13—22no—
217C23×C13—22no—
218C23×C13—41no—
219C23×C13—41no—
220C23×C13—41no—
221C23×C13—41no—
222C23×C13C22×D411yescharacteristic
223D4×C13—22no—
224D4×C13—22no—
225D4×C13—22no—
226D4×C13C4×Q814yes—
227C2×C4×C13—21no—
228C2×C4×C13—21no—
229C2×C4×C13—21no—
230C2×C4×C13—21no—
231C2×C4×C13—22no—
232C2×C4×C13—22no—
233C2×C4×C13—22no—
234C2×C4×C13—41no—
235C2×C4×C13—41no—
236C2×C4×C13—41no—
237C2×C4×C13—41no—
238C2×C4×C13—41no—
239C2×C4×C13—41no—
240C2×C4×C13—84no—
241C2×C4×C13C4×D412yes—
242C2×C4×C13C4×D412yes—
243C2×C4×C13C23×C411yescharacteristic
244C2×C4×C13C22×Q811yescharacteristic
245C2×C4×C13C2×#4111yescharacteristic
246C2×C4×C13C2×#4111yescharacteristic
247C2×C4×C13H(5,𝔽2)11yescharacteristic
248C2×C4×C132−1+411yescharacteristic
249C2×Dic13—21no—
250C2×Dic13—21no—
251C2×Dic13—21no—
252C2×Dic13—21no—
253C2×Dic13—21no—
254C2×Dic13—21no—
255C2×Dic13—22no—
256C2×Dic13—21no—
257C2×Dic13—21no—
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13—41no—
263C2×Dic13—41no—
264C2×Dic13—41no—
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13—41no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13—41no—
271C2×Dic13—84no—
272C2×Dic13—84no—
273C2×Dic13#13512yes—
274C2×Dic13#13512yes—
275C8×C13—22no—
276C8×C13—22no—
277#1055—22no—
278#1055—41no—
279C2×#125C2×D1311yescharacteristic
280C2×C4×D4—131no—
281C2×C4×D4—131no—
282C2×C4×D4C2×D1311yescharacteristic
283#478Dic1318yescomplemented
284C2×#107—131no—
285C2×#107C2×D1311yescharacteristic
286C2×#106—131no—
287C2×#106C2×D1311yescharacteristic
288C2×C4⋊C8C2×D1311yescharacteristic
289C2×#102—131no—
290C2×#102—131no—
291C2×#102C2×D1311yescharacteristic
292C2×#102C2×D1311yescharacteristic
293C2×C4×C8—131no—
294C2×C4×C8—131no—
295C2×C4×C8—131no—
296#389—131no—
297#389—131no—
298#389—131no—
299#389—131no—
300#389—261no—
301#387—261no—
302#387—261no—
303#385—261no—
304#385—261no—
305#382—131no—
306#380—261no—
307C4×C4⋊C4—131no—
308C4×C4⋊C4—131no—
309C4×C4⋊C4—131no—
310C4×C4⋊C4—131no—
311C4×C4⋊C4—261no—
312C2×#95—261no—
313#340—131no—
314#340—131no—
315#340—131no—
316#340—131no—
317#340—131no—
318#340—131no—
319#340—131no—
320#340—131no—
321#336—131no—
322#336—131no—
323C22×C13—24no—
324C22×C13—42no—
325C22×C13—42no—
326C22×C13—42no—
327C22×C13—42no—
328C22×C13—42no—
329C22×C13—42no—
330C22×C13C2×C4×D411yescharacteristic
331C22×C13#54611yescharacteristic
332C22×C13D4×Q811yescharacteristic
333C22×C13C2×#13511yescharacteristic
334C22×C13C2×#13511yescharacteristic
335C22×C13#57611yescharacteristic
336C22×C13#57711yescharacteristic
337C4×C13—22no—
338C4×C13—22no—
339C4×C13—42no—
340C4×C13—42no—
341C4×C13—42no—
342C4×C13C2×C4×Q811yescharacteristic
343C4×C13#52011yescharacteristic
344C4×C13#55211yescharacteristic
345C4×C13#55411yescharacteristic
346Dic13—22no—
347Dic13—22no—
348Dic13—42no—
349Dic13—41no—
350Dic13—41no—
351Dic13—81no—
352Dic13—81no—
353Dic13—81no—
354C22×D4C22×D1311yescharacteristic
355C23×C4—21no—
356C23×C4—261no—
357C23×C4—261no—
358C22×C8—131no—
359C22×C8—131no—
360C22×C8C22×D1311yescharacteristic
361C22×C8C22×D1311yescharacteristic
362#125—22no—
363#125C2×Dic1314yescomplemented
364C4×D4—22no—
365C4×D4—262no—
366C4×D4—262no—
367C4×D4—521no—
368C4×D4—521no—
369C4×D4C2×Dic1314yes—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—131no—
373C2×C4⋊C4—131no—
374C2×C4⋊C4—131no—
375C2×C4⋊C4—131no—
376C2×C4⋊C4—131no—
377C2×C4⋊C4—131no—
378C2×C4⋊C4—261no—
379C2×C4⋊C4—261no—
380C2×C4⋊C4—261no—
381C2×C4⋊C4—261no—
382C2×C4⋊C4C22×D1311yescharacteristic
383C2×C4⋊C4C22×D1311yescharacteristic
384C2×C4⋊C4C22×D1311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—261no—
387C2×C22⋊C4—261no—
388C2×C42—131no—
389C2×C42—131no—
390C2×C42—261no—
391C2×C42—261no—
392C2×C42—261no—
393C2×C42C22×D1311yescharacteristic
394#107—521no—
395#107C2×Dic1314yescomplemented
396#106—521no—
397#106C2×Dic1314yescomplemented
398C4⋊C8C2×Dic1314yescomplemented
399#102—134no—
400#102—262no—
401#102C2×Dic1314yes—
402#102C2×Dic1314yes—
403C4×C8—262no—
404C4×C8—521no—
405C4×C8—521no—
406#95—131no—
407#95—131no—
408#95—261no—
409#95—261no—
410#95—261no—
411#95—261no—
412#95—261no—
413#95—261no—
414#95—261no—
415#95—261no—
416#95—261no—
417C2×C13—44no—
418C2×C13#236011yescharacteristic
419C2×C13C4×#13511yescharacteristic
420C2×C13#294611yescharacteristic
421C2×C13#329611yescharacteristic
422C2×C13#329711yescharacteristic
423C2×C13#338011yescharacteristic
424C2×C13#338111yescharacteristic
425C24—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4Q8×D1312yes—
430C2×D4#613112yes—
431C2×D4C22×Dic1312yes—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—131no—
439C22×C4—131no—
440C22×C4—131no—
441C22×C4—131no—
442C22×C4—261no—
443C22×C4—261no—
444C22×C4—261no—
445C22×C4—261no—
446C22×C4—522no—
447C22×C4—522no—
448C22×C4—522no—
449C22×C4—522no—
450C22×C4—1041no—
451C22×C4—1041no—
452C22×C4D4×D1311yescharacteristic
453C22×C4D4×D1311yescharacteristic
454C22×C4C23×D1311yescharacteristic
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—132no—
460C2×C8—261no—
461C2×C8—261no—
462C2×C8—261no—
463C2×C8—261no—
464C2×C8—261no—
465C2×C8C22×Dic1312yes—
466C2×C8C22×Dic1312yes—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—132no—
473C4⋊C4—262no—
474C4⋊C4—262no—
475C4⋊C4—261no—
476C4⋊C4—261no—
477C4⋊C4—261no—
478C4⋊C4—261no—
479C4⋊C4—261no—
480C4⋊C4—261no—
481C4⋊C4—261no—
482C4⋊C4—522no—
483C4⋊C4—1041no—
484C4⋊C4—1041no—
485C4⋊C4—1041no—
486C4⋊C4C22×Dic1312yes—
487C4⋊C4C22×Dic1312yes—
488C4⋊C4C22×Dic1312yes—
489C22⋊C4—24no—
490C22⋊C4—522no—
491C22⋊C4—522no—
492C42—261no—
493C42—261no—
494C42—261no—
495C42—261no—
496C42—522no—
497C42—1041no—
498C42—1041no—
499C42#612912yes—
500C42C22×Dic1312yes—
501C13#2296611yescharacteristic, complemented, Sylow
502C23—22no—
503C23—22no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C2×D4×D1311yescharacteristic
510D4—22no—
511D4—22no—
512D4—22no—
513D4Q8×Dic1314yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—84no—
528C2×C4—132no—
529C2×C4—132no—
530C2×C4—261no—
531C2×C4—261no—
532C2×C4—261no—
533C2×C4—261no—
534C2×C4—261no—
535C2×C4—261no—
536C2×C4—261no—
537C2×C4—261no—
538C2×C4—261no—
539C2×C4—261no—
540C2×C4—262no—
541C2×C4—521no—
542C2×C4—521no—
543C2×C4—521no—
544C2×C4—521no—
545C2×C4—521no—
546C2×C4—521no—
547C2×C4—521no—
548C2×C4—521no—
549C2×C4—521no—
550C2×C4—521no—
551C2×C4—521no—
552C2×C4—1044no—
553C2×C4—1044no—
554C2×C4D4×Dic1312yes—
555C2×C4D4×Dic1312yes—
556C2×C4C2×#612911yescharacteristic
557C2×C4C2×Q8×D1311yescharacteristic
558C2×C4C2×#613111yescharacteristic
559C2×C4#8825611yescharacteristic
560C2×C4#8825711yescharacteristic
561C2×C4C23×Dic1311yescharacteristic
562C8—22no—
563C8—22no—
564C8—262no—
565C8—521no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22#1171933511yescharacteristic
574C22#1171933711yescharacteristic
575C22D13×#13511yescharacteristic
576C22D13×#13511yescharacteristic
577C22#1171947911yescharacteristic
578C22#1171948011yescharacteristic
579C22C2×D4×Dic1311yescharacteristic
580C4—22no—
581C4—22no—
582C4—42no—
583C4—42no—
584C4—42no—
585C4—262no—
586C4—262no—
587C4—521no—
588C4—522no—
589C4—521no—
590C4—1041no—
591C4—1041no—
592C4—1041no—
593C4#1171934011yescharacteristic
594C4#1171934711yescharacteristic
595C4C2×Q8×Dic1311yescharacteristic
596C4#1171953211yescharacteristic
597C2—44no—
598C2#1311784511yescharacteristic
599C2#135×Dic1311yescharacteristic
600C2#1312366911yescharacteristic
601C2#1312666811yescharacteristic
602C2#1312668311yescharacteristic
603C2#1312769811yescharacteristic
604C2#1312770611yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#478C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
cosocleC22×C4×C13C24
hypercenterC2×#478D13
hypocenterC13#22966
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. #13117936C2semidirect
3. #13115802C2semidirect
4. #13115606C2semidirect
5. #13115570C2semidirect
6. #13115565C2semidirect
7. #13115430C2semidirect
8. #13115423C2semidirect
11. #13114701C2semidirect
20. C13×#478C4semidirect
67. C13×#125C2×C4semidirect
100. C13×#107C2×C4semidirect
101. C13×#106C2×C4semidirect
102. C13×C4⋊C8C2×C4semidirect
283. #478Dic13semidirect
363. #125C2×Dic13semidirect
395. #107C2×Dic13semidirect
397. #106C2×Dic13semidirect
398. C4⋊C8C2×Dic13semidirect
501. C13#22966semidirect

Maximal subgroups

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

subgroupindex
#131179362
#131177462
#131158022
#131156062
#131155702
#131155652
#131154302
#131154232
#131150992
#131150852
#131147012
#102×Dic132
C4×#880722
C2×C13×#4782
#2296613