Finite Group Databasebeta

← #15799627

Subgroups — #15799627

Order 3328.

1934 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88198C211yesmaximal, characteristic
2C2×#11718317C211yesmaximal, characteristic
3C2×#11718640C211yesmaximal, characteristic
4C2×#11718890C211yesmaximal, characteristic
5C2×#11718633C211yesmaximal, characteristic
6C2×#11718637C212yesmaximal
7#13113100C212yesmaximal
8#13113099C212yesmaximal
9#13113064C212yesmaximal
10#95×Dic13C211yesmaximal, characteristic
11C2×C13×#395C211yesmaximal, characteristic
12C23×Q8×C13C2211yescharacteristic
13C23×Dic26C2211yescharacteristic
14C22×C13×C4⋊C4C2211yescharacteristic
15C22×C13×C4⋊C4C2212yes—
16C2×#88198—28no—
17C2×#88198—24no—
18C22×C4⋊Dic13C2212yes—
19C22×#6101C2212yes—
20C22×#6101C2211yescharacteristic
21C22×#6101C2211yescharacteristic
22C22×#6101C2212yes—
23C22×#6101C2212yes—
24C22×#6101C2212yes—
25C22×#6101C2211yescharacteristic
26C22×C4×Dic13C2211yescharacteristic
27C22×C4×Dic13C2212yes—
28C13×#395—24no—
29C2×C13×#95C2212yes—
30C2×C13×#95C2211yescharacteristic
31#11718890—24no—
32C2×#88071C2211yescharacteristic
33C2×#88071C2212yes—
34C2×#88071C2212yes—
35C2×#88071C2211yescharacteristic
36C2×#88071C2212yes—
37C2×#88071C2211yescharacteristic
38C2×#88071C2212yes—
39#11718640—24no—
40#11718637—28no—
41#11718633—24no—
42#11718317—24no—
43C22×Q8×C13—24no—
44C22×Q8×C13—22no—
45C22×Q8×C13—22no—
46C22×Q8×C13—22no—
47C22×Q8×C13—42no—
48C23×C4×C13C2311yescharacteristic
49C23×C4×C13C2312yes—
50C23×C4×C13C2311yescharacteristic
51C23×C4×C13C2311yescharacteristic
52C23×C4×C13C2312yes—
53C23×Dic13C2311yescharacteristic
54C23×Dic13C2312yes—
55C23×Dic13C2311yescharacteristic
56C23×Dic13C2311yescharacteristic
57C23×Dic13C2312yes—
58C23×Dic13C2311yescharacteristic
59C22×Dic26—22no—
60C22×Dic26—24no—
61C22×Dic26—22no—
62C22×Dic26—22no—
63C22×Dic26—42no—
64C2×C13×C4⋊C4—24no—
65C2×C13×C4⋊C4—22no—
66C2×C13×C4⋊C4—22no—
67C2×C13×C4⋊C4—24no—
68C2×C13×C4⋊C4—22no—
69C2×C13×C4⋊C4—42no—
70#88198—232no—
71C2×C4⋊Dic13—24no—
72C2×C4⋊Dic13—24no—
73C2×C4⋊Dic13—42no—
74C2×#6101—24no—
75C2×#6101—24no—
76C2×#6101—24no—
77C2×#6101—24no—
78C2×#6101—24no—
79C2×#6101—22no—
80C2×#6101—22no—
81C2×#6101—22no—
82C2×#6101—22no—
83C2×#6101—22no—
84C2×#6101—24no—
85C2×#6101—22no—
86C2×#6101—24no—
87C2×#6101—24no—
88C2×#6101—24no—
89C2×#6101—24no—
90C2×#6101—42no—
91C2×#6101—42no—
92C2×#6101D418yes—
93C2×#6101Q814yescomplemented
94C2×C4×Dic13—24no—
95C2×C4×Dic13—24no—
96C2×C4×Dic13—24no—
97C2×C4×Dic13D414yes—
98C2×C4×Dic13Q818yescomplemented
99C13×#95—24no—
100C13×#95D414yes—
101#88071—24no—
102#88071—22no—
103#88071—22no—
104#88071—24no—
105#88071—24no—
106#88071D414yes—
107#88071Q818yescomplemented
108#15143—131nomaximal, Sylow
109C24×C13C2411yescharacteristic
110C2×Q8×C13—28no—
111C2×Q8×C13—48no—
112C2×Q8×C13—48no—
113C2×Q8×C13—44no—
114C2×Q8×C13—44no—
115C22×C4×C13—22no—
116C22×C4×C13—22no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—22no—
120C22×C4×C13—22no—
121C22×C4×C13—21no—
122C22×C4×C13—21no—
123C22×C4×C13—22no—
124C22×C4×C13—21no—
125C22×C4×C13—22no—
126C22×C4×C13—22no—
127C22×C4×C13—22no—
128C22×C4×C13—21no—
129C22×C4×C13—22no—
130C22×C4×C13—22no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—22no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—22no—
138C22×C4×C13C2×D414yes—
139C22×C4×C13C2×D412yes—
140C22×C4×C13C2×D412yes—
141C22×C4×C13C2×D412yes—
142C22×C4×C13C2×D414yes—
143C22×C4×C13C2×Q812yes—
144C22×C4×C13C2×Q814yes—
145C22×C4×C13C2×Q814yes—
146C22×Dic13—22no—
147C22×Dic13—22no—
148C22×Dic13—22no—
149C22×Dic13—22no—
150C22×Dic13—21no—
151C22×Dic13—21no—
152C22×Dic13—22no—
153C22×Dic13—21no—
154C22×Dic13—21no—
155C22×Dic13—21no—
156C22×Dic13—22no—
157C22×Dic13—22no—
158C22×Dic13—22no—
159C22×Dic13—22no—
160C22×Dic13—21no—
161C22×Dic13—22no—
162C22×Dic13—22no—
163C22×Dic13—21no—
164C22×Dic13—21no—
165C22×Dic13—21no—
166C22×Dic13—21no—
167C22×Dic13—22no—
168C22×Dic13—21no—
169C22×Dic13—22no—
170C22×Dic13C2×D414yes—
171C22×Dic13C2×D412yes—
172C22×Dic13C2×D414yes—
173C22×Dic13C2×D414yes—
174C22×Dic13C2×D412yes—
175C22×Dic13C2×Q812yes—
176C22×Dic13C2×Q814yes—
177C22×Dic13C2×Q814yes—
178C22×Dic13C2×Q814yes—
179C22×Dic13C2×Q814yes—
180C22×Dic13C2×Q812yes—
181C22×Dic13#4112yes—
182C2×Dic26—28no—
183C2×Dic26—48no—
184C2×Dic26—48no—
185C2×Dic26—44no—
186C2×Dic26—44no—
187C13×C4⋊C4—48no—
188C13×C4⋊C4—44no—
189C4⋊Dic13—48no—
190#6101—216no—
191#6101—216no—
192#6101—28no—
193#6101—48no—
194#6101—48no—
195#6101—44no—
196#6101—44no—
197C4×Dic13—28no—
198C4×Dic13—216no—
199C22×C4⋊Q8—131no—
200C2×#395—131no—
201C2×#395—131no—
202C2×#395—131no—
203C2×#395D1311yescharacteristic
204C2×#389—131no—
205C2×#384—132no—
206#1444—132no—
207#1444—132no—
208C4⋊#95—132no—
209C4×#95—131no—
210C23×C13C22×D411yescharacteristic
211C23×C13C22×D412yes—
212C23×C13C22×D411yescharacteristic
213C23×C13C22×D411yescharacteristic
214C23×C13C22×Q812yes—
215C23×C13C22×Q811yescharacteristic
216C23×C13C22×Q812yes—
217C23×C13C2×#4111yescharacteristic
218C23×C13H(5,𝔽2)12yes—
219C23×C132−1+411yescharacteristic
220C23×C132−1+411yescharacteristic
221Q8×C13—416no—
222C2×C4×C13—22no—
223C2×C4×C13—24no—
224C2×C4×C13—22no—
225C2×C4×C13—24no—
226C2×C4×C13—24no—
227C2×C4×C13—24no—
228C2×C4×C13—24no—
229C2×C4×C13—24no—
230C2×C4×C13—24no—
231C2×C4×C13—24no—
232C2×C4×C13—24no—
233C2×C4×C13—22no—
234C2×C4×C13—24no—
235C2×C4×C13—24no—
236C2×C4×C13—22no—
237C2×C4×C13—22no—
238C2×C4×C13—41no—
239C2×C4×C13—42no—
240C2×C4×C13—42no—
241C2×C4×C13—41no—
242C2×C4×C13—42no—
243C2×C4×C13—42no—
244C2×C4×C13—41no—
245C2×C4×C13—41no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—41no—
249C2×C4×C13—42no—
250C2×C4×C13—42no—
251C2×C4×C13#12714yes—
252C2×C4×C13C4⋊Q818yescomplemented
253C2×Dic13—22no—
254C2×Dic13—24no—
255C2×Dic13—22no—
256C2×Dic13—24no—
257C2×Dic13—24no—
258C2×Dic13—24no—
259C2×Dic13—24no—
260C2×Dic13—24no—
261C2×Dic13—24no—
262C2×Dic13—24no—
263C2×Dic13—24no—
264C2×Dic13—22no—
265C2×Dic13—24no—
266C2×Dic13—24no—
267C2×Dic13—22no—
268C2×Dic13—22no—
269C2×Dic13—41no—
270C2×Dic13—42no—
271C2×Dic13—42no—
272C2×Dic13—41no—
273C2×Dic13—42no—
274C2×Dic13—42no—
275C2×Dic13—41no—
276C2×Dic13—41no—
277C2×Dic13—41no—
278C2×Dic13—41no—
279C2×Dic13—41no—
280C2×Dic13—42no—
281C2×Dic13—42no—
282C2×Dic13#12014yes—
283C2×Dic13C22⋊Q818yes—
284C2×Dic13C22⋊Q814yes—
285C2×Dic13C4⋊Q814yes—
286C2×Dic13C4⋊Q814yes—
287C2×Dic13C4⋊Q818yes—
288C2×Dic13C4⋊Q818yescomplemented
289Dic26—416no—
290C23×Q8—131no—
291C23×Q8C2×D1311yescharacteristic
292C2×C4⋊Q8—264no—
293C2×C4⋊Q8—268no—
294C22×C4⋊C4—132no—
295C22×C4⋊C4—131no—
296C22×C4⋊C4—131no—
297C22×C4⋊C4—132no—
298C22×C4⋊C4—131no—
299C22×C4⋊C4—132no—
300C22×C4⋊C4—132no—
301C22×C4⋊C4—132no—
302C22×C4⋊C4C2×D1311yescharacteristic
303C22×C4⋊C4C2×D1312yes—
304C22×C42—131no—
305C22×C42—132no—
306#395—24no—
307#395—264no—
308#395—264no—
309#395—264no—
310#389—264no—
311#384—268no—
312C2×#95—131no—
313C2×#95—132no—
314C2×#95—132no—
315C2×#95—132no—
316C2×#95—132no—
317C2×#95—131no—
318C2×#95—131no—
319C2×#95C2×D1312yes—
320C2×#95C2×D1311yescharacteristic
321C22×C13C2×#12011yescharacteristic
322C22×C13C2×C22⋊Q811yescharacteristic
323C22×C13C2×C22⋊Q812yes—
324C22×C13C2×#12711yescharacteristic
325C22×C13C2×C4⋊Q811yescharacteristic
326C22×C13C2×C4⋊Q811yescharacteristic
327C22×C13C2×C4⋊Q812yes—
328C22×C13C2×C4⋊Q812yes—
329C22×C13C2×C4⋊Q812yes—
330C22×C13#53511yescharacteristic
331C22×C13#53611yescharacteristic
332C22×C13#54412yes—
333C22×C13#54412yes—
334C22×C13D4212yes—
335C22×C13D4212yes—
336C22×C13#54711yescharacteristic
337C22×C13#54711yescharacteristic
338C22×C13D4×Q812yes—
339C22×C13D4×Q811yescharacteristic
340C22×C13D4×Q812yes—
341C22×C13D4×Q811yescharacteristic
342C22×C13#55412yes—
343C22×C13Q8212yes—
344C4×C13—24no—
345C4×C13—28no—
346C4×C13—44no—
347C4×C13—42no—
348C4×C13—81no—
349Dic13—24no—
350Dic13—28no—
351Dic13—44no—
352Dic13—42no—
353Dic13—81no—
354Dic13#39518yescomplemented
355C22×Q8—24no—
356C22×Q8—22no—
357C22×Q8—22no—
358C22×Q8—22no—
359C22×Q8—42no—
360C22×Q8—264no—
361C22×Q8—262no—
362C22×Q8—262no—
363C22×Q8—262no—
364C22×Q8—522no—
365C23×C4—131no—
366C23×C4—132no—
367C23×C4—131no—
368C23×C4—132no—
369C23×C4—131no—
370C23×C4—131no—
371C23×C4C22×D1311yescharacteristic
372C23×C4C22×D1312yes—
373C23×C4C22×D1311yescharacteristic
374C23×C4C22×D1311yescharacteristic
375C23×C4C22×D1312yes—
376C4⋊Q8—2632no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—42no—
383C2×C4⋊C4—138no—
384C2×C4⋊C4—134no—
385C2×C4⋊C4—264no—
386C2×C4⋊C4—264no—
387C2×C4⋊C4—264no—
388C2×C4⋊C4—264no—
389C2×C4⋊C4—262no—
390C2×C4⋊C4—262no—
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—264no—
398C2×C4⋊C4—264no—
399C2×C4⋊C4—262no—
400C2×C4⋊C4—262no—
401C2×C4⋊C4—262no—
402C2×C4⋊C4—264no—
403C2×C4⋊C4—522no—
404C2×C4⋊C4—522no—
405C2×C4⋊C4—522no—
406C2×C42—134no—
407C2×C42—138no—
408C2×C42—264no—
409C2×C42—264no—
410C2×C42—264no—
411#95—24no—
412#95—134no—
413#95—138no—
414#95—262no—
415#95—262no—
416#95—264no—
417#95—264no—
418#95—264no—
419#95C22⋊D1314yes—
420C2×C13C2×#39511yescharacteristic
421C2×C13#243311yescharacteristic
422C2×C13#243411yescharacteristic
423C2×C13#244012yes—
424C2×C13#244012yes—
425C2×C13#254211yescharacteristic
426C2×C13#255112yes—
427C2×C13#255112yes—
428C2×C13#256711yescharacteristic
429C2×C13#266011yescharacteristic
430C2×C13#286811yescharacteristic
431C24C23×D1311yescharacteristic
432C2×Q8—28no—
433C2×Q8—48no—
434C2×Q8—48no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—268no—
438C2×Q8—528no—
439C2×Q8—528no—
440C2×Q8—524no—
441C2×Q8—524no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—132no—
466C22×C4—134no—
467C22×C4—134no—
468C22×C4—132no—
469C22×C4—132no—
470C22×C4—134no—
471C22×C4—132no—
472C22×C4—134no—
473C22×C4—134no—
474C22×C4—134no—
475C22×C4—134no—
476C22×C4—132no—
477C22×C4—262no—
478C22×C4—262no—
479C22×C4—262no—
480C22×C4—261no—
481C22×C4—262no—
482C22×C4—262no—
483C22×C4—261no—
484C22×C4—261no—
485C22×C4—261no—
486C22×C4—262no—
487C22×C4—262no—
488C22×C4—262no—
489C22×C4—262no—
490C22×C4—262no—
491C22×C4—261no—
492C22×C4—261no—
493C22×C4—261no—
494C22×C4—261no—
495C22×C4—262no—
496C22×C4—261no—
497C22×C4—262no—
498C22×C4—261no—
499C22×C4—262no—
500C22×C4—261no—
501C22×C4D4×D1314yes—
502C22×C4D4×D1312yes—
503C22×C4D4×D1312yes—
504C22×C4Q8×D1312yes—
505C22×C4Q8×D1314yes—
506C22×C4Q8×D1314yes—
507C22×C4C2×C22⋊D1312yes—
508C22×C4C2×C22⋊D1314yes—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—2616no—
512C4⋊C4—268no—
513C4⋊C4—2616no—
514C4⋊C4—528no—
515C4⋊C4—528no—
516C4⋊C4—524no—
517C4⋊C4—528no—
518C4⋊C4—524no—
519C42—268no—
520C42—2616no—
521C13#1514311yescharacteristic, complemented, Sylow
522C23C2×D4×D1311yescharacteristic
523C23C2×D4×D1311yescharacteristic
524C23C2×D4×D1312yes—
525C23#8825112yes—
526C23C2×Q8×D1311yescharacteristic
527C23C2×Q8×D1312yes—
528C23C2×Q8×D1312yes—
529C23#8825411yescharacteristic
530C23#41×D1311yescharacteristic
531C23#8825711yescharacteristic
532C23C22×C22⋊D1311yescharacteristic
533Q8—416no—
534Q8—5216no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—138no—
565C2×C4—134no—
566C2×C4—134no—
567C2×C4—138no—
568C2×C4—134no—
569C2×C4—134no—
570C2×C4—138no—
571C2×C4—264no—
572C2×C4—264no—
573C2×C4—264no—
574C2×C4—262no—
575C2×C4—264no—
576C2×C4—264no—
577C2×C4—262no—
578C2×C4—264no—
579C2×C4—264no—
580C2×C4—264no—
581C2×C4—264no—
582C2×C4—262no—
583C2×C4—264no—
584C2×C4—264no—
585C2×C4—262no—
586C2×C4—262no—
587C2×C4—522no—
588C2×C4—521no—
589C2×C4—522no—
590C2×C4—521no—
591C2×C4—522no—
592C2×C4—522no—
593C2×C4—521no—
594C2×C4—521no—
595C2×C4—522no—
596C2×C4—521no—
597C2×C4—521no—
598C2×C4—522no—
599C2×C4—521no—
600C2×C4#8819414yes—
601C2×C4#8819818yescomplemented
602C22D13×#12011yescharacteristic
603C22#1171929512yes—
604C22#1171929711yescharacteristic
605C22#1171932812yes—
606C22D13×C22⋊Q811yescharacteristic
607C22D13×C22⋊Q812yes—
608C22#1171933311yescharacteristic
609C22#1171933611yescharacteristic
610C22#1171933712yes—
611C22#1171933711yescharacteristic
612C22#1171934112yes—
613C22#1171942712yes—
614C22D13×C4⋊Q811yescharacteristic
615C22D13×C4⋊Q812yes—
616C22D13×C4⋊Q812yes—
617C22D13×C4⋊Q811yescharacteristic
618C22#1171943212yes—
619C22#1171943412yes—
620C22D4×C22⋊D1312yes—
621C22C2×#8819411yescharacteristic
622C22C2×#8819812yes—
623C22Q8×C22⋊D1311yescharacteristic
624C22#1171952911yescharacteristic
625C4—24no—
626C4—28no—
627C4—44no—
628C4—42no—
629C4—81no—
630C4—138no—
631C4—264no—
632C4—268no—
633C4—524no—
634C4—522no—
635C4—1041no—
636C2D13×#39511yescharacteristic
637C2#1311847111yescharacteristic
638C2#1311847511yescharacteristic
639C2#1311853112yes—
640C2#1311853512yes—
641C2#1311978611yescharacteristic
642C2#1311986112yes—
643C2#1311986312yes—
644C2#1312004011yescharacteristic
645C2#1312095411yescharacteristic
646C2#1312323211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutator, socle, prime radical, cosocleC24×C13C24
FittingC2×C13×#395C2
hypercenterC2×#395D13
hypocenterC13#15143
LCSG ▷ C24×C13 ▷ C13
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
93. C2×#6101Q8semidirect
98. C2×C4×Dic13Q8semidirect
107. #88071Q8semidirect
252. C2×C4×C13C4⋊Q8semidirect
288. C2×Dic13C4⋊Q8semidirect
354. Dic13#395semidirect
521. C13#15143semidirect
601. C2×C4#88198semidirect

Maximal subgroups

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

subgroupindex
C22×#881982
C2×#117183172
C2×#117186402
C2×#117188902
C2×#117186332
C2×#117186372
#131131002
#131130992
#131130642
#95×Dic132
C2×C13×#3952
#1514313