Finite Group Databasebeta

← #15784255

Subgroups — #15784255

Order 3328.

1710 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718605C212yesmaximal
2C2×C4×C22⋊Dic13C211yesmaximal, characteristic
3#13113073C212yesmaximal
4#13113071C212yesmaximal, complemented
5#13113064C211yesmaximal, characteristic, complemented
6#13113062C211yesmaximal, characteristic, complemented
7#13113062C212yesmaximal, complemented
8#13113059C211yesmaximal, characteristic
9#95×Dic13C211yesmaximal, characteristic, complemented
10#95×Dic13C211yesmaximal, characteristic, complemented
11C13×C22⋊#95C211yesmaximal, characteristic
12C24×C4×C13C2211yescharacteristic
13C22×C13×C22⋊C4C2212yes—
14C22×C22⋊Dic13C2211yescharacteristic
15C22×C22⋊Dic13C2212yes—
16C22×C22⋊Dic13C2211yescharacteristic
17C22×#6101C2212yes—
18C22×C4×Dic13C2212yes—
19C22×C4×Dic13C2211yescharacteristic
20C22×C4×Dic13C2211yescharacteristic
21C22×C4×Dic13C2212yes—
22C2×C13×#95C2211yescharacteristic
23C2×C13×#95C2211yescharacteristic
24C2×C13×#95C2212yes—
25C4×C22⋊Dic13C418yescomplemented
26C2×#88071C2211yescharacteristic
27C2×#88071C2211yescharacteristic
28C2×#88071C2212yes—
29C2×#88071C2212yes—
30C2×#88071C2212yes—
31C2×#88071C2212yes—
32C2×#88071C2211yescharacteristic
33C2×#88071C2211yescharacteristic
34C2×#88071C2212yes—
35C2×#88071C2212yes—
36#11718605C4116yescomplemented
37C25×C13C2311yescharacteristic
38C23×C4×C13—22no—
39C23×C4×C13—22no—
40C23×C4×C13—42no—
41C23×C4×C13—42no—
42C23×C4×C13C2×C414yes—
43C23×C4×C13C2312yes—
44C23×C4×C13C2312yes—
45C23×C4×C13C2311yescharacteristic
46C23×C4×C13C2311yescharacteristic
47C23×Dic13C2312yes—
48C23×Dic13C2312yes—
49C23×Dic13C2311yescharacteristic
50C23×Dic13C2311yescharacteristic
51C23×Dic13C2311yescharacteristic
52C23×Dic13C2311yescharacteristic
53C2×C13×C22⋊C4—42no—
54C2×C13×C22⋊C4—42no—
55C2×C13×C22⋊C4C2×C418yes—
56C2×C22⋊Dic13—24no—
57C2×C22⋊Dic13—42no—
58C2×C22⋊Dic13—42no—
59C2×C22⋊Dic13C2×C414yes—
60C2×C22⋊Dic13C2×C418yes—
61C2×C22⋊Dic13C2×C418yes—
62C2×C22⋊Dic13C2×C414yes—
63C2×#6101—42no—
64C2×#6101—42no—
65C2×#6101C2×C418yescomplemented
66C2×C4×Dic13—24no—
67C2×C4×Dic13—42no—
68C2×C4×Dic13—42no—
69C2×C4×Dic13—42no—
70C2×C4×Dic13—42no—
71C2×C4×Dic13—42no—
72C2×C4×Dic13—42no—
73C2×C4×Dic13C2×C414yes—
74C2×C4×Dic13C2×C414yes—
75C2×C4×Dic13C2×C418yescomplemented
76C13×#95—22no—
77C13×#95—22no—
78C13×#95—42no—
79#88071—22no—
80#88071—22no—
81#88071—24no—
82#88071—42no—
83#88071—42no—
84#88071—42no—
85#88071C2×C414yes—
86#88071C2×C414yes—
87#88071C2×C418yescomplemented
88#88071C2×C418yescomplemented
89#13788—131nomaximal, Sylow
90C24×C13—22no—
91C24×C13—21no—
92C24×C13—21no—
93C24×C13—21no—
94C24×C13—22no—
95C24×C13—21no—
96C24×C13—22no—
97C24×C13—21no—
98C24×C13—21no—
99C24×C13C22×C414yes—
100C24×C13C22×C412yes—
101C24×C13C2411yescharacteristic
102C22×C4×C13—22no—
103C22×C4×C13—22no—
104C22×C4×C13—22no—
105C22×C4×C13—22no—
106C22×C4×C13—22no—
107C22×C4×C13—22no—
108C22×C4×C13—21no—
109C22×C4×C13—21no—
110C22×C4×C13—22no—
111C22×C4×C13—22no—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
114C22×C4×C13—22no—
115C22×C4×C13—22no—
116C22×C4×C13—22no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—22no—
120C22×C4×C13—28no—
121C22×C4×C13—44no—
122C22×C4×C13—44no—
123C22×C4×C13—48no—
124C22×C4×C13—48no—
125C22×C4×C13C22×C412yes—
126C22×C4×C13C22×C414yes—
127C22×C4×C13C22×C414yes—
128C22×C4×C13C22×C412yes—
129C22×C4×C13#4112yes—
130C22×C4×C13#4112yes—
131C22×Dic13—22no—
132C22×Dic13—21no—
133C22×Dic13—22no—
134C22×Dic13—21no—
135C22×Dic13—22no—
136C22×Dic13—21no—
137C22×Dic13—21no—
138C22×Dic13—21no—
139C22×Dic13—21no—
140C22×Dic13—22no—
141C22×Dic13—22no—
142C22×Dic13—22no—
143C22×Dic13—22no—
144C22×Dic13—22no—
145C22×Dic13—21no—
146C22×Dic13—22no—
147C22×Dic13—22no—
148C22×Dic13—22no—
149C22×Dic13—22no—
150C22×Dic13—22no—
151C22×Dic13—22no—
152C22×Dic13—22no—
153C22×Dic13—22no—
154C22×Dic13—21no—
155C22×Dic13—22no—
156C22×Dic13C22×C412yes—
157C22×Dic13C22×C414yes—
158C22×Dic13C22×C414yes—
159C22×Dic13C22×C412yes—
160C22×Dic13C22×C412yes—
161C22×Dic13C22×C414yes—
162C22×Dic13C22×C414yes—
163C22×Dic13C22×C412yes—
164C22×Dic13#4114yes—
165C13×C22⋊C4—48no—
166C22⋊Dic13—28no—
167C22⋊Dic13—48no—
168C22⋊Dic13C42116yescomplemented
169#6101—48no—
170C4×Dic13—44no—
171C4×Dic13—44no—
172C4×Dic13—48no—
173C4×Dic13—48no—
174C2×#388—132no—
175C2×C4×C22⋊C4—131no—
176#1437—132no—
177#1436—132no—
178C4⋊#95—131no—
179#1434—132no—
180#1434—131no—
181C22⋊#95—131no—
182C22⋊#95D1311yescharacteristic
183C4×#95—131no—
184C4×#95—131no—
185C23×C13—22no—
186C23×C13—22no—
187C23×C13—24no—
188C23×C13—24no—
189C23×C13—24no—
190C23×C13—24no—
191C23×C13—22no—
192C23×C13—22no—
193C23×C13—22no—
194C23×C13—22no—
195C23×C13—24no—
196C23×C13—24no—
197C23×C13—42no—
198C23×C13—42no—
199C23×C13—42no—
200C23×C13—42no—
201C23×C13—41no—
202C23×C13—42no—
203C23×C13—41no—
204C23×C13—41no—
205C23×C13—41no—
206C23×C13—42no—
207C23×C13C2×C4214yes—
208C23×C13C23×C411yescharacteristic
209C23×C13C23×C412yes—
210C23×C13C2×#4111yescharacteristic
211C23×C13C2×#4111yescharacteristic
212C23×C13C2×#4112yes—
213C23×C13H(5,𝔽2)12yes—
214C23×C13H(5,𝔽2)12yes—
215C23×C13H(5,𝔽2)11yescharacteristic
216C23×C13H(5,𝔽2)11yescharacteristic
217C23×C132−1+411yescharacteristic
218C23×C132−1+411yescharacteristic
219C2×C4×C13—22no—
220C2×C4×C13—22no—
221C2×C4×C13—24no—
222C2×C4×C13—24no—
223C2×C4×C13—22no—
224C2×C4×C13—24no—
225C2×C4×C13—24no—
226C2×C4×C13—24no—
227C2×C4×C13—24no—
228C2×C4×C13—24no—
229C2×C4×C13—22no—
230C2×C4×C13—22no—
231C2×C4×C13—24no—
232C2×C4×C13—416no—
233C2×C4×C13—42no—
234C2×C4×C13—42no—
235C2×C4×C13—42no—
236C2×C4×C13—42no—
237C2×C4×C13—42no—
238C2×C4×C13—42no—
239C2×C4×C13—42no—
240C2×C4×C13—42no—
241C2×C4×C13—42no—
242C2×C4×C13—42no—
243C2×C4×C13#11714yes—
244C2×Dic13—22no—
245C2×Dic13—22no—
246C2×Dic13—22no—
247C2×Dic13—24no—
248C2×Dic13—24no—
249C2×Dic13—22no—
250C2×Dic13—22no—
251C2×Dic13—24no—
252C2×Dic13—22no—
253C2×Dic13—24no—
254C2×Dic13—24no—
255C2×Dic13—24no—
256C2×Dic13—22no—
257C2×Dic13—24no—
258C2×Dic13—24no—
259C2×Dic13—22no—
260C2×Dic13—24no—
261C2×Dic13—24no—
262C2×Dic13—24no—
263C2×Dic13—22no—
264C2×Dic13—24no—
265C2×Dic13—42no—
266C2×Dic13—42no—
267C2×Dic13—42no—
268C2×Dic13—42no—
269C2×Dic13—42no—
270C2×Dic13—42no—
271C2×Dic13—42no—
272C2×Dic13—42no—
273C2×Dic13—42no—
274C2×Dic13—42no—
275C2×Dic13C2×C4214yes—
276C2×Dic13C2×C4214yes—
277C2×Dic13#11714yes—
278C24×C4C2×D1311yescharacteristic
279C22×C4⋊C4—132no—
280C22×C22⋊C4—131no—
281C22×C22⋊C4—131no—
282C22×C22⋊C4—132no—
283C22×C22⋊C4C2×D1312yes—
284C22×C42—131no—
285C22×C42—132no—
286C22×C42—131no—
287C22×C42—132no—
288#388—1316no—
289C4×C22⋊C4—138no—
290C2×#95—132no—
291C2×#95—131no—
292C2×#95—132no—
293C2×#95—131no—
294C2×#95—131no—
295C2×#95—131no—
296C2×#95—132no—
297C2×#95—132no—
298C2×#95—132no—
299C2×#95—132no—
300C2×#95C2×D1311yescharacteristic
301C2×#95C2×D1311yescharacteristic
302C2×#95C2×D1312yes—
303C22×C13—24no—
304C22×C13—24no—
305C22×C13—24no—
306C22×C13—42no—
307C22×C13—42no—
308C22×C13—42no—
309C22×C13—42no—
310C22×C13—44no—
311C22×C13—44no—
312C22×C13—44no—
313C22×C13—44no—
314C22×C13C22×C4211yescharacteristic
315C22×C13C2×#11711yescharacteristic
316C22×C13C2×#11711yescharacteristic
317C22×C13C4×#4112yes—
318C22×C13C4×#4112yes—
319C22×C13#51811yescharacteristic
320C22×C13#51811yescharacteristic
321C22×C13#51812yes—
322C22×C13#51812yes—
323C22×C13#52011yescharacteristic
324C22×C13#52012yes—
325C22×C13#52011yescharacteristic
326C22×C13#52012yes—
327C22×C13#53812yes—
328C22×C13#53812yes—
329C22×C13#53812yes—
330C22×C13#53912yes—
331C22×C13#54012yes—
332C22×C13#54011yescharacteristic
333C22×C13#54111yescharacteristic
334C22×C13#54211yescharacteristic
335C22×C13#54212yes—
336C22×C13#54211yescharacteristic
337C4×C13—24no—
338C4×C13—42no—
339C4×C13—44no—
340C4×C13—44no—
341Dic13—24no—
342Dic13—24no—
343Dic13—24no—
344Dic13—42no—
345Dic13—44no—
346Dic13—44no—
347C25C22×D1311yescharacteristic
348C23×C4—22no—
349C23×C4—22no—
350C23×C4—42no—
351C23×C4—42no—
352C23×C4—131no—
353C23×C4—131no—
354C23×C4—131no—
355C23×C4—132no—
356C23×C4—132no—
357C23×C4—131no—
358C23×C4C4×D1314yes—
359C23×C4C22×D1312yes—
360C23×C4C22×D1312yes—
361C23×C4C22×D1311yescharacteristic
362C23×C4C22×D1311yescharacteristic
363C2×C4⋊C4—138no—
364C2×C4⋊C4—522no—
365C2×C4⋊C4—522no—
366C2×C22⋊C4—42no—
367C2×C22⋊C4—42no—
368C2×C22⋊C4—134no—
369C2×C22⋊C4—138no—
370C2×C22⋊C4—134no—
371C2×C22⋊C4—138no—
372C2×C22⋊C4—264no—
373C2×C22⋊C4—522no—
374C2×C22⋊C4—522no—
375C2×C22⋊C4C4×D1318yes—
376C2×C42—134no—
377C2×C42—134no—
378C2×C42—138no—
379C2×C42—264no—
380C2×C42—522no—
381C2×C42—522no—
382C2×C42—522no—
383C2×C42—522no—
384C2×C42—522no—
385C2×C42—522no—
386#95—22no—
387#95—22no—
388#95—42no—
389#95—134no—
390#95—134no—
391#95—138no—
392#95—138no—
393#95—264no—
394#95—262no—
395#95—262no—
396#95—522no—
397#95—522no—
398#95—522no—
399C2×C13—44no—
400C2×C13C4×#11711yescharacteristic
401C2×C13#227511yescharacteristic
402C2×C13#227511yescharacteristic
403C2×C13#231211yescharacteristic
404C2×C13#231411yescharacteristic
405C2×C13#231811yescharacteristic
406C2×C13#231911yescharacteristic
407C2×C13#233312yes—
408C2×C13#233412yes—
409C2×C13#233512yes—
410C2×C13#233712yes—
411C24—21no—
412C24—21no—
413C24—21no—
414C24—22no—
415C24—22no—
416C24—21no—
417C24—22no—
418C24—21no—
419C24—21no—
420C24C2×C4×D1314yes—
421C24C2×C4×D1312yes—
422C24C23×D1311yescharacteristic
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—28no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—132no—
447C22×C4—134no—
448C22×C4—132no—
449C22×C4—132no—
450C22×C4—134no—
451C22×C4—134no—
452C22×C4—134no—
453C22×C4—132no—
454C22×C4—134no—
455C22×C4—262no—
456C22×C4—261no—
457C22×C4—262no—
458C22×C4—262no—
459C22×C4—261no—
460C22×C4—261no—
461C22×C4—261no—
462C22×C4—261no—
463C22×C4—261no—
464C22×C4—262no—
465C22×C4—262no—
466C22×C4—261no—
467C22×C4—262no—
468C22×C4—262no—
469C22×C4—262no—
470C22×C4—262no—
471C22×C4—262no—
472C22×C4—262no—
473C22×C4—261no—
474C22×C4—262no—
475C22×C4—262no—
476C22×C4—262no—
477C22×C4—262no—
478C22×C4—262no—
479C22×C4—262no—
480C22×C4C2×C4×D1312yes—
481C22×C4C2×C4×D1314yes—
482C22×C4C2×C4×D1314yes—
483C22×C4C2×C4×D1312yes—
484C22×C4#612912yes—
485C22×C4#612912yes—
486C4⋊C4—528no—
487C22⋊C4—48no—
488C22⋊C4—1316no—
489C22⋊C4—268no—
490C22⋊C4—528no—
491C42—524no—
492C42—524no—
493C42—528no—
494C42—528no—
495C13#1378811yescharacteristic, complemented, Sylow
496C23—22no—
497C23—24no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23C42×D1314yes—
519C23C22×C4×D1311yescharacteristic
520C23C22×C4×D1312yes—
521C23C2×#612911yescharacteristic
522C23C2×#612911yescharacteristic
523C23#8825112yes—
524C23#8825111yescharacteristic
525C23#8825112yes—
526C23#8825411yescharacteristic
527C23#41×D1312yes—
528C23#8825611yescharacteristic
529C23#8825711yescharacteristic
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—416no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—134no—
555C2×C4—134no—
556C2×C4—134no—
557C2×C4—262no—
558C2×C4—264no—
559C2×C4—262no—
560C2×C4—262no—
561C2×C4—262no—
562C2×C4—264no—
563C2×C4—262no—
564C2×C4—262no—
565C2×C4—264no—
566C2×C4—264no—
567C2×C4—264no—
568C2×C4—264no—
569C2×C4—262no—
570C2×C4—264no—
571C2×C4—262no—
572C2×C4—264no—
573C2×C4—262no—
574C2×C4—264no—
575C2×C4—264no—
576C2×C4—264no—
577C2×C4—264no—
578C2×C4—522no—
579C2×C4—522no—
580C2×C4—522no—
581C2×C4—522no—
582C2×C4—522no—
583C2×C4—522no—
584C2×C4—522no—
585C2×C4—522no—
586C2×C4—522no—
587C2×C4—522no—
588C2×C4#8813114yes—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—42no—
600C22C2×C42×D1311yescharacteristic
601C22C2×#8813111yescharacteristic
602C22#1171919111yescharacteristic
603C22#1171919112yes—
604C22#1171919111yescharacteristic
605C22#1171920911yescharacteristic
606C22#1171920912yes—
607C22D13×#11711yescharacteristic
608C22#1171922811yescharacteristic
609C22C4×#612912yes—
610C22C4×#612912yes—
611C22#1171925012yes—
612C22#1171925112yes—
613C22#1171929812yes—
614C22#1171929812yes—
615C22#1171930811yescharacteristic
616C22#1171932012yes—
617C22#1171932411yescharacteristic
618C22#1171933011yescharacteristic
619C22#1171934512yes—
620C22#1171934811yescharacteristic
621C22#1171936212yes—
622C22#1171936512yes—
623C4—24no—
624C4—44no—
625C4—44no—
626C4—42no—
627C4—264no—
628C4—264no—
629C4—264no—
630C4—522no—
631C4—524no—
632C4—524no—
633C2—44no—
634C2C4×#8813111yescharacteristic
635C2#1311719411yescharacteristic
636C2#1311719511yescharacteristic
637C2#1311743211yescharacteristic
638C2#1311744311yescharacteristic
639C2#1311746511yescharacteristic
640C2#1311747811yescharacteristic
641C2#1311759712yes—
642C2#1311760612yes—
643C2#1311762712yes—
644C2#1311765012yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC22×C13C22×C42
FittingC13×C22⋊#95C2
socle, cosocleC24×C13C24
prime radicalC25×C13C23
hypercenterC22⋊#95D13
hypocenterC13#13788
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
4. #13113071C2semidirect
5. #13113064C2semidirect
6. #13113062C2semidirect
7. #13113062C2semidirect
9. #95×Dic13C2semidirect
10. #95×Dic13C2semidirect
25. C4×C22⋊Dic13C4semidirect
36. #11718605C4semidirect
65. C2×#6101C2×C4semidirect
75. C2×C4×Dic13C2×C4semidirect
87. #88071C2×C4semidirect
88. #88071C2×C4semidirect
168. C22⋊Dic13C42semidirect
495. C13#13788semidirect

Maximal subgroups

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

subgroupindex
C2×#117186052
C2×C4×C22⋊Dic132
#131130732
#131130712
#131130642
#131130622
#131130592
#95×Dic132
C13×C22⋊#952
#1378813