Finite Group Databasebeta

← #15925855

Subgroups — #15925855

Order 3328.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718518C211yesmaximal, characteristic, complemented
2#13118554C211yesmaximal, characteristic
3#13117863C211yesmaximal, characteristic
4#13116381C211yesmaximal, characteristic, complemented
5#13116330C211yesmaximal, characteristic, complemented
6#13116226C211yesmaximal, characteristic, complemented
7#13116172C211yesmaximal, characteristic, complemented
8#13115605C211yesmaximal, characteristic, complemented
9#13115416C211yesmaximal, characteristic, complemented
10#13115269C211yesmaximal, characteristic
11#13115143C211yesmaximal, characteristic
12#13115101C211yesmaximal, characteristic, complemented
13#13115035C211yesmaximal, characteristic
14#13115019C211yesmaximal, characteristic
15C2×C13×#428C211yesmaximal, characteristic
16C22×C13×M16C2211yescharacteristic
17C2×C13×#117C2211yescharacteristic
18C22×C13×C4⋊C4C2211yescharacteristic
19C2×#88178C2211yescharacteristic
20C2×#88144C2211yescharacteristic
21C2×#88141C2211yescharacteristic
22C13×#428—24no—
23C2×C13×#107C2211yescharacteristic
24C2×C13×#107C2211yescharacteristic
25C2×C13×#106C2211yescharacteristic
26C2×C13×#106C2211yescharacteristic
27#11718873—21no—
28#11718873C2211yescharacteristic
29#11718871—21no—
30C4×C22⋊Dic13C2211yescharacteristic
31C2×C22⋊#1055C2211yescharacteristic
32C2×#88056C2211yescharacteristic
33C2×#88056C2211yescharacteristic
34C2×#88054C2211yescharacteristic
35C2×#88054C2211yescharacteristic
36#11718647—21no—
37#11718646C2211yescharacteristic
38#11718645—21no—
39#11718638C2211yescharacteristic
40C4⋊C4×Dic13C2211yescharacteristic
41#11718635C2211yescharacteristic
42#11718634C2211yescharacteristic
43#11718634C2211yescharacteristic
44C2×#88050C2211yescharacteristic
45C2×#88050C2211yescharacteristic
46C2×#88048C2211yescharacteristic
47C2×#88047C2211yescharacteristic
48#11718606—21no—
49#11718605—21no—
50#11718594—21no—
51C4×#6101—21no—
52#11718518—24no—
53#11718143C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60#11718121C2211yescharacteristic
61C23×C4×C13—21no—
62C23×C4×C13C2311yescharacteristic
63C22×Dic26C2311yescharacteristic
64C2×C13×M16—22no—
65C2×C13×M16—22no—
66C2×C13×M16—22no—
67C22×C8×C13C2311yescharacteristic
68C22×C8×C13C2311yescharacteristic
69C13×#117—22no—
70C13×#117—41no—
71C2×C13×C4⋊C4—22no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4—41no—
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C4⋊C4C2311yescharacteristic
77C2×C13×C4⋊C4C2311yescharacteristic
78C2×C13×C22⋊C4—21no—
79C2×C42×C13—21no—
80C2×C22⋊Dic13—21no—
81C2×C22⋊Dic13—21no—
82C2×C22⋊Dic13D411yescharacteristic
83C2×C22⋊Dic13D411yescharacteristic
84C2×C4⋊Dic13—21no—
85C2×C4⋊Dic13—21no—
86C2×C4⋊Dic13C2311yescharacteristic
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89#88178—22no—
90#88178Q814yes—
91C2×#6101—21no—
92C2×#6101—21no—
93C2×#6101—21no—
94C2×#6101—21no—
95C2×#6101—21no—
96C2×C4×Dic13—21no—
97C2×C4×Dic13—21no—
98C2×C4×Dic13C2311yescharacteristic
99C2×C4×Dic13C2311yescharacteristic
100C22×#1055C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102#88144—22no—
103#88144—41no—
104#88141—22no—
105#88141—41no—
106C13×#107—22no—
107C13×#107—22no—
108C13×#106—22no—
109C13×#106D414yes—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88071—21no—
118#88071—21no—
119#88071D411yescharacteristic
120#88071D411yescharacteristic
121C22⋊#1055—22no—
122#88056—22no—
123#88056—22no—
124#88054—22no—
125#88054—22no—
126#88050—22no—
127#88050—41no—
128#88048—41no—
129#88047—22no—
130#26367—131nomaximal, Sylow
131C24×C13C2×D411yescharacteristic
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13—42no—
141C22×C4×C13—42no—
142C22×C4×C13—81no—
143C22×C4×C13C2×D411yescharacteristic
144C22×C4×C13C2×Q812yes—
145C22×C4×C13#4112yes—
146C22×C4×C13C2411yescharacteristic
147C22×Dic13—21no—
148C22×Dic13—21no—
149C22×Dic13—21no—
150C22×Dic13—21no—
151C22×Dic13C2×D411yescharacteristic
152C22×Dic13C2×D411yescharacteristic
153C22×Dic13C2×D411yescharacteristic
154C22×Dic13C2×D411yescharacteristic
155C2×Dic26—21no—
156C2×Dic26—21no—
157C2×Dic26—21no—
158C2×Dic26—21no—
159C2×Dic26—21no—
160C2×Dic26C2×Q812yes—
161C13×M16—28no—
162C2×C8×C13—21no—
163C2×C8×C13—21no—
164C2×C8×C13—21no—
165C2×C8×C13—21no—
166C2×C8×C13C2×D412yes—
167C2×C8×C13#4112yes—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—21no—
170C13×C4⋊C4—21no—
171C13×C4⋊C4—21no—
172C13×C4⋊C4—21no—
173C13×C4⋊C4—21no—
174C13×C4⋊C4—42no—
175C13×C4⋊C4C2×D412yes—
176C13×C4⋊C4C2×D412yes—
177C13×C22⋊C4—42no—
178C42×C13—42no—
179C22⋊Dic13—24no—
180C22⋊Dic13—41no—
181C22⋊Dic13—41no—
182C22⋊Dic13—42no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—42no—
189C4⋊Dic13—81no—
190C4⋊Dic13C2×Q812yes—
191#6101—24no—
192#6101—42no—
193#6101—42no—
194#6101—42no—
195#6101—81no—
196C4×Dic13—21no—
197C4×Dic13—21no—
198C4×Dic13—21no—
199C4×Dic13—21no—
200C4×Dic13—42no—
201C4×Dic13—42no—
202C2×#1055—21no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#479—131no—
209C2×#428D1311yescharacteristic
210#2442—131no—
211#2362—131no—
212#2097—131no—
213#2087—131no—
214#2068—131no—
215#2059—131no—
216#1949—131no—
217#1919—131no—
218#1894—131no—
219#1872—131no—
220#1865—131no—
221#1854—131no—
222#1851—131no—
223C23×C13—21no—
224C23×C13—21no—
225C23×C13—21no—
226C23×C13—21no—
227C23×C13—21no—
228C23×C13—21no—
229C23×C13C22⋊Q812yes—
230C23×C13C22×D411yescharacteristic
231C2×C4×C13—21no—
232C2×C4×C13—21no—
233C2×C4×C13—22no—
234C2×C4×C13—22no—
235C2×C4×C13—22no—
236C2×C4×C13—22no—
237C2×C4×C13—22no—
238C2×C4×C13—22no—
239C2×C4×C13—41no—
240C2×C4×C13—41no—
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—84no—
250C2×C4×C13C22⋊Q812yes—
251C2×C4×C13C22×D411yescharacteristic
252C2×C4×C13C22×Q811yescharacteristic
253C2×C4×C13C2×#4111yescharacteristic
254C2×C4×C13C2×#4111yescharacteristic
255C2×C4×C13H(5,𝔽2)11yescharacteristic
256C2×C4×C132−1+411yescharacteristic
257C2×Dic13—22no—
258C2×Dic13—22no—
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—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276C2×Dic13—41no—
277C2×Dic13—41no—
278C2×Dic13—41no—
279C2×Dic13—41no—
280C2×Dic13—41no—
281Dic26—22no—
282Dic26—22no—
283Dic26—41no—
284Dic26—41no—
285C8×C13—22no—
286C8×C13—22no—
287#1055—41no—
288#1055—41no—
289C22×M16C2×D1311yescharacteristic
290C2×#125—131no—
291C2×C22⋊Q8—131no—
292C2×C4×Q8—131no—
293C2×#117C2×D1311yescharacteristic
294C22×C4⋊C4C2×D1311yescharacteristic
295#479—264no—
296#428—24no—
297C2×#107—131no—
298C2×#107C2×D1311yescharacteristic
299C2×#107C2×D1311yescharacteristic
300C2×#106—131no—
301C2×#106C2×D1311yescharacteristic
302C2×#106C2×D1311yescharacteristic
303C2×C4⋊C8—131no—
304C2×C4⋊C8—131no—
305C2×#103—131no—
306C2×#103—131no—
307C2×#103—131no—
308C2×#103—131no—
309C2×C22⋊C8—131no—
310#400—261no—
311#398—261no—
312#391—131no—
313#389—131no—
314#389—131no—
315#389—131no—
316#388—261no—
317#388—261no—
318#387—261no—
319#385—261no—
320#382—131no—
321#380—131no—
322#380—261no—
323C4×C4⋊C4—131no—
324C4×C4⋊C4—261no—
325C4×C22⋊C4—131no—
326#340—131no—
327#340—131no—
328#340—131no—
329#340—131no—
330#340—131no—
331#340—131no—
332#336—131no—
333#336—131no—
334C22×C13—22no—
335C22×C13—22no—
336C22×C13—22no—
337C22×C13—41no—
338C22×C13—41no—
339C22×C13—41no—
340C22×C13—41no—
341C22×C13C2×C22⋊Q811yescharacteristic
342C22×C13D4211yescharacteristic
343C22×C13#54711yescharacteristic
344C22×C13#57511yescharacteristic
345C22×C13#57511yescharacteristic
346C22×C13#57711yescharacteristic
347C22×C13#57811yescharacteristic
348C4×C13—22no—
349C4×C13—42no—
350C4×C13—42no—
351C4×C13—81no—
352C4×C13—81no—
353C4×C13#52511yescharacteristic
354C4×C13#53911yescharacteristic
355C4×C13D4×Q811yescharacteristic
356C4×C13#55411yescharacteristic
357Dic13—42no—
358Dic13—41no—
359Dic13—41no—
360Dic13—42no—
361Dic13—41no—
362Dic13—41no—
363Dic13—81no—
364Dic13—81no—
365C22×Q8—131no—
366C23×C4—21no—
367C23×C4C22×D1311yescharacteristic
368C2×M16—22no—
369C2×M16—22no—
370C2×M16—22no—
371C22×C8—131no—
372C22×C8—131no—
373C22×C8C22×D1311yescharacteristic
374C22×C8C22×D1311yescharacteristic
375#125—262no—
376#125—521no—
377C22⋊Q8—134no—
378C22⋊Q8—262no—
379C4×Q8—262no—
380C4×Q8—521no—
381#117—22no—
382#117—41no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—41no—
386C2×C4⋊C4—131no—
387C2×C4⋊C4—131no—
388C2×C4⋊C4—131no—
389C2×C4⋊C4—261no—
390C2×C4⋊C4—261no—
391C2×C4⋊C4—261no—
392C2×C4⋊C4—261no—
393C2×C4⋊C4—261no—
394C2×C4⋊C4—261no—
395C2×C4⋊C4—261no—
396C2×C4⋊C4C22×D1311yescharacteristic
397C2×C4⋊C4C22×D1311yescharacteristic
398C2×C4⋊C4C22×D1311yescharacteristic
399C2×C4⋊C4C22×D1311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—131no—
402C2×C22⋊C4—131no—
403C2×C22⋊C4—261no—
404C2×C22⋊C4—261no—
405C2×C42—21no—
406C2×C42—131no—
407C2×C42—131no—
408C2×C42—261no—
409C2×C42—261no—
410#107—22no—
411#107—22no—
412#107—262no—
413#106—22no—
414#106—521no—
415#106C22⋊D1314yes—
416C4⋊C8—262no—
417C4⋊C8—262no—
418#103—262no—
419#103—262no—
420#103—262no—
421#103—521no—
422C22⋊C8—262no—
423#95—131no—
424#95—131no—
425#95—261no—
426#95—261no—
427#95—261no—
428#95—261no—
429#95—261no—
430#95—261no—
431#95—261no—
432#95—261no—
433#95—261no—
434C2×C13—42no—
435C2×C13#244911yescharacteristic
436C2×C13#308011yescharacteristic
437C2×C13#308211yescharacteristic
438C2×C13#326911yescharacteristic
439C2×C13#327511yescharacteristic
440C2×C13#334911yescharacteristic
441C2×C13#335111yescharacteristic
442C24D4×D1311yescharacteristic
443C2×Q8—132no—
444C2×Q8—261no—
445C2×Q8—261no—
446C2×Q8—261no—
447C2×Q8—261no—
448C2×Q8—261no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—81no—
460C22×C4—131no—
461C22×C4—131no—
462C22×C4—131no—
463C22×C4—131no—
464C22×C4—261no—
465C22×C4—261no—
466C22×C4—261no—
467C22×C4—261no—
468C22×C4#612712yes—
469C22×C4D4×D1311yescharacteristic
470C22×C4Q8×D1312yes—
471C22×C4C23×D1311yescharacteristic
472M16—28no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—261no—
478C2×C8—261no—
479C2×C8—261no—
480C2×C8—261no—
481C2×C8—261no—
482C2×C8—261no—
483C2×C8#612712yes—
484C2×C8C2×C22⋊D1312yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—132no—
493C4⋊C4—264no—
494C4⋊C4—261no—
495C4⋊C4—261no—
496C4⋊C4—261no—
497C4⋊C4—261no—
498C4⋊C4—261no—
499C4⋊C4—522no—
500C4⋊C4—522no—
501C4⋊C4—522no—
502C4⋊C4—522no—
503C4⋊C4—1041no—
504C4⋊C4—1041no—
505C4⋊C4C2×C22⋊D1312yes—
506C4⋊C4C2×C22⋊D1312yes—
507C22⋊C4—42no—
508C22⋊C4—264no—
509C22⋊C4—521no—
510C22⋊C4—521no—
511C22⋊C4—522no—
512C42—42no—
513C42—261no—
514C42—261no—
515C42—261no—
516C42—261no—
517C42—522no—
518C42—522no—
519C13#2636711yescharacteristic, complemented, Sylow
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#8815012yes—
527C23C2×D4×D1311yescharacteristic
528Q8—262no—
529Q8—262no—
530Q8—521no—
531Q8—521no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—84no—
551C2×C4—261no—
552C2×C4—261no—
553C2×C4—262no—
554C2×C4—262no—
555C2×C4—261no—
556C2×C4—261no—
557C2×C4—261no—
558C2×C4—261no—
559C2×C4—261no—
560C2×C4—261no—
561C2×C4—261no—
562C2×C4—261no—
563C2×C4—261no—
564C2×C4—261no—
565C2×C4—521no—
566C2×C4—521no—
567C2×C4—521no—
568C2×C4—521no—
569C2×C4—521no—
570C2×C4—521no—
571C2×C4—521no—
572C2×C4—521no—
573C2×C4—521no—
574C2×C4—521no—
575C2×C4#8815012yes—
576C2×C4C2×#612711yescharacteristic
577C2×C4C2×#612711yescharacteristic
578C2×C4#8825111yescharacteristic
579C2×C4C2×Q8×D1311yescharacteristic
580C2×C4#8825411yescharacteristic
581C2×C4C22×C22⋊D1311yescharacteristic
582C8—22no—
583C8—22no—
584C8—521no—
585C8—521no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22#1171921211yescharacteristic
594C22C2×#8815011yescharacteristic
595C22#1171947811yescharacteristic
596C22#1171948011yescharacteristic
597C22#1171948211yescharacteristic
598C22#1171948411yescharacteristic
599C22D4×C22⋊D1311yescharacteristic
600C4—22no—
601C4—42no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4—522no—
606C4—521no—
607C4—522no—
608C4—521no—
609C4—521no—
610C4—521no—
611C4—1041no—
612C4—1041no—
613C4#1171921611yescharacteristic
614C4#1171922011yescharacteristic
615C4#1171949611yescharacteristic
616C4Q8×C22⋊D1311yescharacteristic
617C2—42no—
618C2#1311864511yescharacteristic
619C2#1312465711yescharacteristic
620C2#1312467611yescharacteristic
621C2#1312638511yescharacteristic
622C2#1312646711yescharacteristic
623C2#1312736711yescharacteristic
624C2#1312739711yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC2×C13×#428C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenterC2×#428D13
hypocenterC13#26367
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#11718518C2semidirect
4. #13116381C2semidirect
5. #13116330C2semidirect
6. #13116226C2semidirect
7. #13116172C2semidirect
8. #13115605C2semidirect
9. #13115416C2semidirect
12. #13115101C2semidirect
519. C13#26367semidirect

Maximal subgroups

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

subgroupindex
C2×#117185182
#131185542
#131178632
#131163812
#131163302
#131162262
#131161722
#131156052
#131154162
#131152692
#131151432
#131151012
#131150352
#131150192
C2×C13×#4282
#2636713