Finite Group Databasebeta

← #15879621

Subgroups — #15879621

Order 3328.

1004 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#6127C211yesmaximal, characteristic
2#13117314C211yesmaximal, characteristic, complemented
3#13115816C212yesmaximal, complemented
4#13115812C211yesmaximal, characteristic, complemented
5#13115780C212yesmaximal, complemented
6#13115780C211yesmaximal, characteristic, complemented
7#13114853C212yesmaximal
8#13114853C211yesmaximal, characteristic
9#13114649C212yesmaximal
10C4×#88065C211yesmaximal, characteristic
11Q8×C13×M16C211yesmaximal, characteristic, complemented
12C2×C4×Q8×C13C2211yescharacteristic
13Q8×C22⋊D13—21no—
14#11719284C411yescharacteristic, complemented
15#11719284C412yescomplemented
16#11719283—21no—
17#11719283—22no—
18Q8×D52C411yescharacteristic, complemented
19C4×Q8×D13—21no—
20#11719272C411yescharacteristic, complemented
21#11719272C412yescomplemented
22Q8×Dic26C411yescharacteristic, complemented
23#11719216—22no—
24#11719216—21no—
25C4×#6127C2211yescharacteristic
26C4×#6127C2212yes—
27#11719175C2211yescharacteristic
28#11719175C2212yes—
29C13×#446—21no—
30C13×#446—22no—
31C8×Q8×C13—21no—
32C13×#423C2211yescharacteristic
33C13×#423C2212yes—
34C4×C13×M16C2211yescharacteristic
35C4×C13×M16C2212yes—
36#11718891—21no—
37C2×#88065C2211yescharacteristic
38C2×#88065C2212yes—
39C2×#88065C2211yescharacteristic
40#11718646—21no—
41#11718642—22no—
42C4×C4⋊Dic13C2212yes—
43C4×C4⋊Dic13C2211yescharacteristic
44#11718593C2212yes—
45C42×Dic13C2211yescharacteristic
46#11718147C2211yescharacteristic
47#11718147C2211yescharacteristic
48#11718147C2212yes—
49#11718147C2212yes—
50#11718147C2212yes—
51#11718147C2212yes—
52#11718147C2212yes—
53C22×Q8×C13D411yescharacteristic
54C2×Q8×D13—21no—
55C2×#6127C2311yescharacteristic
56C2×C13×M16C2311yescharacteristic
57C2×C13×M16C2312yes—
58C2×C13×M16C2311yescharacteristic
59C4×Q8×C13C2×C411yescharacteristic
60C4×Q8×C13C2×C411yescharacteristic
61C4×Q8×C13C2×C412yes—
62C4×Q8×C13D411yescharacteristic
63C4×Q8×C13D411yescharacteristic
64C4×Q8×C13D412yes—
65C2×C13×C4⋊C4D412yes—
66C2×C13×C4⋊C4D411yescharacteristic
67C2×C42×C13C2311yescharacteristic
68C2×C42×C13C2312yes—
69#88200—21no—
70#88200—22no—
71Q8×Dic13—21no—
72#88198—21no—
73#88198—22no—
74C4×C22⋊D13—21no—
75C4×C22⋊D13—22no—
76C4×C22⋊D13—21no—
77C4×C22⋊D13—22no—
78#88180C2311yescharacteristic
79#88180C2312yes—
80C2×C4⋊Dic13—21no—
81C2×C4⋊Dic13—22no—
82#88178—21no—
83#88178—22no—
84#88178—21no—
85#88178—22no—
86C2×C4×Dic13—21no—
87C2×C4×Dic13—21no—
88C2×C4×Dic13C2311yescharacteristic
89C2×C4×Dic13C2312yes—
90C2×C4×Dic13C2311yescharacteristic
91#88151—22no—
92#88151—21no—
93#88150—22no—
94#88150—22no—
95#88150—22no—
96C4⋊C4×D13—22no—
97C4⋊C4×D13—21no—
98#88143—22no—
99#88143—22no—
100#88143—22no—
101#88142—22no—
102#88142—21no—
103#88141—22no—
104#88141—21no—
105C4×D52C2×C411yescharacteristic
106C4×D52C2×C412yes—
107C4×D52C2×C411yescharacteristic
108C4×D52C2×C412yes—
109#88126—22no—
110#88126—21no—
111C42×D13—21no—
112C42×D13—22no—
113#88124C2×C412yes—
114#88124C2×C412yes—
115#88124C2×C412yes—
116#88123C2×C411yescharacteristic
117#88123C2×C412yes—
118#88123C2×C411yescharacteristic
119#88123C2×C412yes—
120C4×Dic26—21no—
121C4×Dic26—22no—
122C4×Dic26—22no—
123C4×Dic26—21no—
124C4×Dic26C2×C411yescharacteristic
125C4×Dic26C2×C412yes—
126C4×Dic26C2×C411yescharacteristic
127C4×Dic26C2×C412yes—
128C13×C4⋊C8—21no—
129C13×C4⋊C8—22no—
130C13×C4⋊C8—22no—
131C13×C4⋊C8—21no—
132C13×#97—22no—
133C13×#97—21no—
134C4×C8×C13—21no—
135C4×C8×C13—22no—
136#88071—22no—
137#88071—22no—
138#88071—22no—
139#88071—21no—
140#88065—22no—
141#88065—24no—
142#88065Q814yescomplemented
143#22529—131nomaximal, Sylow
144C2×Q8×C13—42no—
145C2×Q8×C13C22⋊C411yescharacteristic
146C2×Q8×C13C22⋊C411yescharacteristic
147C2×Q8×C13C22⋊C412yes—
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13C2×D412yes—
150C22×C4×C13C2×D411yescharacteristic
151C22×C4×C13C2×D412yes—
152C22×C4×C13C2411yescharacteristic
153C2×C22⋊D13—21no—
154C22×Dic13—21no—
155C22×Dic13—21no—
156C22×Dic13—22no—
157Q8×D13—44no—
158#6127—21no—
159#6127—21no—
160#6127C2×Q812yes—
161#6127#4112yes—
162C2×D52C22×C411yescharacteristic
163C2×C4×D13—21no—
164C2×C4×D13—22no—
165C2×C4×D13—21no—
166C2×C4×D13—21no—
167C2×C4×D13—22no—
168C2×Dic26—22no—
169C2×Dic26—21no—
170C2×Dic26—22no—
171C2×Dic26—21no—
172C2×Dic26C22×C411yescharacteristic
173C13×M16—21no—
174C13×M16—21no—
175C13×M16—22no—
176C13×M16—21no—
177C13×M16—22no—
178C13×M16C2×Q812yescomplemented
179C2×C8×C13—21no—
180C2×C8×C13—21no—
181C2×C8×C13—22no—
182C13×C4⋊C4C22⋊C412yes—
183C13×C4⋊C4C22⋊C411yescharacteristic
184C13×C4⋊C4C22⋊C412yes—
185C13×C4⋊C4C22⋊C411yescharacteristic
186C13×C4⋊C4C22⋊C412yes—
187C13×C4⋊C4C22⋊C412yes—
188C13×C4⋊C4C22⋊C412yes—
189C42×C13C22×C412yes—
190C42×C13C22×C411yescharacteristic
191C42×C13C22×C411yescharacteristic
192C42×C13C22×C412yes—
193C42×C13C2×D411yescharacteristic
194C42×C13C2×D412yes—
195C42×C13C2×D412yes—
196C42×C13C2×D411yescharacteristic
197C22⋊Dic13—22no—
198C22⋊Dic13—21no—
199#6103—22no—
200#6103—21no—
201#6103—21no—
202#6103—22no—
203C4⋊Dic13—22no—
204C4⋊Dic13—21no—
205C4⋊Dic13—22no—
206C4⋊Dic13—21no—
207C4⋊Dic13—44no—
208C4⋊Dic13—42no—
209C4⋊Dic13C22×C411yescharacteristic
210C4⋊Dic13C22×C412yes—
211C4⋊Dic13C22×C411yescharacteristic
212C4⋊Dic13C22×C412yes—
213#6101—22no—
214#6101—22no—
215#6101—22no—
216#6101—22no—
217#6101—22no—
218#6101—22no—
219#6101—22no—
220#6101—21no—
221#6101—21no—
222#6101—22no—
223C4×Dic13—21no—
224C4×Dic13—22no—
225C4×Dic13—22no—
226C4×Dic13—21no—
227C4×Dic13—22no—
228C4×Dic13—21no—
229C4×Dic13—21no—
230C4×Dic13—21no—
231C4×Dic13—22no—
232C4×Dic13—21no—
233C4×Dic13—22no—
234C4×Dic13—42no—
235C4×Dic13—42no—
236C4×Dic13C2×Q812yescomplemented
237Q8×#41—131no—
238Q8×M16D1311yescharacteristic, complemented
239#2295—131no—
240#1995—132no—
241#1994—131no—
242#1986—131no—
243#1986—132no—
244#1806—131no—
245#1806—132no—
246#1759—132no—
247C4×#104—131no—
248C23×C13C22×D411yescharacteristic
249C22×D13—21no—
250Q8×C13—22no—
251Q8×C13—22no—
252Q8×C13—42no—
253C2×C4×C13—22no—
254C2×C4×C13—41no—
255C2×C4×C13—42no—
256C2×C4×C13—42no—
257C2×C4×C13C2×C22⋊C411yescharacteristic
258C2×C4×C13C2×C22⋊C412yes—
259C2×C4×C13C2×C22⋊C412yes—
260C2×C4×C13C2×C22⋊C411yescharacteristic
261C2×C4×C13C2×C22⋊C412yes—
262C2×C4×C13C2×C22⋊C412yes—
263C2×C4×C13C2×C22⋊C411yescharacteristic
264C2×C4×C13C2×C22⋊C411yescharacteristic
265C2×C4×C13C23×C411yescharacteristic
266C2×C4×C13C22×D411yescharacteristic
267C2×C4×C13C22×Q811yescharacteristic
268C2×C4×C13C2×#4111yescharacteristic
269C2×C4×C132−1+411yescharacteristic
270C2×C4×C132−1+411yescharacteristic
271C22⋊D13—22no—
272C22⋊D13—22no—
273C2×Dic13—21no—
274C2×Dic13—22no—
275C2×Dic13—21no—
276C2×Dic13—22no—
277C2×Dic13—22no—
278C2×Dic13—21no—
279C2×Dic13—22no—
280C2×Dic13—41no—
281C2×Dic13—41no—
282C2×Dic13—41no—
283C2×Dic13—41no—
284C2×Dic13—42no—
285C2×Dic13—42no—
286C2×Dic13—42no—
287D52—21no—
288D52C4×Q812yes—
289C4×D13—22no—
290C4×D13—22no—
291C4×D13—44no—
292C4×D13—42no—
293C4×D13—81no—
294C4×D13—82no—
295Dic26—21no—
296Dic26—44no—
297Dic26—42no—
298Dic26—81no—
299Dic26—82no—
300Dic26C4×Q812yes—
301C8×C13—22no—
302C8×C13—41no—
303C8×C13—42no—
304Q82—131no—
305#557—132no—
306#557—131no—
307#554—131no—
308#554—132no—
309#554—262no—
310#554—261no—
311D4×Q8—131no—
312D4×Q8—261no—
313#533—131no—
314#533—132no—
315#533—261no—
316#533—262no—
317C4×#41—131no—
318C4×#41—132no—
319C2×C4×Q8—261no—
320C2×C4×Q8C2×D1311yescharacteristic
321#446—21no—
322#446—22no—
323C8×Q8—21no—
324#423C2×D1311yescharacteristic
325#423C2×D1312yes—
326C2×#104—131no—
327C2×#104—131no—
328C2×#104—132no—
329C4×M16C2×D1311yescharacteristic
330C4×M16C2×D1312yes—
331#391—261no—
332#389—261no—
333#387—262no—
334C4×C4⋊C4—131no—
335C4×C4⋊C4—132no—
336#376—132no—
337C43—131no—
338#339—132no—
339#339—132no—
340#339—132no—
341#339—131no—
342#339—132no—
343#339—132no—
344#339—131no—
345C22×C13—21no—
346C22×C13—21no—
347C22×C13C22×C22⋊C411yescharacteristic
348C22×C13#54811yescharacteristic
349C22×C13D4×Q811yescharacteristic
350C2×D13—22no—
351C2×D13—41no—
352C2×D13—41no—
353C4×C13—22no—
354C4×C13—21no—
355C4×C13—22no—
356C4×C13—21no—
357C4×C13—22no—
358C4×C13—21no—
359C4×C13—22no—
360C4×C13#42112yes—
361C4×C13C2×C4×Q811yescharacteristic
362C4×C13#51911yescharacteristic
363C4×C13#54811yescharacteristic
364C4×C13D4×Q811yescharacteristic
365Dic13—22no—
366Dic13—42no—
367Dic13—42no—
368Dic13—41no—
369Dic13—42no—
370Dic13—42no—
371Dic13—41no—
372Dic13—82no—
373C2×#41—131no—
374C22×Q8—261no—
375C22×Q8D5211yescharacteristic
376C2×M16C22×D1312yes—
377C2×M16C22×D1311yescharacteristic
378C2×M16C22×D1311yescharacteristic
379C4⋊Q8—131no—
380C4⋊Q8—132no—
381C4⋊Q8—131no—
382C4⋊Q8—132no—
383C4⋊Q8—261no—
384C4⋊Q8—262no—
385C4⋊Q8—262no—
386C4⋊Q8—261no—
387#125—132no—
388#125—132no—
389#125—132no—
390#125—262no—
391#125—262no—
392#125—262no—
393C22⋊Q8—261no—
394C22⋊Q8—262no—
395C22⋊Q8—261no—
396C22⋊Q8—262no—
397C22⋊Q8—261no—
398C22⋊Q8—262no—
399C22⋊Q8—262no—
400C22⋊Q8—261no—
401C22⋊Q8—262no—
402C22⋊Q8—262no—
403C22⋊Q8—262no—
404C4×Q8—131no—
405C4×Q8—132no—
406C4×Q8—131no—
407C4×Q8—132no—
408C4×Q8—261no—
409C4×Q8—262no—
410C4×Q8—261no—
411C4×Q8—261no—
412C4×Q8—262no—
413C4×Q8—262no—
414C4×Q8—261no—
415C4×Q8C4×D1311yescharacteristic
416C4×Q8C4×D1311yescharacteristic
417C4×Q8C4×D1312yes—
418C4×Q8C22⋊D1311yescharacteristic
419C4×Q8C22⋊D1311yescharacteristic
420C4×Q8C22⋊D1312yes—
421C4×D4—131no—
422C4×D4—132no—
423C4×D4—131no—
424C4×D4—132no—
425C4×D4—261no—
426C4×D4—262no—
427C4×D4—261no—
428C4×D4—262no—
429#117—131no—
430#117—132no—
431#117—262no—
432#117—261no—
433C2×C4⋊C4—262no—
434C2×C4⋊C4—261no—
435C2×C4⋊C4—262no—
436C2×C4⋊C4—261no—
437C2×C4⋊C4D5212yes—
438C2×C4⋊C4D5211yescharacteristic
439C2×C42—131no—
440C2×C42—131no—
441C2×C42—132no—
442C2×C42—261no—
443C2×C42—262no—
444C2×C42—261no—
445C2×C42—261no—
446C2×C42C22×D1311yescharacteristic
447C2×C42C22×D1312yes—
448C4⋊C8—21no—
449C4⋊C8—22no—
450C4⋊C8—22no—
451C4⋊C8—21no—
452#104—134no—
453#104—262no—
454#104—264no—
455#97—22no—
456#97—21no—
457C4×C8—21no—
458C4×C8—22no—
459#95—262no—
460#95—262no—
461#95—261no—
462#95—262no—
463C2×C13—21no—
464C2×C13—21no—
465C2×C13Q8×C22⋊C411yescharacteristic
466C2×C13C2×#42111yescharacteristic
467C2×C13#290111yescharacteristic
468D13—42no—
469#41—132no—
470#41—132no—
471#41—261no—
472#41—261no—
473C2×Q8—42no—
474C2×Q8—131no—
475C2×Q8—262no—
476C2×Q8—261no—
477C2×Q8—262no—
478C2×Q8—261no—
479C2×Q8—524no—
480C2×Q8#610311yescharacteristic
481C2×Q8#610311yescharacteristic
482C2×Q8#610312yes—
483C2×D4—131no—
484C2×D4—261no—
485C22×C4—261no—
486C22×C4—261no—
487C22×C4—261no—
488C22×C4—262no—
489C22×C4—261no—
490C22×C4—261no—
491C22×C4—262no—
492C22×C4—262no—
493C22×C4C2×D5211yescharacteristic
494C22×C4C2×D5212yes—
495C22×C4C2×D5211yescharacteristic
496C22×C4C2×D5212yes—
497C22×C4C23×D1311yescharacteristic
498M16—21no—
499M16—21no—
500M16—22no—
501M16—22no—
502M16—21no—
503M16Q8×D1312yescomplemented
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—22no—
507C4⋊C4—131no—
508C4⋊C4—132no—
509C4⋊C4—131no—
510C4⋊C4—132no—
511C4⋊C4—262no—
512C4⋊C4—262no—
513C4⋊C4—262no—
514C4⋊C4—262no—
515C4⋊C4—261no—
516C4⋊C4—261no—
517C4⋊C4—262no—
518C4⋊C4—262no—
519C4⋊C4—261no—
520C4⋊C4—262no—
521C4⋊C4—262no—
522C4⋊C4—262no—
523C4⋊C4—262no—
524C4⋊C4—261no—
525C4⋊C4—522no—
526C4⋊C4—524no—
527C4⋊C4#610312yes—
528C4⋊C4#610311yescharacteristic
529C4⋊C4#610312yes—
530C4⋊C4#610312yes—
531C4⋊C4#610312yes—
532C4⋊C4#610312yes—
533C4⋊C4#610311yescharacteristic
534C22⋊C4—261no—
535C22⋊C4—262no—
536C22⋊C4—261no—
537C22⋊C4—262no—
538C22⋊C4—262no—
539C22⋊C4—261no—
540C42—132no—
541C42—261no—
542C42—262no—
543C42—262no—
544C42—261no—
545C42—261no—
546C42—261no—
547C42—262no—
548C42—261no—
549C42—262no—
550C42—261no—
551C42—262no—
552C42—522no—
553C42—522no—
554C42C2×C4×D1311yescharacteristic
555C42C2×C4×D1312yes—
556C42C2×C4×D1312yes—
557C42C2×C4×D1311yescharacteristic
558C42C2×C22⋊D1311yescharacteristic
559C42C2×C22⋊D1312yes—
560C42C2×C22⋊D1312yes—
561C42C2×C22⋊D1311yescharacteristic
562C13#2252911yescharacteristic, complemented, Sylow
563C23—261no—
564C23C22×D5211yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—42no—
568Q8—132no—
569Q8—261no—
570Q8—524no—
571Q8—522no—
572Q8—1041no—
573Q8—1042no—
574D4—132no—
575D4—262no—
576D4—262no—
577D4—261no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—261no—
583C2×C4—262no—
584C2×C4—262no—
585C2×C4—262no—
586C2×C4—262no—
587C2×C4—261no—
588C2×C4—262no—
589C2×C4—261no—
590C2×C4—262no—
591C2×C4—524no—
592C2×C4—521no—
593C2×C4—521no—
594C2×C4—521no—
595C2×C4—521no—
596C2×C4—522no—
597C2×C4—522no—
598C2×C4—522no—
599C2×C4—522no—
600C2×C4—1041no—
601C2×C4—1042no—
602C2×C4C2×#610311yescharacteristic
603C2×C4C2×#610312yes—
604C2×C4C2×#610312yes—
605C2×C4C2×#610311yescharacteristic
606C2×C4C2×#610312yes—
607C2×C4C2×#610312yes—
608C2×C4C2×#610311yescharacteristic
609C2×C4C2×#610311yescharacteristic
610C2×C4C22×C4×D1311yescharacteristic
611C2×C4C2×Q8×D1311yescharacteristic
612C2×C4#8825411yescharacteristic
613C2×C4#41×D1311yescharacteristic
614C2×C4#8825711yescharacteristic
615C2×C4C22×C22⋊D1311yescharacteristic
616C8—22no—
617C8—41no—
618C8—42no—
619C22—21no—
620C22—21no—
621C22—262no—
622C22—521no—
623C22—521no—
624C22Q8×D5211yescharacteristic
625C22#1171928111yescharacteristic
626C22C22×#610311yescharacteristic
627C4—21no—
628C4—22no—
629C4—22no—
630C4—21no—
631C4—22no—
632C4—21no—
633C4—22no—
634C4—262no—
635C4—522no—
636C4—522no—
637C4—522no—
638C4—521no—
639C4—522no—
640C4—521no—
641C4—1042no—
642C4#1171880012yes—
643C4C4×Q8×D1311yescharacteristic
644C4#1171927711yescharacteristic
645C4Q8×C22⋊D1311yescharacteristic
646C4#1171953611yescharacteristic
647C2—21no—
648C2—21no—
649C2—522no—
650C2Q8×#610311yescharacteristic
651C2C2×#1171880011yescharacteristic
652C2#1312338111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingQ8×C13×M16C2
socleC22×C13C22×C22⋊C4
prime radicalC2×#6127C23
cosocleC22×C4×C13C24
hypercenterQ8×M16D13
hypocenterC13#22529
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. #13117314C2semidirect
3. #13115816C2semidirect
4. #13115812C2semidirect
5. #13115780C2semidirect
6. #13115780C2semidirect
11. Q8×C13×M16C2semidirect
14. #11719284C4semidirect
15. #11719284C4semidirect
18. Q8×D52C4semidirect
20. #11719272C4semidirect
21. #11719272C4semidirect
22. Q8×Dic26C4semidirect
142. #88065Q8semidirect
178. C13×M16C2×Q8semidirect
236. C4×Dic13C2×Q8semidirect
238. Q8×M16D13semidirect
503. M16Q8×D13semidirect
562. C13#22529semidirect

Maximal subgroups

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

subgroupindex
Q8×#61272
#131173142
#131158162
#131158122
#131157802
#131148532
#131146492
C4×#880652
Q8×C13×M162
#2252913