Finite Group Databasebeta

← #16802265

Subgroups — #16802265

Order 3328.

722 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131007C211yesmaximal, characteristic, complemented
2#13131006C211yesmaximal, characteristic, complemented
3#13130935C211yesmaximal, characteristic, complemented
4#13130933C211yesmaximal, characteristic, complemented
5#13130892C211yesmaximal, characteristic, complemented
6#13130892C211yesmaximal, characteristic, complemented
7#13130891C211yesmaximal, characteristic, complemented
8#13130889C211yesmaximal, characteristic, complemented
9#13130783C211yesmaximal, characteristic
10#13130783C211yesmaximal, characteristic
11#13130694C211yesmaximal, characteristic
12C22⋊C4×#1057C211yesmaximal, characteristic
13C22⋊C4×#1057C211yesmaximal, characteristic
14#13130502C211yesmaximal, characteristic
15#13118111C211yesmaximal, characteristic
16C22×#6122C2211yescharacteristic
17C23×#1057C2211yescharacteristic
18C2×#88141C2211yescharacteristic
19C2×#88131C2211yescharacteristic
20C2×C22⋊#1057C2211yescharacteristic
21C2×C22⋊#1057C2211yescharacteristic
22C2×#88112C2211yescharacteristic
23C2×#88112C2211yescharacteristic
24C2×#88112C2211yescharacteristic
25C2×C4⋊#1057C2211yescharacteristic
26C2×C4⋊#1057C2211yescharacteristic
27C2×C4⋊#1057C2211yescharacteristic
28C2×C4×#1057C2211yescharacteristic
29C2×C4×#1057C2211yescharacteristic
30C2×C4×#1057C2211yescharacteristic
31C2×C4×#1057C2211yescharacteristic
32C13×#396C411yescharacteristic
33#11718645C411yescharacteristic
34#11718644C411yescharacteristic
35#11718640C411yescharacteristic
36C4⋊C4×Dic13C2211yescharacteristic
37C4⋊C4×Dic13C2211yescharacteristic
38#11718610C411yescharacteristic
39#11718609C411yescharacteristic
40#11718608C411yescharacteristic
41C22⋊C4×Dic13C2211yescharacteristic
42C22⋊C4×Dic13C2211yescharacteristic
43#11718319C411yescharacteristic
44#11718310C2211yescharacteristic
45#11718290C2211yescharacteristic
46#11718290C2211yescharacteristic
47#11718290C2211yescharacteristic
48#11718290C2211yescharacteristic
49#11718290C2211yescharacteristic
50#11718290C2211yescharacteristic
51#11718290C2211yescharacteristic
52#11718290C2211yescharacteristic
53#11718290C2211yescharacteristic
54#11718290C2211yescharacteristic
55#11718290C2211yescharacteristic
56#11718290C2211yescharacteristic
57#11718290C2211yescharacteristic
58#11718290C2211yescharacteristic
59C23×Dic13C2311yescharacteristic
60C22×Dic26C2×C411yescharacteristic
61C2×#6122—22no—
62C2×#6122—41no—
63C2×#6122—41no—
64C22×#1057—41no—
65C22×#1057—41no—
66C22×#1057Q814yescomplemented
67C22×#1057C2311yescharacteristic
68C22×#1057C2311yescharacteristic
69C22×#1057C2311yescharacteristic
70C22×#1057C2311yescharacteristic
71C22×#1057C2311yescharacteristic
72C22×#1057C2311yescharacteristic
73C22×#1057C2311yescharacteristic
74C22×#1057C2311yescharacteristic
75C2×C13×C4⋊C4C2×C411yescharacteristic
76C2×C13×C4⋊C4C2×C411yescharacteristic
77C2×C13×C4⋊C4C2×C411yescharacteristic
78C2×C13×C22⋊C4C2×C411yescharacteristic
79C2×C13×C22⋊C4C2×C411yescharacteristic
80C2×C13×C22⋊C4C2×C411yescharacteristic
81C2×C22⋊Dic13C2×C411yescharacteristic
82C2×C22⋊Dic13C2×C411yescharacteristic
83C2×C22⋊Dic13C2×C411yescharacteristic
84C2×C4⋊Dic13C2×C411yescharacteristic
85C2×C4⋊Dic13C2×C411yescharacteristic
86C2×#6101C2×C411yescharacteristic
87C2×#6101C2×C411yescharacteristic
88C2×#6101C2×C411yescharacteristic
89C2×#6101C2×C411yescharacteristic
90C2×#6101C2×C411yescharacteristic
91C2×#6101C2×C411yescharacteristic
92C2×C4×Dic13C2311yescharacteristic
93C2×C4×Dic13C2311yescharacteristic
94C2×C4×Dic13C2311yescharacteristic
95C2×C4×Dic13C2311yescharacteristic
96C2×C4×Dic13C2311yescharacteristic
97C2×C4×Dic13C2311yescharacteristic
98#88141—41no—
99#88141—41no—
100#88131—41no—
101#88131—41no—
102C22⋊#1057—22no—
103C22⋊#1057—22no—
104#88112—41no—
105#88112—41no—
106#88112—41no—
107C4⋊#1057—41no—
108C4⋊#1057—41no—
109C4⋊#1057—41no—
110C4×#1057—41no—
111C4×#1057—41no—
112C4×#1057—41no—
113C4×#1057—41no—
114C13×#95C2×C411yescharacteristic
115#88071C2×C411yescharacteristic
116#88071C2×C411yescharacteristic
117#88071C2×C411yescharacteristic
118#88071C2×C411yescharacteristic
119#88071C2×C411yescharacteristic
120#88071C2×C411yescharacteristic
121#88071C2×C411yescharacteristic
122#88071C2×C411yescharacteristic
123#88071C2×C411yescharacteristic
124#22288—131nomaximal, Sylow
125C24×C13C22×C411yescharacteristic
126C22×C4×C13C22×C411yescharacteristic
127C22×C4×C13C22×C411yescharacteristic
128C22×C4×C13C22×C411yescharacteristic
129C22×C4×C13C22×C411yescharacteristic
130C22×C4×C13C22×C411yescharacteristic
131C22×C4×C13C22×C411yescharacteristic
132C22×Dic13—21no—
133C22×Dic13—21no—
134C22×Dic13—21no—
135C22×Dic13—21no—
136C22×Dic13C22×C411yescharacteristic
137C22×Dic13C22×C411yescharacteristic
138C22×Dic13C22×C411yescharacteristic
139C22×Dic13C22×C411yescharacteristic
140C22×Dic13C22×C411yescharacteristic
141C22×Dic13C22×C411yescharacteristic
142C22×Dic13C22×C411yescharacteristic
143C22×Dic13C2×Q812yes—
144C22×Dic13#4112yes—
145C22×Dic13C2411yescharacteristic
146C2×Dic26—22no—
147C2×Dic26—22no—
148C2×Dic26—41no—
149#6122—44no—
150C2×#1057—21no—
151C2×#1057—21no—
152C2×#1057—21no—
153C2×#1057—21no—
154C2×#1057—21no—
155C2×#1057—21no—
156C2×#1057—21no—
157C2×#1057—21no—
158C2×#1057—21no—
159C2×#1057—21no—
160C2×#1057—21no—
161C2×#1057—21no—
162C2×#1057—21no—
163C2×#1057—21no—
164C2×#1057—21no—
165C2×#1057—21no—
166C2×#1057—21no—
167C2×#1057—21no—
168C2×#1057—21no—
169C2×#1057—21no—
170C2×#1057—21no—
171C2×#1057—21no—
172C2×#1057—44no—
173C2×#1057C2×Q812yes—
174C2×#1057C2×Q812yes—
175C13×C4⋊C4—22no—
176C13×C4⋊C4—22no—
177C13×C4⋊C4—41no—
178C13×C22⋊C4—22no—
179C13×C22⋊C4—22no—
180C13×C22⋊C4—41no—
181C22⋊Dic13—22no—
182C22⋊Dic13—22no—
183C22⋊Dic13—41no—
184C4⋊Dic13—22no—
185C4⋊Dic13—22no—
186#6101—22no—
187#6101—22no—
188#6101—41no—
189#6101—41no—
190#6101—41no—
191#6101—41no—
192C4×Dic13—21no—
193C4×Dic13—21no—
194C4×Dic13—21no—
195C4×Dic13—21no—
196C4×Dic13—21no—
197C4×Dic13—21no—
198C4×Dic13—21no—
199C4×Dic13—21no—
200C4×Dic13—21no—
201C4×Dic13—21no—
202C4×Dic13—21no—
203C4×Dic13—21no—
204#2396—131no—
205#1984—131no—
206#1984—131no—
207#1939—131no—
208#1939—131no—
209#1916—131no—
210#1916—131no—
211#1916—131no—
212#1916—131no—
213#1851—131no—
214#1851—131no—
215#1798—131no—
216C8×C22⋊C4—131no—
217C8×C22⋊C4—131no—
218#1729—131no—
219C23×C13—21no—
220C23×C13—21no—
221C23×C13—21no—
222C23×C13—21no—
223C23×C13—21no—
224C23×C13—21no—
225C23×C13C4×Q812yes—
226C23×C13C23×C411yescharacteristic
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—21no—
242C2×C4×C13—21no—
243C2×C4×C13#13112yes—
244C2×C4×C13#13112yes—
245C2×Dic13—21no—
246C2×Dic13—21no—
247C2×Dic13—21no—
248C2×Dic13—21no—
249C2×Dic13—21no—
250C2×Dic13—21no—
251C2×Dic13—22no—
252C2×Dic13—22no—
253C2×Dic13—22no—
254C2×Dic13—22no—
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—21no—
264C2×Dic13—21no—
265C2×Dic13—21no—
266C2×Dic13—21no—
267C2×Dic13C4×Q812yes—
268C2×Dic13#13112yes—
269C2×Dic13#13112yes—
270C2×Dic13C22×Q811yescharacteristic
271C2×Dic13C2×#4111yescharacteristic
272C2×Dic13H(5,𝔽2)11yescharacteristic
273C2×Dic13H(5,𝔽2)11yescharacteristic
274C2×Dic13H(5,𝔽2)11yescharacteristic
275C2×Dic132−1+411yescharacteristic
276Dic26—44no—
277#1057—22no—
278#1057—22no—
279#1057—41no—
280#1057—41no—
281#1057—41no—
282#1057—41no—
283#1057—41no—
284#1057—41no—
285C22×M16—131no—
286C23×C8—131no—
287C2×C4×Q8—131no—
288C2×#117—131no—
289C2×C4⋊C8—131no—
290C2×C4⋊C8—131no—
291C2×C4⋊C8—131no—
292C2×C4⋊C8—131no—
293C2×C4⋊C8—131no—
294C2×C4⋊C8—131no—
295C2×C22⋊C8—131no—
296C2×C22⋊C8—131no—
297C2×C4×C8—131no—
298C2×C4×C8—131no—
299C2×C4×C8—131no—
300C2×C4×C8—131no—
301#400—131no—
302#399—131no—
303#398—131no—
304#398—131no—
305#396—131no—
306#396#26111yescharacteristic
307#395—131no—
308#393—131no—
309C4×C4⋊C4—131no—
310C4×C4⋊C4—131no—
311C4×C4⋊C4—131no—
312C4×C22⋊C4—131no—
313C4×C22⋊C4—131no—
314#336—131no—
315#336—131no—
316#336—131no—
317#336—131no—
318#336—131no—
319#336—131no—
320#336—131no—
321#336—131no—
322#336—131no—
323#336—131no—
324#336—131no—
325#336—131no—
326#336—131no—
327#336—131no—
328C22×C13—22no—
329C22×C13—22no—
330C22×C13—22no—
331C22×C13—41no—
332C22×C13—41no—
333C22×C13—41no—
334C22×C13—41no—
335C22×C13C2×C4×Q811yescharacteristic
336C22×C13#51811yescharacteristic
337C22×C13#52011yescharacteristic
338C22×C13C2×#13111yescharacteristic
339C22×C13C2×#13111yescharacteristic
340C22×C13C2×#13111yescharacteristic
341C22×C13C2×#13111yescharacteristic
342C4×C13—22no—
343C4×C13—22no—
344C4×C13—41no—
345C4×C13—41no—
346C4×C13—41no—
347C4×C13—41no—
348Dic13—22no—
349Dic13—22no—
350Dic13—22no—
351Dic13—41no—
352Dic13—41no—
353Dic13—41no—
354Dic13—41no—
355Dic13#53811yescharacteristic
356Dic13#53911yescharacteristic
357Dic13#54311yescharacteristic
358Dic13#54411yescharacteristic
359C22×Q8—131no—
360C23×C4—131no—
361C2×M16—262no—
362C2×M16—521no—
363C2×M16—521no—
364C22×C8—131no—
365C22×C8—131no—
366C22×C8—134no—
367C22×C8—131no—
368C22×C8—131no—
369C22×C8—131no—
370C22×C8—131no—
371C22×C8—131no—
372C22×C8—131no—
373C22×C8—521no—
374C22×C8—521no—
375C4×Q8—521no—
376C4×Q8—521no—
377#117—521no—
378#117—521no—
379C2×C4⋊C4—131no—
380C2×C4⋊C4—131no—
381C2×C4⋊C4—131no—
382C2×C4⋊C4—131no—
383C2×C4⋊C4—131no—
384C2×C4⋊C4—131no—
385C2×C4⋊C4—131no—
386C2×C4⋊C4—131no—
387C2×C4⋊C4C2×#26111yescharacteristic
388C2×C4⋊C4C2×#26111yescharacteristic
389C2×C4⋊C4C2×#26111yescharacteristic
390C2×C22⋊C4—131no—
391C2×C22⋊C4—131no—
392C2×C22⋊C4—131no—
393C2×C22⋊C4C2×#26111yescharacteristic
394C2×C22⋊C4C2×#26111yescharacteristic
395C2×C22⋊C4C2×#26111yescharacteristic
396C2×C42—131no—
397C2×C42—131no—
398C2×C42—131no—
399C2×C42—131no—
400C2×C42—131no—
401C2×C42—131no—
402C4⋊C8—521no—
403C4⋊C8—521no—
404C4⋊C8—521no—
405C4⋊C8—521no—
406C4⋊C8—521no—
407C4⋊C8—521no—
408C22⋊C8—262no—
409C22⋊C8—262no—
410C4×C8—521no—
411C4×C8—521no—
412C4×C8—521no—
413C4×C8—521no—
414#95—131no—
415#95—131no—
416#95—131no—
417#95—131no—
418#95—131no—
419#95—131no—
420#95—131no—
421#95—131no—
422#95—131no—
423#95C2×#26111yescharacteristic
424C2×C13—42no—
425C2×C13#232911yescharacteristic
426C2×C13#296011yescharacteristic
427C2×C13#296011yescharacteristic
428C2×C13#297711yescharacteristic
429C2×C13#297911yescharacteristic
430C2×C13#299411yescharacteristic
431C2×C13#299411yescharacteristic
432C24C22×#26111yescharacteristic
433C2×Q8—262no—
434C2×Q8—262no—
435C2×Q8—521no—
436C22×C4—131no—
437C22×C4—132no—
438C22×C4—132no—
439C22×C4—131no—
440C22×C4—131no—
441C22×C4—131no—
442C22×C4—131no—
443C22×C4—131no—
444C22×C4—131no—
445C22×C4—131no—
446C22×C4—261no—
447C22×C4—261no—
448C22×C4—261no—
449C22×C4—261no—
450C22×C4C22×#26111yescharacteristic
451C22×C4C22×#26111yescharacteristic
452C22×C4C22×#26111yescharacteristic
453C22×C4C22×#26111yescharacteristic
454C22×C4C22×#26111yescharacteristic
455C22×C4C22×#26111yescharacteristic
456M16—524no—
457C2×C8—132no—
458C2×C8—132no—
459C2×C8—261no—
460C2×C8—261no—
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—
471C2×C8—261no—
472C2×C8—261no—
473C2×C8—261no—
474C2×C8—261no—
475C2×C8—261no—
476C2×C8—261no—
477C2×C8—261no—
478C2×C8—261no—
479C2×C8—261no—
480C2×C8—261no—
481C2×C8—524no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—41no—
485C4⋊C4—262no—
486C4⋊C4—262no—
487C4⋊C4—262no—
488C4⋊C4—262no—
489C4⋊C4—521no—
490C4⋊C4—521no—
491C4⋊C4—521no—
492C4⋊C4—521no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—41no—
496C22⋊C4—262no—
497C22⋊C4—262no—
498C22⋊C4—521no—
499C42—261no—
500C42—261no—
501C42—261no—
502C42—261no—
503C42—261no—
504C42—261no—
505C42—261no—
506C42—261no—
507C42—261no—
508C42—261no—
509C42—261no—
510C42—261no—
511C13#2228811yescharacteristic, complemented, Sylow
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23Q8×#26112yes—
519C23C23×#26111yescharacteristic
520Q8—524no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
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—131no—
538C2×C4—131no—
539C2×C4—131no—
540C2×C4—131no—
541C2×C4—131no—
542C2×C4—131no—
543C2×C4—132no—
544C2×C4—132no—
545C2×C4—132no—
546C2×C4—262no—
547C2×C4—262no—
548C2×C4—262no—
549C2×C4—262no—
550C2×C4—261no—
551C2×C4—261no—
552C2×C4—261no—
553C2×C4—261no—
554C2×C4—261no—
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—261no—
566C2×C4—261no—
567C2×C4—261no—
568C2×C4#8823812yes—
569C2×C4#8823812yes—
570C8—262no—
571C8—262no—
572C8—521no—
573C8—521no—
574C8—521no—
575C8—521no—
576C8—521no—
577C8—521no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×#8823811yescharacteristic
586C22C2×#8823811yescharacteristic
587C22#1171962311yescharacteristic
588C22C2×Q8×#26111yescharacteristic
589C22D13⋊#13111yescharacteristic
590C22D13⋊#13111yescharacteristic
591C22#1171963111yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—131no—
599C4—131no—
600C4—131no—
601C4—131no—
602C4—262no—
603C4—262no—
604C4—262no—
605C4—521no—
606C4—521no—
607C4—521no—
608C4—521no—
609C2—42no—
610C2#1313132911yescharacteristic
611C2#1313162411yescharacteristic
612C2#1313162411yescharacteristic
613C2#1313170911yescharacteristic
614C2#1313171611yescharacteristic
615C2#1313178311yescharacteristic
616C2#1313178511yescharacteristic
subgroupG/H
center, FrattiniC23C23×#261
commutator, socleC23×C13C23×C4
FittingC13×#396C4
prime radicalC24×C13C22×C4
cosocleC22×Dic13C24
hypercenter#396#261
hypocenterC13#22288
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. #13131007C2semidirect
2. #13131006C2semidirect
3. #13130935C2semidirect
4. #13130933C2semidirect
5. #13130892C2semidirect
6. #13130892C2semidirect
7. #13130891C2semidirect
8. #13130889C2semidirect
66. C22×#1057Q8semidirect
511. C13#22288semidirect

Maximal subgroups

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

subgroupindex
#131310072
#131310062
#131309352
#131309332
#131308922
#131308912
#131308892
#131307832
#131306942
C22⋊C4×#10572
#131305022
#131181112
#2228813