Finite Group Databasebeta

← #15914699

Subgroups — #15914699

Order 3328.

812 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718741C211yesmaximal, characteristic, complemented
2C2×#11718670C211yesmaximal, characteristic
3#13117930C211yesmaximal, characteristic, complemented
4#13116313C211yesmaximal, characteristic, complemented
5#13116308C211yesmaximal, characteristic, complemented
6#13116224C211yesmaximal, characteristic
7#13116210C211yesmaximal, characteristic
8#13115785C211yesmaximal, characteristic, complemented
9#13115597C211yesmaximal, characteristic, complemented
10#13115592C211yesmaximal, characteristic, complemented
11#13115560C211yesmaximal, characteristic, complemented
12#13115460C211yesmaximal, characteristic
13#13115453C211yesmaximal, characteristic
14#13114661C211yesmaximal, characteristic
15C13×#2542C211yesmaximal, characteristic
16C2×C13×C4⋊Q8C2211yescharacteristic
17C2×C13×C22⋊Q8—21no—
18C2×C13×C22⋊Q8—21no—
19C2×C4×D4×C13C2211yescharacteristic
20C2×#88124C2211yescharacteristic
21C13×#399—21no—
22C13×#399—21no—
23C13×#390C2211yescharacteristic
24C13×#390C2211yescharacteristic
25C13×#384C2211yescharacteristic
26C13×#384C2211yescharacteristic
27C4×C13×C4⋊C4C2211yescharacteristic
28C2×#88072C2211yescharacteristic
29C2×#88072C2211yescharacteristic
30C2×#88072C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32#11718741—24no—
33#11718670—24no—
34#11718639—21no—
35#11718633—21no—
36C2×#88048C2211yescharacteristic
37C2×#88048C2211yescharacteristic
38C2×#88048C2211yescharacteristic
39C2×#88047C2211yescharacteristic
40C2×#88047C2211yescharacteristic
41C2×#88047C2211yescharacteristic
42C4×C4⋊Dic13C2211yescharacteristic
43C4×C4⋊Dic13C2211yescharacteristic
44C4×C4⋊Dic13C2211yescharacteristic
45#11718589C2211yescharacteristic
46#11718589C2211yescharacteristic
47C2×C4⋊#1055C2211yescharacteristic
48C2×C4⋊#1055C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50C2×#88043C2211yescharacteristic
51#11718313—21no—
52#11718310—21no—
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C22×Q8×C13—21no—
62C22×D4×C13C2311yescharacteristic
63C23×C4×C13—21no—
64C13×C4⋊Q8—22no—
65C13×C4⋊Q8—22no—
66C13×C22⋊Q8—44no—
67C13×C22⋊Q8—44no—
68C4×D4×C13—22no—
69C4×D4×C13—22no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4—21no—
74C2×C13×C4⋊C4—21no—
75C2×C13×C4⋊C4D411yescharacteristic
76C2×C13×C4⋊C4D411yescharacteristic
77C2×C13×C4⋊C4C2311yescharacteristic
78C2×C13×C4⋊C4C2311yescharacteristic
79C2×C13×C4⋊C4C2311yescharacteristic
80C2×C13×C22⋊C4—21no—
81C2×C13×C22⋊C4—21no—
82C2×C13×C22⋊C4—21no—
83C2×C13×C22⋊C4—21no—
84C2×C13×C22⋊C4—21no—
85C2×C42×C13C2311yescharacteristic
86C2×C42×C13C2311yescharacteristic
87C2×C42×C13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×#6101—21no—
93C2×#6101—21no—
94C2×#6101—21no—
95C2×#6101—21no—
96C2×C4×Dic13—21no—
97C2×C4×Dic13—21no—
98C2×C4×Dic13—21no—
99C22×#1055C2311yescharacteristic
100C22×#1055C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102C22×#1055C2311yescharacteristic
103#88124—22no—
104#88124D414yes—
105C13×#95D411yescharacteristic
106C13×#95D411yescharacteristic
107#88072—22no—
108#88072—22no—
109#88072—22no—
110#88072Q814yescomplemented
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88048—22no—
117#88048—22no—
118#88048—22no—
119#88047—22no—
120#88047—22no—
121#88047—22no—
122C4⋊#1055—22no—
123C4⋊#1055—22no—
124C4⋊#1055—22no—
125#88043—22no—
126#25465—131nomaximal, Sylow
127C24×C13—21no—
128C2×Q8×C13—42no—
129C2×Q8×C13—42no—
130C2×Q8×C13—42no—
131C2×D4×C13—21no—
132C2×D4×C13—21no—
133C2×D4×C13—21no—
134C2×D4×C13—21no—
135C2×D4×C13—21no—
136C2×D4×C13C2×Q812yes—
137C22×C4×C13—21no—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13—42no—
141C22×C4×C13—42no—
142C22×C4×C13—42no—
143C22×C4×C13C2×D411yescharacteristic
144C22×C4×C13C2×D411yescharacteristic
145C22×C4×C13C2×D411yescharacteristic
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2×D411yescharacteristic
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13C2411yescharacteristic
150C22×Dic13—21no—
151C22×Dic13—21no—
152C22×Dic13—21no—
153C22×Dic13—21no—
154C13×C4⋊C4—22no—
155C13×C4⋊C4—22no—
156C13×C4⋊C4—21no—
157C13×C4⋊C4—21no—
158C13×C4⋊C4—21no—
159C13×C4⋊C4—21no—
160C13×C4⋊C4—21no—
161C13×C4⋊C4—21no—
162C13×C4⋊C4—42no—
163C13×C4⋊C4—42no—
164C13×C4⋊C4—42no—
165C13×C4⋊C4—42no—
166C13×C4⋊C4—42no—
167C13×C22⋊C4—42no—
168C13×C22⋊C4—42no—
169C13×C22⋊C4—42no—
170C13×C22⋊C4—42no—
171C13×C22⋊C4—42no—
172C42×C13—21no—
173C42×C13—21no—
174C42×C13—21no—
175C42×C13—21no—
176C42×C13C2×D412yes—
177C42×C13#4112yes—
178C4⋊Dic13—21no—
179C4⋊Dic13—21no—
180C4⋊Dic13—21no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13C2×D412yes—
184C4⋊Dic13C2×D412yes—
185C4⋊Dic13C2×Q812yes—
186#6101—24no—
187#6101—24no—
188#6101—42no—
189#6101—81no—
190C4×Dic13—42no—
191C4×Dic13—81no—
192C4×Dic13—81no—
193C2×#1055—21no—
194C2×#1055—21no—
195C2×#1055—21no—
196C2×#1055—21no—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202C2×#1055—21no—
203C2×#1055C2×Q812yes—
204C2×#1055#4112yes—
205C2×#501—131no—
206C2×#478—131no—
207#2542D1311yescharacteristic
208#2370—131no—
209#2084—131no—
210#2083—131no—
211#2067—131no—
212#2065—131no—
213#1987—131no—
214#1948—131no—
215#1947—131no—
216#1940—131no—
217#1926—131no—
218#1925—131no—
219#1762—131no—
220C23×C13—22no—
221C23×C13—41no—
222C23×C13—41no—
223C23×C13—41no—
224C23×C13—41no—
225C23×C13—41no—
226C23×C13—41no—
227C23×C13C22×D411yescharacteristic
228Q8×C13—84no—
229D4×C13—22no—
230D4×C13—22no—
231D4×C13—41no—
232D4×C13—41no—
233C2×C4×C13—21no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
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—41no—
248C2×C4×C13—41no—
249C2×C4×C13—41no—
250C2×C4×C13—41no—
251C2×C4×C13—41no—
252C2×C4×C13—41no—
253C2×C4×C13—41no—
254C2×C4×C13—41no—
255C2×C4×C13—41no—
256C2×C4×C13—84no—
257C2×C4×C13#12112yes—
258C2×C4×C13#12112yes—
259C2×C4×C13AGL(1,ℤ/8)12yes—
260C2×C4×C13AGL(1,ℤ/8)12yes—
261C2×C4×C13C22×D411yescharacteristic
262C2×C4×C13C22×Q811yescharacteristic
263C2×C4×C13C2×#4111yescharacteristic
264C2×C4×C13C2×#4111yescharacteristic
265C2×C4×C132−1+411yescharacteristic
266C2×C4×C132−1+411yescharacteristic
267C2×Dic13—22no—
268C2×Dic13—22no—
269C2×Dic13—22no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276C2×Dic13—41no—
277C2×Dic13—41no—
278C2×Dic13—41no—
279#1055—22no—
280#1055—22no—
281#1055—41no—
282#1055—41no—
283C2×C4⋊Q8C2×D1311yescharacteristic
284C2×#125—131no—
285C2×C22⋊Q8—21no—
286C2×C22⋊Q8—21no—
287C2×C4×D4C2×D1311yescharacteristic
288#501—264no—
289#478—264no—
290C2×#107—131no—
291C2×#107—131no—
292C2×#107—131no—
293C2×#106—131no—
294C2×#106—131no—
295C2×#106—131no—
296C2×C4⋊C8—131no—
297C2×C4⋊C8—131no—
298C2×C4⋊C8—131no—
299C2×#102—131no—
300C2×#102—131no—
301C2×#102—131no—
302C2×#102—131no—
303C2×#97—131no—
304#399—21no—
305#399—21no—
306#390C2×D1311yescharacteristic
307#390C2×D1311yescharacteristic
308#389—131no—
309#389—131no—
310#389—261no—
311#387—261no—
312#387—261no—
313#384C2×D1311yescharacteristic
314#384C2×D1311yescharacteristic
315C4×C4⋊C4—131no—
316C4×C4⋊C4—131no—
317C4×C4⋊C4—131no—
318C4×C4⋊C4—261no—
319C4×C4⋊C4C2×D1311yescharacteristic
320#340—131no—
321#340—131no—
322#340—131no—
323#340—131no—
324#340—131no—
325#340—131no—
326#336—131no—
327#336—131no—
328C22×C13—42no—
329C22×C13—42no—
330C22×C13—42no—
331C22×C13—81no—
332C22×C13—81no—
333C22×C13—81no—
334C22×C13—81no—
335C22×C13C2×#12111yescharacteristic
336C22×C13#54711yescharacteristic
337C22×C13D4×Q811yescharacteristic
338C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
339C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
340C22×C13#57711yescharacteristic
341C22×C13#57811yescharacteristic
342C4×C13—22no—
343C4×C13—22no—
344C4×C13—22no—
345C4×C13—22no—
346C4×C13—41no—
347C4×C13—41no—
348C4×C13—81no—
349C4×C13—81no—
350C4×C13—81no—
351C4×C13#53311yescharacteristic
352C4×C13#54111yescharacteristic
353C4×C13#54811yescharacteristic
354C4×C13D4×Q811yescharacteristic
355Dic13—42no—
356Dic13—42no—
357Dic13—42no—
358Dic13—81no—
359C22×Q8—21no—
360C22×D4C22×D1311yescharacteristic
361C23×C4—21no—
362C22×C8—131no—
363C22×C8—131no—
364C22×C8—131no—
365C22×C8—131no—
366C4⋊Q8—22no—
367C4⋊Q8—22no—
368#125—134no—
369#125—262no—
370C22⋊Q8—44no—
371C22⋊Q8—44no—
372C4×D4—22no—
373C4×D4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—131no—
380C2×C4⋊C4—131no—
381C2×C4⋊C4—131no—
382C2×C4⋊C4—131no—
383C2×C4⋊C4—261no—
384C2×C4⋊C4—261no—
385C2×C4⋊C4—261no—
386C2×C4⋊C4—261no—
387C2×C4⋊C4C22⋊D1311yescharacteristic
388C2×C4⋊C4C22⋊D1311yescharacteristic
389C2×C4⋊C4C22×D1311yescharacteristic
390C2×C4⋊C4C22×D1311yescharacteristic
391C2×C4⋊C4C22×D1311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C42—261no—
398C2×C42—261no—
399C2×C42—261no—
400C2×C42C22×D1311yescharacteristic
401C2×C42C22×D1311yescharacteristic
402C2×C42C22×D1311yescharacteristic
403#107—262no—
404#107—262no—
405#107—262no—
406#106—262no—
407#106—262no—
408#106—262no—
409C4⋊C8—262no—
410C4⋊C8—262no—
411C4⋊C8—262no—
412#102—134no—
413#102—262no—
414#102—262no—
415#102—262no—
416#97—262no—
417#95—261no—
418#95—261no—
419#95—261no—
420#95—261no—
421#95—261no—
422#95C22⋊D1311yescharacteristic
423#95C22⋊D1311yescharacteristic
424C2×C13—82no—
425C2×C13#242911yescharacteristic
426C2×C13#303811yescharacteristic
427C2×C13#304611yescharacteristic
428C2×C13#334311yescharacteristic
429C2×C13#334511yescharacteristic
430C2×C13#338411yescharacteristic
431C2×C13#338611yescharacteristic
432C24—21no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4Q8×D1312yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—261no—
449C22×C4—261no—
450C22×C4—261no—
451C22×C4—261no—
452C22×C4C2×C22⋊D1311yescharacteristic
453C22×C4C2×C22⋊D1311yescharacteristic
454C22×C4C2×C22⋊D1311yescharacteristic
455C22×C4C2×C22⋊D1311yescharacteristic
456C22×C4C2×C22⋊D1311yescharacteristic
457C22×C4C2×C22⋊D1311yescharacteristic
458C22×C4C23×D1311yescharacteristic
459C2×C8—132no—
460C2×C8—132no—
461C2×C8—261no—
462C2×C8—261no—
463C2×C8—261no—
464C2×C8—261no—
465C2×C8—261no—
466C2×C8—261no—
467C2×C8—261no—
468C2×C8—261no—
469C2×C8—261no—
470C2×C8—261no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—132no—
485C4⋊C4—132no—
486C4⋊C4—132no—
487C4⋊C4—264no—
488C4⋊C4—264no—
489C4⋊C4—261no—
490C4⋊C4—261no—
491C4⋊C4—261no—
492C4⋊C4—261no—
493C4⋊C4—261no—
494C4⋊C4—522no—
495C4⋊C4—1041no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—522no—
506C42—1041no—
507C42—1041no—
508C42D4×D1312yes—
509C42#612912yes—
510C13#2546511yescharacteristic, complemented, Sylow
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C22×C22⋊D1311yescharacteristic
519Q8—84no—
520D4—22no—
521D4—22no—
522D4—41no—
523D4—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—84no—
548C2×C4—262no—
549C2×C4—262no—
550C2×C4—262no—
551C2×C4—521no—
552C2×C4—521no—
553C2×C4—521no—
554C2×C4—521no—
555C2×C4—521no—
556C2×C4—521no—
557C2×C4—521no—
558C2×C4—521no—
559C2×C4—521no—
560C2×C4#8818612yes—
561C2×C4#8818612yes—
562C2×C4#8819212yes—
563C2×C4#8819212yes—
564C2×C4C2×D4×D1311yescharacteristic
565C2×C4C2×#612911yescharacteristic
566C2×C4C2×Q8×D1311yescharacteristic
567C2×C4#8825411yescharacteristic
568C2×C4#41×D1311yescharacteristic
569C2×C4#8825711yescharacteristic
570C8—262no—
571C8—262no—
572C8—521no—
573C8—521no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×#8818611yescharacteristic
582C22C2×#8818611yescharacteristic
583C22C2×#8819211yescharacteristic
584C22Q8×C22⋊D1311yescharacteristic
585C22#1171952911yescharacteristic
586C22#1171954211yescharacteristic
587C22#1171954311yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—522no—
598C4—522no—
599C4—522no—
600C4—1041no—
601C4#1171932711yescharacteristic
602C4#1171933011yescharacteristic
603C4#1171933711yescharacteristic
604C4#1171933811yescharacteristic
605C2—82no—
606C2#1311843111yescharacteristic
607C2#1312431211yescharacteristic
608C2#1312438211yescharacteristic
609C2#1312727011yescharacteristic
610C2#1312730611yescharacteristic
611C2#1312773111yescharacteristic
612C2#1312775711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2542C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
hypercenter#2542D13
hypocenterC13#25465
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2542

Complements

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

NH ≅ G/Ntype
1. C2×#11718741C2semidirect
3. #13117930C2semidirect
4. #13116313C2semidirect
5. #13116308C2semidirect
8. #13115785C2semidirect
9. #13115597C2semidirect
10. #13115592C2semidirect
11. #13115560C2semidirect
110. #88072Q8semidirect
510. C13#25465semidirect

Maximal subgroups

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

subgroupindex
C2×#117187412
C2×#117186702
#131179302
#131163132
#131163082
#131162242
#131162102
#131157852
#131155972
#131155922
#131155602
#131154602
#131154532
#131146612
C13×#25422
#2546513