Finite Group Databasebeta

← #15914594

Subgroups — #15914594

Order 3328.

754 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718725C211yesmaximal, characteristic, complemented
2C2×#11718670C211yesmaximal, characteristic
3#13117930C211yesmaximal, characteristic, complemented
4#13116318C211yesmaximal, characteristic, complemented
5#13116308C211yesmaximal, characteristic, complemented
6#13116210C211yesmaximal, characteristic
7#13116210C211yesmaximal, characteristic
8#13115794C211yesmaximal, characteristic, complemented
9#13115602C211yesmaximal, characteristic, complemented
10#13115597C211yesmaximal, characteristic, complemented
11#13115555C211yesmaximal, characteristic, complemented
12#13115453C211yesmaximal, characteristic
13#13115453C211yesmaximal, characteristic
14C4×#88072C211yesmaximal, characteristic
15C13×#2566C211yesmaximal, characteristic
16C2×C13×#125C2211yescharacteristic
17C2×C4×D4×C13C2211yescharacteristic
18C2×#88124C2211yescharacteristic
19C13×#396—21no—
20C13×#396—21no—
21C13×#390C2211yescharacteristic
22C13×#390C2211yescharacteristic
23C13×#385—21no—
24C13×#385—21no—
25C13×#384C2211yescharacteristic
26C13×#382C2211yescharacteristic
27C4×C13×C4⋊C4C2211yescharacteristic
28C2×#88072C2211yescharacteristic
29C2×#88072C2211yescharacteristic
30C2×#88072C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32#11718725—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×C4×#1055C2211yescharacteristic
51#11718313—21no—
52#11718310—21no—
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C22×D4×C13C2311yescharacteristic
62C23×C4×C13—21no—
63C13×#125—22no—
64C13×#125—22no—
65C4×D4×C13—22no—
66C4×D4×C13—22no—
67C2×C13×C4⋊C4—21no—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4D411yescharacteristic
72C2×C13×C4⋊C4D411yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C22⋊C4—21no—
77C2×C13×C22⋊C4—21no—
78C2×C13×C22⋊C4—21no—
79C2×C13×C22⋊C4—21no—
80C2×C13×C22⋊C4—21no—
81C2×C42×C13C2311yescharacteristic
82C2×C42×C13C2311yescharacteristic
83C2×C42×C13C2311yescharacteristic
84C2×C4⋊Dic13C2311yescharacteristic
85C2×C4⋊Dic13C2311yescharacteristic
86C2×C4⋊Dic13C2311yescharacteristic
87C2×C4⋊Dic13C2311yescharacteristic
88C2×#6101—21no—
89C2×#6101—21no—
90C2×#6101—21no—
91C2×#6101—21no—
92C2×C4×Dic13—21no—
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C22×#1055C2311yescharacteristic
96C22×#1055C2311yescharacteristic
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99#88124—22no—
100#88124D414yes—
101C13×#95—21no—
102C13×#95—21no—
103C13×#95D411yescharacteristic
104C13×#95D411yescharacteristic
105#88072—22no—
106#88072—22no—
107#88072—22no—
108#88072Q814yes—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88048—22no—
115#88048—22no—
116#88048—22no—
117#88047—22no—
118#88047—22no—
119#88047—22no—
120C4⋊#1055—22no—
121C4⋊#1055—22no—
122C4⋊#1055—22no—
123C4×#1055—22no—
124#25458—131nomaximal, Sylow
125C24×C13—21no—
126C2×D4×C13—21no—
127C2×D4×C13—21no—
128C2×D4×C13—21no—
129C2×D4×C13—21no—
130C2×D4×C13—21no—
131C2×D4×C13C2×Q812yes—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—42no—
136C22×C4×C13—81no—
137C22×C4×C13—81no—
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2×D411yescharacteristic
142C22×C4×C13C2×D411yescharacteristic
143C22×C4×C13C2×D411yescharacteristic
144C22×C4×C13C2411yescharacteristic
145C22×Dic13—21no—
146C22×Dic13—21no—
147C22×Dic13—21no—
148C22×Dic13—21no—
149C13×C4⋊C4—22no—
150C13×C4⋊C4—21no—
151C13×C4⋊C4—21no—
152C13×C4⋊C4—21no—
153C13×C4⋊C4—21no—
154C13×C4⋊C4—21no—
155C13×C4⋊C4—21no—
156C13×C4⋊C4—41no—
157C13×C4⋊C4—42no—
158C13×C4⋊C4—42no—
159C13×C4⋊C4—81no—
160C13×C4⋊C4—81no—
161C13×C22⋊C4—42no—
162C13×C22⋊C4—42no—
163C13×C22⋊C4—42no—
164C13×C22⋊C4—81no—
165C13×C22⋊C4—81no—
166C42×C13—21no—
167C42×C13—21no—
168C42×C13—21no—
169C42×C13—21no—
170C42×C13C2×D412yes—
171C42×C13#4112yes—
172C4⋊Dic13—21no—
173C4⋊Dic13—21no—
174C4⋊Dic13—21no—
175C4⋊Dic13—21no—
176C4⋊Dic13—21no—
177C4⋊Dic13C2×D412yes—
178C4⋊Dic13C2×D412yes—
179C4⋊Dic13C2×Q812yes—
180#6101—24no—
181#6101—24no—
182#6101—42no—
183#6101—81no—
184C4×Dic13—42no—
185C4×Dic13—81no—
186C4×Dic13—81no—
187C2×#1055—21no—
188C2×#1055—21no—
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×#1055C2×Q812yes—
198C2×#1055#4112yes—
199C2×#499—131no—
200C2×#478—131no—
201#2566D1311yescharacteristic
202#2370—131no—
203#2085—131no—
204#2083—131no—
205#2065—131no—
206#2065—131no—
207#1989—131no—
208#1949—131no—
209#1948—131no—
210#1939—131no—
211#1925—131no—
212#1925—131no—
213C4×#102—131no—
214C23×C13—22no—
215C23×C13—41no—
216C23×C13—41no—
217C23×C13—41no—
218C23×C13—41no—
219C23×C13—41no—
220C23×C13—41no—
221C23×C13C22×D411yescharacteristic
222D4×C13—22no—
223D4×C13—22no—
224D4×C13—41no—
225D4×C13—41no—
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—21no—
236C2×C4×C13—21no—
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—84no—
251C2×C4×C13#12112yes—
252C2×C4×C13#12112yes—
253C2×C4×C13#13512yes—
254C2×C4×C13C22×D411yescharacteristic
255C2×C4×C13C22×Q811yescharacteristic
256C2×C4×C13C2×#4111yescharacteristic
257C2×C4×C13C2×#4111yescharacteristic
258C2×C4×C132−1+411yescharacteristic
259C2×C4×C132−1+411yescharacteristic
260C2×Dic13—22no—
261C2×Dic13—22no—
262C2×Dic13—22no—
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—
272#1055—22no—
273#1055—22no—
274#1055—41no—
275#1055—41no—
276C2×#125—131no—
277C2×#125C2×D1311yescharacteristic
278C2×C4×D4C2×D1311yescharacteristic
279#499—264no—
280#478—264no—
281C2×#107—131no—
282C2×#107—131no—
283C2×#107—131no—
284C2×#106—131no—
285C2×#106—131no—
286C2×#106—131no—
287C2×C4⋊C8—131no—
288C2×C4⋊C8—131no—
289C2×C4⋊C8—131no—
290C2×#102—131no—
291C2×#102—131no—
292C2×#102—131no—
293C2×#102—131no—
294C2×C4×C8—131no—
295#396—21no—
296#396—21no—
297#390C2×D1311yescharacteristic
298#390C2×D1311yescharacteristic
299#389—131no—
300#389—131no—
301#389—261no—
302#387—261no—
303#387—261no—
304#385—21no—
305#385—21no—
306#384C2×D1311yescharacteristic
307#382C2×D1311yescharacteristic
308C4×C4⋊C4—131no—
309C4×C4⋊C4—131no—
310C4×C4⋊C4—131no—
311C4×C4⋊C4—261no—
312C4×C4⋊C4C2×D1311yescharacteristic
313#340—131no—
314#340—131no—
315#340—131no—
316#340—131no—
317#340—131no—
318#340—131no—
319#336—131no—
320#336—131no—
321C22×C13—42no—
322C22×C13—42no—
323C22×C13—42no—
324C22×C13—81no—
325C22×C13—81no—
326C22×C13—81no—
327C22×C13—81no—
328C22×C13C2×#12111yescharacteristic
329C22×C13#54711yescharacteristic
330C22×C13D4×Q811yescharacteristic
331C22×C13C2×#13511yescharacteristic
332C22×C13#57511yescharacteristic
333C22×C13#57611yescharacteristic
334C22×C13#57711yescharacteristic
335C4×C13—22no—
336C4×C13—22no—
337C4×C13—22no—
338C4×C13—41no—
339C4×C13—41no—
340C4×C13—41no—
341C4×C13—81no—
342C4×C13—81no—
343C4×C13—81no—
344C4×C13#53311yescharacteristic
345C4×C13#54111yescharacteristic
346C4×C13#54811yescharacteristic
347C4×C13D4×Q811yescharacteristic
348Dic13—42no—
349Dic13—42no—
350Dic13—42no—
351Dic13—81no—
352C22×D4C22×D1311yescharacteristic
353C23×C4—21no—
354C22×C8—131no—
355C22×C8—131no—
356C22×C8—131no—
357C22×C8—131no—
358#125—22no—
359#125—22no—
360#125—134no—
361#125—262no—
362C4×D4—22no—
363C4×D4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—131no—
369C2×C4⋊C4—131no—
370C2×C4⋊C4—131no—
371C2×C4⋊C4—131no—
372C2×C4⋊C4—261no—
373C2×C4⋊C4—261no—
374C2×C4⋊C4—261no—
375C2×C4⋊C4—261no—
376C2×C4⋊C4C22⋊D1311yescharacteristic
377C2×C4⋊C4C22⋊D1311yescharacteristic
378C2×C4⋊C4C22×D1311yescharacteristic
379C2×C4⋊C4C22×D1311yescharacteristic
380C2×C4⋊C4C22×D1311yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C42—261no—
387C2×C42—261no—
388C2×C42—261no—
389C2×C42C22×D1311yescharacteristic
390C2×C42C22×D1311yescharacteristic
391C2×C42C22×D1311yescharacteristic
392#107—262no—
393#107—262no—
394#107—262no—
395#106—262no—
396#106—262no—
397#106—262no—
398C4⋊C8—262no—
399C4⋊C8—262no—
400C4⋊C8—262no—
401#102—134no—
402#102—262no—
403#102—262no—
404#102—262no—
405C4×C8—262no—
406#95—21no—
407#95—21no—
408#95—261no—
409#95—261no—
410#95—261no—
411#95—261no—
412#95—261no—
413#95C22⋊D1311yescharacteristic
414#95C22⋊D1311yescharacteristic
415C2×C13—82no—
416C2×C13#242911yescharacteristic
417C2×C13#303511yescharacteristic
418C2×C13#304511yescharacteristic
419C2×C13#333711yescharacteristic
420C2×C13#335511yescharacteristic
421C2×C13#338011yescharacteristic
422C2×C13#338811yescharacteristic
423C24—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4Q8×D1312yes—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—42no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—261no—
437C22×C4—261no—
438C22×C4—261no—
439C22×C4—261no—
440C22×C4C2×C22⋊D1311yescharacteristic
441C22×C4C2×C22⋊D1311yescharacteristic
442C22×C4C2×C22⋊D1311yescharacteristic
443C22×C4C2×C22⋊D1311yescharacteristic
444C22×C4C2×C22⋊D1311yescharacteristic
445C22×C4C2×C22⋊D1311yescharacteristic
446C22×C4C23×D1311yescharacteristic
447C2×C8—132no—
448C2×C8—132no—
449C2×C8—261no—
450C2×C8—261no—
451C2×C8—261no—
452C2×C8—261no—
453C2×C8—261no—
454C2×C8—261no—
455C2×C8—261no—
456C2×C8—261no—
457C2×C8—261no—
458C2×C8—261no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—41no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—132no—
472C4⋊C4—132no—
473C4⋊C4—132no—
474C4⋊C4—264no—
475C4⋊C4—264no—
476C4⋊C4—261no—
477C4⋊C4—261no—
478C4⋊C4—261no—
479C4⋊C4—261no—
480C4⋊C4—261no—
481C4⋊C4—522no—
482C4⋊C4—1041no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—522no—
493C42—1041no—
494C42—1041no—
495C42D4×D1312yes—
496C42#612912yes—
497C13#2545811yescharacteristic, complemented, Sylow
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23C22×C22⋊D1311yescharacteristic
506D4—22no—
507D4—22no—
508D4—41no—
509D4—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
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—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—84no—
535C2×C4—262no—
536C2×C4—262no—
537C2×C4—262no—
538C2×C4—521no—
539C2×C4—521no—
540C2×C4—521no—
541C2×C4—521no—
542C2×C4—521no—
543C2×C4—521no—
544C2×C4—521no—
545C2×C4—521no—
546C2×C4—521no—
547C2×C4#8819212yes—
548C2×C4#8819212yes—
549C2×C4#8820412yes—
550C2×C4C2×D4×D1311yescharacteristic
551C2×C4C2×#612911yescharacteristic
552C2×C4C2×Q8×D1311yescharacteristic
553C2×C4#8825411yescharacteristic
554C2×C4#41×D1311yescharacteristic
555C2×C4#8825711yescharacteristic
556C8—262no—
557C8—262no—
558C8—521no—
559C8—521no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22C2×#8819211yescharacteristic
568C22Q8×C22⋊D1311yescharacteristic
569C22C2×#8820411yescharacteristic
570C22#1171952711yescharacteristic
571C22#1171952911yescharacteristic
572C22#1171954011yescharacteristic
573C22#1171954111yescharacteristic
574C4—22no—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—522no—
584C4—522no—
585C4—522no—
586C4—1041no—
587C4#1171932711yescharacteristic
588C4#1171933011yescharacteristic
589C4#1171933711yescharacteristic
590C4#1171933811yescharacteristic
591C2—82no—
592C2#1311843111yescharacteristic
593C2#1312428511yescharacteristic
594C2#1312437111yescharacteristic
595C2#1312720911yescharacteristic
596C2#1312743611yescharacteristic
597C2#1312769911yescharacteristic
598C2#1312778111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2566C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
hypercenter#2566D13
hypocenterC13#25458
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2566

Complements

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

NH ≅ G/Ntype
1. C2×#11718725C2semidirect
3. #13117930C2semidirect
4. #13116318C2semidirect
5. #13116308C2semidirect
8. #13115794C2semidirect
9. #13115602C2semidirect
10. #13115597C2semidirect
11. #13115555C2semidirect
497. C13#25458semidirect

Maximal subgroups

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

subgroupindex
C2×#117187252
C2×#117186702
#131179302
#131163182
#131163082
#131162102
#131157942
#131156022
#131155972
#131155552
#131154532
C4×#880722
C13×#25662
#2545813