Finite Group Databasebeta

← #15926605

Subgroups — #15926605

Order 3328.

738 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13118543C211yesmaximal, characteristic
2#13117863C211yesmaximal, characteristic
3#13116401C211yesmaximal, characteristic, complemented
4#13116328C211yesmaximal, characteristic, complemented
5#13116232C211yesmaximal, characteristic, complemented
6#13115986C211yesmaximal, characteristic, complemented
7#13115792C211yesmaximal, characteristic, complemented
8#13115412C211yesmaximal, characteristic, complemented
9#13115259C211yesmaximal, characteristic
10#13115148C211yesmaximal, characteristic
11#13115105C211yesmaximal, characteristic, complemented
12#13115035C211yesmaximal, characteristic
13#13115014C211yesmaximal, characteristic
14#13114840C211yesmaximal, characteristic, complemented
15C13×#1735C211yesmaximal, characteristic
16C23×C8×C13C2211yescharacteristic
17C2×C13×#117C2211yescharacteristic
18C2×C13×#117C2211yescharacteristic
19C2×#88178C2211yescharacteristic
20C2×#88144C2211yescharacteristic
21C2×#88141C2211yescharacteristic
22#11718873C2211yescharacteristic
23C4×C22⋊Dic13C2211yescharacteristic
24C2×C22⋊#1055C2211yescharacteristic
25C2×#88056C2211yescharacteristic
26C2×#88056C2211yescharacteristic
27C2×#88054C2211yescharacteristic
28C2×#88054C2211yescharacteristic
29#11718646C2211yescharacteristic
30#11718637C2211yescharacteristic
31C4⋊C4×Dic13C2211yescharacteristic
32#11718635C2211yescharacteristic
33#11718634C2211yescharacteristic
34#11718634C2211yescharacteristic
35C2×#88050C2211yescharacteristic
36C2×#88050C2211yescharacteristic
37C2×#88048C2211yescharacteristic
38C2×#88047C2211yescharacteristic
39#11718610—21no—
40#11718609—21no—
41#11718606—21no—
42#11718605—21no—
43#11718594—21no—
44#11718594—21no—
45C4×#6101—21no—
46#11718589—21no—
47C13×#340C2211yescharacteristic
48C13×#340C2211yescharacteristic
49C13×#340C2211yescharacteristic
50C13×#340C2211yescharacteristic
51#11718143C2211yescharacteristic
52#11718143C2211yescharacteristic
53#11718142C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59C23×C4×C13C2311yescharacteristic
60C22×Dic26C2311yescharacteristic
61C22×C8×C13—22no—
62C22×C8×C13—41no—
63C22×C8×C13—41no—
64C22×C8×C13C2311yescharacteristic
65C22×C8×C13C2311yescharacteristic
66C13×#117—22no—
67C13×#117—22no—
68C13×#117—41no—
69C13×#117—41no—
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C22⋊C4—21no—
75C2×C13×C22⋊C4—21no—
76C2×C42×C13—21no—
77C2×C42×C13—21no—
78C2×C22⋊Dic13—21no—
79C2×C22⋊Dic13—21no—
80C2×C22⋊Dic13D411yescharacteristic
81C2×C22⋊Dic13D411yescharacteristic
82C2×C4⋊Dic13—21no—
83C2×C4⋊Dic13—21no—
84C2×C4⋊Dic13C2311yescharacteristic
85C2×C4⋊Dic13C2311yescharacteristic
86C2×C4⋊Dic13C2311yescharacteristic
87#88178—22no—
88#88178Q814yes—
89C2×#6101—21no—
90C2×#6101—21no—
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—
106#88071—21no—
107#88071—21no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071D411yescharacteristic
114#88071D411yescharacteristic
115C22⋊#1055—22no—
116#88056—41no—
117#88056—41no—
118#88054—41no—
119#88054—41no—
120#88050—22no—
121#88050—41no—
122#88048—22no—
123#88047—41no—
124#26417—131nomaximal, Sylow
125C24×C13C2×D411yescharacteristic
126C22×C4×C13—21no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13C2×D411yescharacteristic
135C22×C4×C13C2×Q812yes—
136C22×C4×C13#4112yes—
137C22×C4×C13C2411yescharacteristic
138C22×Dic13—21no—
139C22×Dic13—21no—
140C22×Dic13—21no—
141C22×Dic13—21no—
142C22×Dic13C2×D411yescharacteristic
143C22×Dic13C2×D411yescharacteristic
144C22×Dic13C2×D411yescharacteristic
145C22×Dic13C2×D411yescharacteristic
146C2×Dic26—21no—
147C2×Dic26—21no—
148C2×Dic26—21no—
149C2×Dic26—21no—
150C2×Dic26—21no—
151C2×Dic26C2×Q812yes—
152C2×C8×C13—21no—
153C2×C8×C13—21no—
154C2×C8×C13—21no—
155C2×C8×C13—21no—
156C2×C8×C13—21no—
157C2×C8×C13—21no—
158C2×C8×C13—44no—
159C13×C4⋊C4—21no—
160C13×C4⋊C4—21no—
161C13×C4⋊C4—21no—
162C13×C4⋊C4—21no—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4#4112yes—
166C13×C4⋊C4#4112yes—
167C13×C22⋊C4—42no—
168C13×C22⋊C4—42no—
169C42×C13—42no—
170C42×C13—42no—
171C22⋊Dic13—24no—
172C22⋊Dic13—22no—
173C22⋊Dic13—41no—
174C22⋊Dic13—81no—
175C4⋊Dic13—21no—
176C4⋊Dic13—21no—
177C4⋊Dic13—21no—
178C4⋊Dic13—21no—
179C4⋊Dic13—21no—
180C4⋊Dic13—42no—
181C4⋊Dic13—42no—
182C4⋊Dic13C2×Q812yes—
183#6101—24no—
184#6101—42no—
185#6101—42no—
186#6101—42no—
187#6101—42no—
188#6101—81no—
189#6101—81no—
190C4×Dic13—21no—
191C4×Dic13—21no—
192C4×Dic13—21no—
193C4×Dic13—21no—
194C4×Dic13—42no—
195C4×Dic13—42no—
196C2×#1055—21no—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202#2441—131no—
203#2362—131no—
204#2100—131no—
205#2086—131no—
206#2068—131no—
207#2026—131no—
208#1989—131no—
209#1919—131no—
210#1892—131no—
211#1873—131no—
212#1865—131no—
213#1854—131no—
214#1850—131no—
215#1803—131no—
216#1735D1311yescharacteristic
217C23×C13—21no—
218C23×C13—21no—
219C23×C13—21no—
220C23×C13—21no—
221C23×C13—21no—
222C23×C13—21no—
223C23×C13C22⋊Q812yes—
224C23×C13C22×D411yescharacteristic
225C2×C4×C13—21no—
226C2×C4×C13—21no—
227C2×C4×C13—22no—
228C2×C4×C13—22no—
229C2×C4×C13—22no—
230C2×C4×C13—22no—
231C2×C4×C13—22no—
232C2×C4×C13—22no—
233C2×C4×C13—41no—
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—41no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13C22⋊Q812yes—
244C2×C4×C13C22×Q811yescharacteristic
245C2×C4×C13C2×#4111yescharacteristic
246C2×C4×C13C2×#4111yescharacteristic
247C2×C4×C13C2×#4111yescharacteristic
248C2×C4×C13H(5,𝔽2)11yescharacteristic
249C2×C4×C13H(5,𝔽2)11yescharacteristic
250C2×Dic13—21no—
251C2×Dic13—21no—
252C2×Dic13—22no—
253C2×Dic13—22no—
254C2×Dic13—21no—
255C2×Dic13—21no—
256C2×Dic13—21no—
257C2×Dic13—21no—
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—21no—
262C2×Dic13—21no—
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—41no—
272C2×Dic13—41no—
273C2×Dic13#13512yes—
274Dic26—22no—
275Dic26—22no—
276Dic26—41no—
277Dic26—41no—
278C8×C13—41no—
279C8×C13—41no—
280#1055—41no—
281#1055—41no—
282C23×C8C2×D1311yescharacteristic
283C2×#125—131no—
284C2×C22⋊Q8—131no—
285C2×C4×Q8—131no—
286C2×#117C2×D1311yescharacteristic
287C2×#117C2×D1311yescharacteristic
288C2×#107—131no—
289C2×#106—131no—
290C2×C4⋊C8—131no—
291C2×C4⋊C8—131no—
292C2×#103—131no—
293C2×#103—131no—
294C2×#103—131no—
295C2×#103—131no—
296C2×C22⋊C8—131no—
297#399—261no—
298#396—261no—
299#391—131no—
300#389—131no—
301#389—131no—
302#389—131no—
303#389—261no—
304#388—261no—
305#388—261no—
306#387—261no—
307#387—261no—
308#384—131no—
309#380—131no—
310C4×C4⋊C4—131no—
311C4×C4⋊C4—261no—
312C4×C22⋊C4—131no—
313#340—131no—
314#340—131no—
315#340—131no—
316#340—131no—
317#340—131no—
318#340—131no—
319#340C2×D1311yescharacteristic
320#340C2×D1311yescharacteristic
321#340C2×D1311yescharacteristic
322#340C2×D1311yescharacteristic
323#336—131no—
324#336—131no—
325C22×C13—22no—
326C22×C13—22no—
327C22×C13—22no—
328C22×C13—41no—
329C22×C13—41no—
330C22×C13—41no—
331C22×C13—41no—
332C22×C13C2×C22⋊Q811yescharacteristic
333C22×C13#54611yescharacteristic
334C22×C13#54611yescharacteristic
335C22×C13C2×#13511yescharacteristic
336C22×C13#57511yescharacteristic
337C22×C13#57711yescharacteristic
338C22×C13#57811yescharacteristic
339C4×C13—22no—
340C4×C13—42no—
341C4×C13—42no—
342C4×C13—81no—
343C4×C13—81no—
344C4×C13#52911yescharacteristic
345C4×C13#53811yescharacteristic
346C4×C13#55411yescharacteristic
347C4×C13#55411yescharacteristic
348Dic13—22no—
349Dic13—42no—
350Dic13—42no—
351Dic13—41no—
352Dic13—41no—
353Dic13—41no—
354Dic13—81no—
355Dic13—81no—
356C22×Q8—131no—
357C23×C4C22×D1311yescharacteristic
358C22×C8—22no—
359C22×C8—41no—
360C22×C8—41no—
361C22×C8—131no—
362C22×C8—131no—
363C22×C8C22×D1311yescharacteristic
364C22×C8C22×D1311yescharacteristic
365#125—262no—
366#125—521no—
367C22⋊Q8—134no—
368C22⋊Q8—262no—
369C4×Q8—262no—
370C4×Q8—521no—
371#117—22no—
372#117—22no—
373#117—41no—
374#117—41no—
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⋊C4—261no—
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×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—131no—
394C2×C22⋊C4—131no—
395C2×C22⋊C4—261no—
396C2×C22⋊C4—261no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—131no—
400C2×C42—131no—
401C2×C42—261no—
402C2×C42—261no—
403#107—521no—
404#106—262no—
405C4⋊C8—521no—
406C4⋊C8—521no—
407#103—262no—
408#103—521no—
409#103—521no—
410#103—521no—
411C22⋊C8—262no—
412#95—131no—
413#95—131no—
414#95—261no—
415#95—261no—
416#95—261no—
417#95—261no—
418#95—261no—
419#95—261no—
420#95—261no—
421C2×C13—42no—
422C2×C13#245011yescharacteristic
423C2×C13#307611yescharacteristic
424C2×C13#308211yescharacteristic
425C2×C13#330111yescharacteristic
426C2×C13#330211yescharacteristic
427C2×C13#331511yescharacteristic
428C2×C13#331711yescharacteristic
429C24D4×D1311yescharacteristic
430C2×Q8—132no—
431C2×Q8—261no—
432C2×Q8—261no—
433C2×Q8—261no—
434C2×Q8—261no—
435C2×Q8—261no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—131no—
445C22×C4—131no—
446C22×C4—131no—
447C22×C4—131no—
448C22×C4—261no—
449C22×C4—261no—
450C22×C4—261no—
451C22×C4—261no—
452C22×C4#612712yes—
453C22×C4D4×D1311yescharacteristic
454C22×C4Q8×D1312yes—
455C22×C4C23×D1311yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—44no—
463C2×C8—261no—
464C2×C8—261no—
465C2×C8—261no—
466C2×C8—261no—
467C2×C8—261no—
468C2×C8—261no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—132no—
476C4⋊C4—264no—
477C4⋊C4—261no—
478C4⋊C4—261no—
479C4⋊C4—261no—
480C4⋊C4—261no—
481C4⋊C4—261no—
482C4⋊C4—522no—
483C4⋊C4—522no—
484C4⋊C4—522no—
485C4⋊C4—522no—
486C4⋊C4—522no—
487C4⋊C4—522no—
488C4⋊C4—1041no—
489C4⋊C4—1041no—
490C4⋊C4#612712yes—
491C4⋊C4#612712yes—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—262no—
495C22⋊C4—264no—
496C22⋊C4—521no—
497C22⋊C4—1041no—
498C42—42no—
499C42—42no—
500C42—261no—
501C42—261no—
502C42—261no—
503C42—261no—
504C42—522no—
505C42—522no—
506C13#2641711yescharacteristic, complemented, Sylow
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#8815012yes—
514C23C2×D4×D1311yescharacteristic
515Q8—262no—
516Q8—262no—
517Q8—521no—
518Q8—521no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—132no—
538C2×C4—261no—
539C2×C4—261no—
540C2×C4—261no—
541C2×C4—261no—
542C2×C4—261no—
543C2×C4—262no—
544C2×C4—262no—
545C2×C4—261no—
546C2×C4—261no—
547C2×C4—261no—
548C2×C4—261no—
549C2×C4—261no—
550C2×C4—261no—
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—521no—
561C2×C4#8815012yes—
562C2×C4C2×#612711yescharacteristic
563C2×C4C2×#612711yescharacteristic
564C2×C4C2×#612711yescharacteristic
565C2×C4#8825111yescharacteristic
566C2×C4#8825111yescharacteristic
567C2×C4C2×Q8×D1311yescharacteristic
568C8—41no—
569C8—41no—
570C8—521no—
571C8—521no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22#1171919511yescharacteristic
580C22#1171919511yescharacteristic
581C22C2×#8815011yescharacteristic
582C22D13×#13511yescharacteristic
583C22#1171947311yescharacteristic
584C22#1171947511yescharacteristic
585C22#1171947611yescharacteristic
586C4—22no—
587C4—42no—
588C4—42no—
589C4—81no—
590C4—81no—
591C4—262no—
592C4—522no—
593C4—521no—
594C4—521no—
595C4—521no—
596C4—522no—
597C4—1041no—
598C4—1041no—
599C4#1171918411yescharacteristic
600C4#1171919911yescharacteristic
601C4#1171921611yescharacteristic
602C4#1171921611yescharacteristic
603C2—42no—
604C2#1311866011yescharacteristic
605C2#1312462211yescharacteristic
606C2#1312468111yescharacteristic
607C2#1312673911yescharacteristic
608C2#1312674911yescharacteristic
609C2#1312692811yescharacteristic
610C2#1312695811yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#1735C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenter#1735D13
hypocenterC13#26417
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
3. #13116401C2semidirect
4. #13116328C2semidirect
5. #13116232C2semidirect
6. #13115986C2semidirect
7. #13115792C2semidirect
8. #13115412C2semidirect
11. #13115105C2semidirect
14. #13114840C2semidirect
506. C13#26417semidirect

Maximal subgroups

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

subgroupindex
#131185432
#131178632
#131164012
#131163282
#131162322
#131159862
#131157922
#131154122
#131152592
#131151482
#131151052
#131150352
#131150142
#131148402
C13×#17352
#2641713