Finite Group Databasebeta

← #15788220

Subgroups — #15788220

Order 3328.

2062 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic26C211yesmaximal, characteristic, complemented
2C2×#11718608C211yesmaximal, characteristic
3C2×#11718871C212yesmaximal
4#13113106C211yesmaximal, characteristic, complemented
5#13113100C211yesmaximal, characteristic, complemented
6#13113081C211yesmaximal, characteristic
7#13113073C212yesmaximal
8#13113069C212yesmaximal, complemented
9#13113067C212yesmaximal, complemented
10#13113063C211yesmaximal, characteristic, complemented
11C2×C4×C13×C22⋊C4C211yesmaximal, characteristic
12C24×C4×C13C2211yescharacteristic
13C23×Dic26C2211yescharacteristic
14C22×C13×C22⋊C4C2211yescharacteristic
15C22×C13×C22⋊C4C2211yescharacteristic
16C22×C42×C13C2211yescharacteristic
17C22×C42×C13C2211yescharacteristic
18C22×C22⋊Dic13C2212yes—
19C22×C22⋊Dic13C2212yes—
20C22×C4⋊Dic13C2211yescharacteristic
21C22×#6101C2212yes—
22C22×#6101C2212yes—
23C22×C4×Dic13C2212yes—
24C2×C4×Dic26—24no—
25C2×C4×Dic26—28no—
26C4×C13×C22⋊C4—24no—
27C2×C13×#95C2211yescharacteristic
28C2×C13×#95C2211yescharacteristic
29#11718871—28no—
30C2×#88071C2212yes—
31C2×#88071C2211yescharacteristic
32C2×#88071C2212yes—
33C2×#88071C2212yes—
34C2×#88071C2212yes—
35C2×#88071C2212yes—
36C2×#88071C2212yes—
37C2×#88071C2211yescharacteristic
38C2×#88071C2212yes—
39#11718608C418yescomplemented
40C25×C13C2311yescharacteristic
41C23×C4×C13—22no—
42C23×C4×C13—22no—
43C23×C4×C13—22no—
44C23×C4×C13—24no—
45C23×C4×C13—42no—
46C23×C4×C13C2311yescharacteristic
47C23×C4×C13C2311yescharacteristic
48C23×C4×C13C2311yescharacteristic
49C23×C4×C13C2311yescharacteristic
50C23×C4×C13C2311yescharacteristic
51C23×C4×C13C2311yescharacteristic
52C23×Dic13C2312yes—
53C23×Dic13C2312yes—
54C23×Dic13C2312yes—
55C23×Dic13C2312yes—
56C22×Dic26—22no—
57C22×Dic26—22no—
58C22×Dic26—24no—
59C22×Dic26—24no—
60C22×Dic26C2×C414yescomplemented
61C2×C13×C22⋊C4—22no—
62C2×C13×C22⋊C4—22no—
63C2×C13×C22⋊C4—22no—
64C2×C13×C22⋊C4—22no—
65C2×C13×C22⋊C4C2×C414yes—
66C2×C13×C22⋊C4Q814yescomplemented
67C2×C42×C13—24no—
68C2×C42×C13—22no—
69C2×C42×C13—42no—
70C2×C42×C13D414yes—
71C2×C22⋊Dic13—24no—
72C2×C22⋊Dic13—24no—
73C2×C22⋊Dic13—24no—
74C2×C22⋊Dic13—24no—
75C2×C22⋊Dic13—42no—
76C2×C22⋊Dic13C2×C418yes—
77C2×C4⋊Dic13—22no—
78C2×C4⋊Dic13—24no—
79C2×#6101—24no—
80C2×#6101—24no—
81C2×#6101—24no—
82C2×#6101—24no—
83C2×#6101—42no—
84C2×#6101D418yes—
85C2×C4×Dic13—24no—
86C2×C4×Dic13—24no—
87C2×C4×Dic13—24no—
88C4×Dic26—232no—
89C13×#95—22no—
90C13×#95—22no—
91#88071—24no—
92#88071—24no—
93#88071—22no—
94#88071—24no—
95#88071—24no—
96#88071—42no—
97#88071—42no—
98#88071C2×C414yescomplemented
99#88071C2×C418yescomplemented
100#14172—131nomaximal, Sylow
101C24×C13—22no—
102C24×C13—21no—
103C24×C13—22no—
104C24×C13—21no—
105C24×C13—21no—
106C24×C13—21no—
107C24×C13—22no—
108C24×C13—22no—
109C24×C13C22×C412yes—
110C24×C13C2×Q812yes—
111C24×C13#4112yes—
112C24×C13C2411yescharacteristic
113C22×C4×C13—22no—
114C22×C4×C13—22no—
115C22×C4×C13—22no—
116C22×C4×C13—21no—
117C22×C4×C13—21no—
118C22×C4×C13—22no—
119C22×C4×C13—22no—
120C22×C4×C13—28no—
121C22×C4×C13—22no—
122C22×C4×C13—21no—
123C22×C4×C13—21no—
124C22×C4×C13—21no—
125C22×C4×C13—22no—
126C22×C4×C13—21no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—22no—
130C22×C4×C13—22no—
131C22×C4×C13—22no—
132C22×C4×C13—22no—
133C22×C4×C13—48no—
134C22×C4×C13—48no—
135C22×C4×C13—44no—
136C22×C4×C13—44no—
137C22×C4×C13C22×C412yes—
138C22×C4×C13C22×C412yes—
139C22×C4×C13C2×D414yes—
140C22×C4×C13C2×D412yes—
141C22×C4×C13C2×D412yes—
142C22×C4×C13C2×D412yes—
143C22×C4×C13C2×Q812yes—
144C22×C4×C13C2×Q812yes—
145C22×C4×C13#4112yes—
146C22×C4×C13#4112yes—
147C22×C4×C13#4112yes—
148C22×Dic13—22no—
149C22×Dic13—22no—
150C22×Dic13—22no—
151C22×Dic13—22no—
152C22×Dic13—22no—
153C22×Dic13—22no—
154C22×Dic13—22no—
155C22×Dic13—22no—
156C22×Dic13—22no—
157C22×Dic13—22no—
158C22×Dic13—22no—
159C22×Dic13—22no—
160C22×Dic13—22no—
161C22×Dic13—22no—
162C22×Dic13—22no—
163C22×Dic13—22no—
164C22×Dic13—22no—
165C22×Dic13—22no—
166C22×Dic13—22no—
167C22×Dic13—22no—
168C22×Dic13—22no—
169C22×Dic13—22no—
170C22×Dic13—22no—
171C22×Dic13C22×C414yes—
172C22×Dic13C22×C414yes—
173C22×Dic13C2×D414yes—
174C22×Dic13C2×D414yes—
175C22×Dic13#4114yes—
176C2×Dic26—28no—
177C2×Dic26—216no—
178C2×Dic26—28no—
179C2×Dic26—28no—
180C2×Dic26—216no—
181C13×C22⋊C4—28no—
182C13×C22⋊C4—28no—
183C42×C13—28no—
184C42×C13—44no—
185C22⋊Dic13—216no—
186C22⋊Dic13—48no—
187C4⋊Dic13—28no—
188#6101—216no—
189#6101—48no—
190C4×Dic13—216no—
191C22×C4×Q8—131no—
192C2×#393—131no—
193C2×#385—132no—
194C2×C4×C22⋊C4D1311yescharacteristic
195#1446—131no—
196#1444—131no—
197#1439—131no—
198#1437—132no—
199#1436—132no—
200#1436—132no—
201#1434—131no—
202C23×C13—24no—
203C23×C13—22no—
204C23×C13—22no—
205C23×C13—24no—
206C23×C13—22no—
207C23×C13—22no—
208C23×C13—22no—
209C23×C13—22no—
210C23×C13—24no—
211C23×C13—24no—
212C23×C13—24no—
213C23×C13—24no—
214C23×C13—41no—
215C23×C13—41no—
216C23×C13—42no—
217C23×C13—41no—
218C23×C13—41no—
219C23×C13—42no—
220C23×C13—42no—
221C23×C13—42no—
222C23×C13—42no—
223C23×C13—42no—
224C23×C13C4×Q814yescomplemented
225C23×C13C23×C411yescharacteristic
226C23×C13C22×D411yescharacteristic
227C23×C13C22×D412yes—
228C23×C13C22×Q811yescharacteristic
229C23×C13C2×#4111yescharacteristic
230C23×C13C2×#4112yes—
231C23×C13C2×#4111yescharacteristic
232C23×C13C2×#4111yescharacteristic
233C23×C13C2×#4111yescharacteristic
234C23×C13H(5,𝔽2)12yes—
235C23×C132−1+412yes—
236C2×C4×C13—24no—
237C2×C4×C13—22no—
238C2×C4×C13—22no—
239C2×C4×C13—22no—
240C2×C4×C13—24no—
241C2×C4×C13—22no—
242C2×C4×C13—22no—
243C2×C4×C13—22no—
244C2×C4×C13—24no—
245C2×C4×C13—24no—
246C2×C4×C13—24no—
247C2×C4×C13—24no—
248C2×C4×C13—22no—
249C2×C4×C13—24no—
250C2×C4×C13—22no—
251C2×C4×C13—22no—
252C2×C4×C13—24no—
253C2×C4×C13—24no—
254C2×C4×C13—416no—
255C2×C4×C13—41no—
256C2×C4×C13—41no—
257C2×C4×C13—41no—
258C2×C4×C13—41no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
261C2×C4×C13—42no—
262C2×C4×C13—42no—
263C2×C4×C13—42no—
264C2×C4×C13C4×D414yes—
265C2×C4×C13#12014yes—
266C2×C4×C13C22⋊Q814yes—
267C2×Dic13—24no—
268C2×Dic13—24no—
269C2×Dic13—24no—
270C2×Dic13—24no—
271C2×Dic13—24no—
272C2×Dic13—24no—
273C2×Dic13—24no—
274C2×Dic13—24no—
275C2×Dic13—24no—
276C2×Dic13—24no—
277C2×Dic13—24no—
278C2×Dic13—24no—
279C2×Dic13—42no—
280C2×Dic13—42no—
281C2×Dic13—42no—
282C2×Dic13—42no—
283C2×Dic13—42no—
284C2×Dic13—42no—
285C2×Dic13—42no—
286C2×Dic13—42no—
287C2×Dic13—42no—
288C2×Dic13—42no—
289C2×Dic13—42no—
290C2×Dic13—42no—
291C2×Dic13—42no—
292C2×Dic13—42no—
293C2×Dic13—42no—
294C2×Dic13C4×D418yes—
295Dic26—232no—
296C23×Q8—131no—
297C24×C4C2×D1311yescharacteristic
298C2×C4×Q8—268no—
299C2×C4×Q8—264no—
300C22×C4⋊C4—132no—
301C22×C4⋊C4—131no—
302C22×C4⋊C4—132no—
303C22×C22⋊C4—132no—
304C22×C22⋊C4—132no—
305C22×C22⋊C4C2×D1311yescharacteristic
306C22×C22⋊C4C2×D1311yescharacteristic
307C22×C42—132no—
308C22×C42C2×D1311yescharacteristic
309C22×C42C2×D1311yescharacteristic
310#393—138no—
311#385—268no—
312C4×C22⋊C4—24no—
313C2×#95—131no—
314C2×#95—132no—
315C2×#95—132no—
316C2×#95—132no—
317C2×#95—131no—
318C2×#95—132no—
319C2×#95—132no—
320C2×#95—132no—
321C2×#95—132no—
322C2×#95C2×D1311yescharacteristic
323C2×#95C2×D1311yescharacteristic
324C22×C13—28no—
325C22×C13—24no—
326C22×C13—42no—
327C22×C13—42no—
328C22×C13—42no—
329C22×C13—44no—
330C22×C13—42no—
331C22×C13—42no—
332C22×C13—44no—
333C22×C13—44no—
334C22×C13—42no—
335C22×C13C2×C4×D411yescharacteristic
336C22×C13C2×C4×D412yes—
337C22×C13C2×C4×Q811yescharacteristic
338C22×C13C4×#4111yescharacteristic
339C22×C13#52012yes—
340C22×C13C2×#12011yescharacteristic
341C22×C13C2×C22⋊Q811yescharacteristic
342C22×C13#52511yescharacteristic
343C22×C13#52512yes—
344C22×C13#52511yescharacteristic
345C22×C13#52911yescharacteristic
346C22×C13#53311yescharacteristic
347C22×C13#53511yescharacteristic
348C22×C13#53611yescharacteristic
349C22×C13#54212yes—
350C22×C13#54612yes—
351C22×C13#54612yes—
352C22×C13#54812yes—
353C22×C13D4×Q812yes—
354C22×C13#55212yes—
355C22×C13#55412yes—
356C22×C13#55612yes—
357C22×C13#55912yes—
358C4×C13—24no—
359C4×C13—24no—
360C4×C13—24no—
361C4×C13—42no—
362C4×C13—42no—
363C4×C13—42no—
364Dic13—28no—
365Dic13—44no—
366Dic13—44no—
367Dic13—82no—
368C25C22×D1311yescharacteristic
369C22×Q8—134no—
370C22×Q8—264no—
371C22×Q8—264no—
372C22×Q8—262no—
373C22×Q8—262no—
374C23×C4—22no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—24no—
378C23×C4—42no—
379C23×C4—132no—
380C23×C4—132no—
381C23×C4—132no—
382C23×C4—132no—
383C23×C4C22×D1311yescharacteristic
384C23×C4C22×D1311yescharacteristic
385C23×C4C22×D1311yescharacteristic
386C23×C4C22×D1311yescharacteristic
387C23×C4C22×D1311yescharacteristic
388C23×C4C22×D1311yescharacteristic
389C4×Q8—2632no—
390C2×C4⋊C4—138no—
391C2×C4⋊C4—264no—
392C2×C4⋊C4—264no—
393C2×C4⋊C4—264no—
394C2×C4⋊C4—262no—
395C2×C4⋊C4—264no—
396C2×C4⋊C4—264no—
397C2×C4⋊C4—522no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—138no—
403C2×C22⋊C4—264no—
404C2×C22⋊C4—264no—
405C2×C22⋊C4—264no—
406C2×C22⋊C4—264no—
407C2×C22⋊C4—522no—
408C2×C22⋊C4Dic2614yescomplemented
409C2×C22⋊C4C4×D1314yes—
410C2×C42—24no—
411C2×C42—22no—
412C2×C42—42no—
413C2×C42—264no—
414C2×C42—264no—
415C2×C42—264no—
416C2×C42C22⋊D1314yes—
417#95—22no—
418#95—22no—
419#95—138no—
420#95—134no—
421#95—264no—
422#95—264no—
423#95—264no—
424#95—262no—
425#95—264no—
426#95—522no—
427#95—522no—
428C2×C13—44no—
429C2×C13C4×#12011yescharacteristic
430C2×C13C4×C22⋊Q811yescharacteristic
431C2×C13#232611yescharacteristic
432C2×C13#236012yes—
433C2×C13#236712yes—
434C2×C13#240411yescharacteristic
435C2×C13#240911yescharacteristic
436C2×C13#245812yes—
437C2×C13#246512yes—
438C2×C13#258011yescharacteristic
439C2×C13#259911yescharacteristic
440C24—21no—
441C24—21no—
442C24—21no—
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24—21no—
448C24C2×Dic2612yes—
449C24C2×C4×D1312yes—
450C24#612712yes—
451C24C23×D1311yescharacteristic
452C2×Q8—2616no—
453C2×Q8—268no—
454C2×Q8—2616no—
455C2×Q8—268no—
456C2×Q8—268no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—28no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—134no—
482C22×C4—134no—
483C22×C4—134no—
484C22×C4—134no—
485C22×C4—134no—
486C22×C4—262no—
487C22×C4—262no—
488C22×C4—262no—
489C22×C4—262no—
490C22×C4—262no—
491C22×C4—262no—
492C22×C4—262no—
493C22×C4—262no—
494C22×C4—262no—
495C22×C4—262no—
496C22×C4—262no—
497C22×C4—262no—
498C22×C4—262no—
499C22×C4—262no—
500C22×C4—262no—
501C22×C4—262no—
502C22×C4—262no—
503C22×C4—262no—
504C22×C4—262no—
505C22×C4—262no—
506C22×C4—262no—
507C22×C4—262no—
508C22×C4—262no—
509C22×C4C2×Dic2612yes—
510C22×C4C2×Dic2612yes—
511C22×C4C2×C4×D1312yes—
512C22×C4C2×C4×D1312yes—
513C22×C4#612712yes—
514C22×C4#612712yes—
515C22×C4#612712yes—
516C22×C4D4×D1314yes—
517C22×C4C2×C22⋊D1312yes—
518C22×C4C2×C22⋊D1312yes—
519C22×C4C2×C22⋊D1312yes—
520C4⋊C4—268no—
521C4⋊C4—2616no—
522C4⋊C4—528no—
523C22⋊C4—28no—
524C22⋊C4—28no—
525C22⋊C4—2616no—
526C22⋊C4—528no—
527C42—28no—
528C42—44no—
529C42—2616no—
530C13#1417211yescharacteristic, complemented, Sylow
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—22no—
536C23—24no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—41no—
552C23—41no—
553C23C4×Dic2614yescomplemented
554C23C22×Dic2611yescharacteristic
555C23C22×C4×D1311yescharacteristic
556C23C2×#612711yescharacteristic
557C23C2×#612711yescharacteristic
558C23C2×#612711yescharacteristic
559C23C2×#612711yescharacteristic
560C23C2×D4×D1312yes—
561C23#8825112yes—
562C23#41×D1312yes—
563C23#8825712yes—
564C23C22×C22⋊D1311yescharacteristic
565Q8—2632no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—416no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—138no—
595C2×C4—264no—
596C2×C4—264no—
597C2×C4—264no—
598C2×C4—264no—
599C2×C4—264no—
600C2×C4—264no—
601C2×C4—264no—
602C2×C4—264no—
603C2×C4—264no—
604C2×C4—264no—
605C2×C4—264no—
606C2×C4—264no—
607C2×C4—522no—
608C2×C4—522no—
609C2×C4—522no—
610C2×C4—522no—
611C2×C4—522no—
612C2×C4—522no—
613C2×C4—522no—
614C2×C4—522no—
615C2×C4—522no—
616C2×C4—522no—
617C2×C4—522no—
618C2×C4—522no—
619C2×C4—522no—
620C2×C4—522no—
621C2×C4—522no—
622C2×C4#8817814yes—
623C2×C4C4×C22⋊D1314yes—
624C2×C4#8819114yes—
625C22—28no—
626C22—24no—
627C22—44no—
628C22—42no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—44no—
635C22—42no—
636C22C2×C4×Dic2611yescharacteristic
637C22#1171917511yescharacteristic
638C22C4×#612711yescharacteristic
639C22#1171918411yescharacteristic
640C22#1171919512yes—
641C22#1171919512yes—
642C22#1171923612yes—
643C22#1171923812yes—
644C22D4×Dic2612yes—
645C22#1171924412yes—
646C22C4×D4×D1312yes—
647C22#1171925112yes—
648C22#1171925212yes—
649C22#1171925712yes—
650C22#1171926412yes—
651C22#1171926612yes—
652C22C2×#8817811yescharacteristic
653C22C2×C4×C22⋊D1311yescharacteristic
654C22#1171949611yescharacteristic
655C22C2×#8819111yescharacteristic
656C22#1171951011yescharacteristic
657C22#1171953011yescharacteristic
658C22#1171953311yescharacteristic
659C4—24no—
660C4—24no—
661C4—24no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—268no—
666C4—524no—
667C4—524no—
668C4—1042no—
669C2—44no—
670C2C4×#8819111yescharacteristic
671C2C4×#8817811yescharacteristic
672C2#1311754811yescharacteristic
673C2#1311784212yes—
674C2#1311790812yes—
675C2#1311815511yescharacteristic
676C2#1311820111yescharacteristic
677C2#1311875812yes—
678C2#1311885712yes—
679C2#1312019011yescharacteristic
680C2#1312041311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC2×C4×C13×C22⋊C4C2
socle, cosocleC24×C13C24
prime radicalC25×C13C23
hypercenterC2×C4×C22⋊C4D13
hypocenterC13#14172
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic26C2semidirect
4. #13113106C2semidirect
5. #13113100C2semidirect
8. #13113069C2semidirect
9. #13113067C2semidirect
10. #13113063C2semidirect
39. #11718608C4semidirect
60. C22×Dic26C2×C4semidirect
66. C2×C13×C22⋊C4Q8semidirect
98. #88071C2×C4semidirect
99. #88071C2×C4semidirect
224. C23×C13C4×Q8semidirect
408. C2×C22⋊C4Dic26semidirect
530. C13#14172semidirect
553. C23C4×Dic26semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic262
C2×#117186082
C2×#117188712
#131131062
#131131002
#131130812
#131130732
#131130692
#131130672
#131130632
C2×C4×C13×C22⋊C42
#1417213