Finite Group Databasebeta

← #15914813

Subgroups — #15914813

Order 3328.

774 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
1C2×#11718727C211yesmaximal, characteristic
2C2×Q8×#1055C211yesmaximal, characteristic, complemented
3#13120194C211yesmaximal, characteristic
4#13116337C211yesmaximal, characteristic, complemented
5#13116332C211yesmaximal, characteristic, complemented
6#13116221C211yesmaximal, characteristic
7#13116207C211yesmaximal, characteristic
8#13115802C211yesmaximal, characteristic, complemented
9#13115604C211yesmaximal, characteristic, complemented
10#13115566C211yesmaximal, characteristic, complemented
11#13115561C211yesmaximal, characteristic, complemented
12#13115518C211yesmaximal, characteristic
13#13115518C211yesmaximal, characteristic
14C4×#88049C211yesmaximal, characteristic
15C13×#2584C211yesmaximal, characteristic, complemented
16C2×C13×#125C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×D52C2211yescharacteristic
19C13×#398—21no—
20C13×#395—21no—
21C13×#389—21no—
22C13×#389—21no—
23C13×#389C2211yescharacteristic
24C13×#389C2211yescharacteristic
25C13×#389C2211yescharacteristic
26C13×#389C2211yescharacteristic
27C4×C13×C4⋊C4C2211yescharacteristic
28#11718892—21no—
29#11718727—24no—
30Q8×#1055—24no—
31#11718653—21no—
32#11718650—21no—
33#11718650—21no—
34C2×#88049C2211yescharacteristic
35C2×#88049C2211yescharacteristic
36C2×#88049C2211yescharacteristic
37C2×#88049C2211yescharacteristic
38C2×#88048C2211yescharacteristic
39C2×#88048C2211yescharacteristic
40C2×#88047C2211yescharacteristic
41C2×#88047C2211yescharacteristic
42#11718600C2211yescharacteristic
43#11718600C2211yescharacteristic
44#11718597C2211yescharacteristic
45#11718589C2211yescharacteristic
46#11718589C2211yescharacteristic
47C2×C4⋊#1055C2211yescharacteristic
48C2×C4⋊#1055C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50C2×C4×#1055C2211yescharacteristic
51C2×C4×#1055C2211yescharacteristic
52C2×C4×#1055C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60#11718121C2211yescharacteristic
61C22×Q8×C13D411yescharacteristic
62C22×D52C2311yescharacteristic
63C22×C4×D13—21no—
64C13×#125—22no—
65C13×#125—22no—
66C4×Q8×C13—22no—
67C4×Q8×C13—22no—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4—21no—
74C2×C13×C4⋊C4D411yescharacteristic
75C2×C13×C4⋊C4D411yescharacteristic
76C2×C13×C4⋊C4D411yescharacteristic
77C2×C13×C4⋊C4C2311yescharacteristic
78C2×C13×C4⋊C4C2311yescharacteristic
79C2×C13×C4⋊C4C2311yescharacteristic
80C2×C13×C4⋊C4C2311yescharacteristic
81C2×C42×C13—21no—
82C2×C42×C13C2311yescharacteristic
83C2×C42×C13C2311yescharacteristic
84C2×C42×C13C2311yescharacteristic
85C2×#6103—21no—
86C2×#6103—21no—
87C2×#6103—21no—
88C2×#6103—21no—
89C2×#6103—21no—
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×#6101—21no—
94C2×#6101—21no—
95C22×#1055C2311yescharacteristic
96C22×#1055C2311yescharacteristic
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C4×D52—22no—
100C4×D52—22no—
101C13×#95—21no—
102C13×#95—21no—
103C13×#95—21no—
104C13×#95—21no—
105C13×#95—21no—
106#88071—21no—
107#88071—21no—
108#88071—21no—
109#88071—21no—
110#88049—22no—
111#88049—22no—
112#88049—22no—
113#88049Q814yescomplemented
114#88048—22no—
115#88048—22no—
116#88047—22no—
117#88047—22no—
118C4⋊#1055—22no—
119C4⋊#1055—22no—
120C4⋊#1055D414yes—
121C4×#1055—22no—
122C4×#1055—22no—
123C4×#1055—22no—
124#25473—131nomaximal, Sylow
125C23×D13—21no—
126C2×Q8×C13—22no—
127C2×Q8×C13—22no—
128C2×Q8×C13—22no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13C2×D411yescharacteristic
134C22×C4×C13C2×D411yescharacteristic
135C22×C4×C13C2×D411yescharacteristic
136C22×C4×C13C2×D411yescharacteristic
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2411yescharacteristic
140C22×Dic13—21no—
141C22×Dic13—21no—
142C22×Dic13—21no—
143C2×D52—21no—
144C2×D52—21no—
145C2×D52—21no—
146C2×D52—21no—
147C2×D52—21no—
148C2×D52C2×Q812yes—
149C2×C4×D13—42no—
150C2×C4×D13—81no—
151C2×C4×D13—81no—
152C13×C4⋊C4—22no—
153C13×C4⋊C4—22no—
154C13×C4⋊C4—22no—
155C13×C4⋊C4—21no—
156C13×C4⋊C4—21no—
157C13×C4⋊C4—21no—
158C13×C4⋊C4—21no—
159C13×C4⋊C4—21no—
160C13×C4⋊C4—21no—
161C13×C4⋊C4—42no—
162C13×C4⋊C4—42no—
163C13×C4⋊C4—42no—
164C13×C4⋊C4—42no—
165C13×C4⋊C4—81no—
166C13×C4⋊C4—81no—
167C13×C4⋊C4C2×Q812yes—
168C13×C4⋊C4#4112yes—
169C42×C13—21no—
170C42×C13—21no—
171C42×C13—21no—
172C42×C13—21no—
173C42×C13—81no—
174C42×C13C2×D412yes—
175C42×C13#4112yes—
176#6103—42no—
177#6103—42no—
178#6103—42no—
179#6103—81no—
180#6103—81no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187#6101—81no—
188#6101—81no—
189C2×#1055—21no—
190C2×#1055—21no—
191C2×#1055—21no—
192C2×#1055—21no—
193C2×#1055—21no—
194C2×#1055—21no—
195C2×#1055—21no—
196C2×#1055—21no—
197C2×#1055—21no—
198C2×#1055C2×D412yes—
199C2×#1055C2×D412yes—
200C2×#1055C2×Q812yes—
201C2×#478—131no—
202C2×C8×Q8—131no—
203#2584D1311yescharacteristic, complemented
204#2580—131no—
205#2088—131no—
206#2087—131no—
207#2067—131no—
208#2065—131no—
209#1991—131no—
210#1949—131no—
211#1942—131no—
212#1941—131no—
213#1934—131no—
214#1934—131no—
215C4×#102—131no—
216C23×C13C22×D411yescharacteristic
217C22×D13—22no—
218C22×D13—41no—
219C22×D13—41no—
220C22×D13—41no—
221C22×D13—41no—
222C22×D13—41no—
223C22×D13—41no—
224Q8×C13—28no—
225C2×C4×C13—21no—
226C2×C4×C13—21no—
227C2×C4×C13—21no—
228C2×C4×C13—21no—
229C2×C4×C13—21no—
230C2×C4×C13—21no—
231C2×C4×C13—21no—
232C2×C4×C13—21no—
233C2×C4×C13—21no—
234C2×C4×C13—21no—
235C2×C4×C13—22no—
236C2×C4×C13—22no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—21no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13—41no—
244C2×C4×C13—41no—
245C2×C4×C13—41no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—41no—
249C2×C4×C13—41no—
250C2×C4×C13—41no—
251C2×C4×C13#12112yes—
252C2×C4×C13#12112yes—
253C2×C4×C13#13512yes—
254C2×C4×C13#13512yes—
255C2×C4×C13C22×D411yescharacteristic
256C2×C4×C13C22×Q811yescharacteristic
257C2×C4×C13C2×#4111yescharacteristic
258C2×C4×C13C2×#4111yescharacteristic
259C2×C4×C132−1+411yescharacteristic
260C2×C4×C132−1+411yescharacteristic
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—
270D52—22no—
271D52—22no—
272D52—41no—
273D52—41no—
274C4×D13—84no—
275#1055—22no—
276#1055—22no—
277#1055—22no—
278#1055—41no—
279C2×#125C2×D1311yescharacteristic
280C2×C4×Q8C2×D1311yescharacteristic
281C2×C4×D4—131no—
282#478—264no—
283C8×Q8—264no—
284C2×#107—131no—
285C2×#107—131no—
286C2×#106—131no—
287C2×#106—131no—
288C2×C4⋊C8—131no—
289C2×C4⋊C8—131no—
290C2×C4⋊C8—131no—
291C2×#102—131no—
292C2×#102—131no—
293C2×#102—131no—
294C2×#102—131no—
295C2×C4×C8—131no—
296C2×C4×C8—131no—
297C2×C4×C8—131no—
298#399—261no—
299#398—21no—
300#395—21no—
301#393—261no—
302#390—131no—
303#390—131no—
304#389—21no—
305#389—21no—
306#389—131no—
307#389—131no—
308#389C2×D1311yescharacteristic
309#389C2×D1311yescharacteristic
310#389C2×D1311yescharacteristic
311#389C2×D1311yescharacteristic
312#385—261no—
313#385—261no—
314#382—131no—
315C4×C4⋊C4C2×D1311yescharacteristic
316#340—131no—
317#340—131no—
318#340—131no—
319#340—131no—
320#340—131no—
321#340—131no—
322#340—131no—
323#340—131no—
324C22×C13C2×#12111yescharacteristic
325C22×C13#54811yescharacteristic
326C22×C13D4×Q811yescharacteristic
327C22×C13C2×#13511yescharacteristic
328C22×C13C2×#13511yescharacteristic
329C22×C13#57611yescharacteristic
330C22×C13#57711yescharacteristic
331C2×D13—42no—
332C2×D13—42no—
333C2×D13—42no—
334C2×D13—81no—
335C2×D13—81no—
336C2×D13—81no—
337C2×D13—81no—
338C4×C13—22no—
339C4×C13—22no—
340C4×C13—22no—
341C4×C13—22no—
342C4×C13—42no—
343C4×C13—42no—
344C4×C13—41no—
345C4×C13—41no—
346C4×C13—81no—
347C4×C13—81no—
348C4×C13#53311yescharacteristic
349C4×C13#54111yescharacteristic
350C4×C13#54711yescharacteristic
351C4×C13D4×Q811yescharacteristic
352Dic13—81no—
353Dic13—81no—
354Dic13—81no—
355C22×Q8C22⋊D1311yescharacteristic
356C22×D4—131no—
357C23×C4—261no—
358C22×C8—131no—
359C22×C8—131no—
360C22×C8—131no—
361C22×C8—131no—
362#125—22no—
363#125—22no—
364C4×Q8—22no—
365C4×Q8—22no—
366C4×D4—262no—
367C4×D4—262no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—131no—
375C2×C4⋊C4—131no—
376C2×C4⋊C4—131no—
377C2×C4⋊C4—261no—
378C2×C4⋊C4—261no—
379C2×C4⋊C4C22⋊D1311yescharacteristic
380C2×C4⋊C4C22⋊D1311yescharacteristic
381C2×C4⋊C4C22⋊D1311yescharacteristic
382C2×C4⋊C4C22×D1311yescharacteristic
383C2×C4⋊C4C22×D1311yescharacteristic
384C2×C4⋊C4C22×D1311yescharacteristic
385C2×C4⋊C4C22×D1311yescharacteristic
386C2×C22⋊C4—261no—
387C2×C22⋊C4—261no—
388C2×C22⋊C4—261no—
389C2×C22⋊C4—261no—
390C2×C22⋊C4—261no—
391C2×C42—21no—
392C2×C42C22×D1311yescharacteristic
393C2×C42C22×D1311yescharacteristic
394C2×C42C22×D1311yescharacteristic
395#107—262no—
396#107—262no—
397#106—262no—
398#106—262no—
399C4⋊C8—134no—
400C4⋊C8—262no—
401C4⋊C8—262no—
402#102—134no—
403#102—262no—
404#102—262no—
405#102—262no—
406C4×C8—262no—
407C4×C8—262no—
408C4×C8—262no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—261no—
415#95—261no—
416#95—261no—
417#95—261no—
418C2×C13#242911yescharacteristic
419C2×C13#305011yescharacteristic
420C2×C13#306211yescharacteristic
421C2×C13#336111yescharacteristic
422C2×C13#336511yescharacteristic
423C2×C13#338011yescharacteristic
424C2×C13#338111yescharacteristic
425D13—82no—
426C24—261no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×D4—132no—
431C2×D4—261no—
432C2×D4—261no—
433C2×D4—261no—
434C2×D4—261no—
435C2×D4—261no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—261no—
441C22×C4—261no—
442C22×C4—261no—
443C22×C4—522no—
444C22×C4—1041no—
445C22×C4—1041no—
446C22×C4C2×C22⋊D1311yescharacteristic
447C22×C4C2×C22⋊D1311yescharacteristic
448C22×C4C2×C22⋊D1311yescharacteristic
449C22×C4C2×C22⋊D1311yescharacteristic
450C22×C4C2×C22⋊D1311yescharacteristic
451C22×C4C2×C22⋊D1311yescharacteristic
452C22×C4C23×D1311yescharacteristic
453C2×C8—132no—
454C2×C8—132no—
455C2×C8—132no—
456C2×C8—261no—
457C2×C8—261no—
458C2×C8—261no—
459C2×C8—261no—
460C2×C8—261no—
461C2×C8—261no—
462C2×C8—261no—
463C2×C8—261no—
464C2×C8—261no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—261no—
481C4⋊C4—261no—
482C4⋊C4—261no—
483C4⋊C4—261no—
484C4⋊C4—261no—
485C4⋊C4—261no—
486C4⋊C4—1041no—
487C4⋊C4—1041no—
488C4⋊C4Q8×D1312yes—
489C4⋊C4#613112yes—
490C22⋊C4—522no—
491C22⋊C4—522no—
492C22⋊C4—522no—
493C22⋊C4—1041no—
494C22⋊C4—1041no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—81no—
500C42D4×D1312yes—
501C42#612912yes—
502C13#2547311yescharacteristic, complemented, Sylow
503C23—262no—
504C23—521no—
505C23—521no—
506C23—521no—
507C23—521no—
508C23—521no—
509C23—521no—
510C23C22×C22⋊D1311yescharacteristic
511Q8—28no—
512D4—262no—
513D4—262no—
514D4—521no—
515D4—521no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
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—1044no—
552C2×C4#8819312yes—
553C2×C4#8819312yes—
554C2×C4#8820412yes—
555C2×C4#8820412yes—
556C2×C4C2×D4×D1311yescharacteristic
557C2×C4C2×#612911yescharacteristic
558C2×C4C2×Q8×D1311yescharacteristic
559C2×C4C2×#613111yescharacteristic
560C2×C4#8825411yescharacteristic
561C2×C4#8825411yescharacteristic
562C8—262no—
563C8—262no—
564C8—262no—
565C8—521no—
566C22—522no—
567C22—522no—
568C22—522no—
569C22—1041no—
570C22—1041no—
571C22—1041no—
572C22—1041no—
573C22C2×#8819311yescharacteristic
574C22Q8×C22⋊D1311yescharacteristic
575C22#1171952211yescharacteristic
576C22C2×#8820411yescharacteristic
577C22C2×#8820411yescharacteristic
578C22#1171954011yescharacteristic
579C22#1171954211yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—1041no—
591C4—1041no—
592C4—1041no—
593C4#1171943111yescharacteristic
594C4#1171943211yescharacteristic
595C4#1171943311yescharacteristic
596C4#1171943611yescharacteristic
597C2—1042no—
598C2#1311843811yescharacteristic
599C2#1312441411yescharacteristic
600C2#1312451811yescharacteristic
601C2#1312749611yescharacteristic
602C2#1312753211yescharacteristic
603C2#1312769911yescharacteristic
604C2#1312770711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2584C2
socleC23×C13C22×D4
prime radicalC22×D52C23
hypercenter#2584D13
hypocenterC13#25473
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2584

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#1055C2semidirect
4. #13116337C2semidirect
5. #13116332C2semidirect
8. #13115802C2semidirect
9. #13115604C2semidirect
10. #13115566C2semidirect
11. #13115561C2semidirect
15. C13×#2584C2semidirect
113. #88049Q8semidirect
203. #2584D13semidirect
502. C13#25473semidirect

Maximal subgroups

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

subgroupindex
C2×#117187272
C2×Q8×#10552
#131201942
#131163372
#131163322
#131162212
#131162072
#131158022
#131156042
#131155662
#131155612
#131155182
C4×#880492
C13×#25842
#2547313