Finite Group Databasebeta

← #16747984

Subgroups — #16747984

Order 3328.

2086 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13128688C211yesmaximal, characteristic
2#13127130C214yesmaximal, complemented
3#13127119C214yesmaximal, complemented
4#13126498C214yesmaximal, complemented
5#13126482C214yesmaximal, complemented
6#13125784C211yesmaximal, characteristic, complemented
7#13125761C211yesmaximal, characteristic, complemented
8#13125666C214yesmaximal, complemented
9#13124875C211yesmaximal, characteristic, complemented
10#13124867C211yesmaximal, characteristic, complemented
11#13124070C214yesmaximal, complemented
12#13123877C211yesmaximal, characteristic, complemented
13C2×D4×Q8×C13C211yesmaximal, characteristic, complemented
14C23×Q8×C13C2212yes—
15D4×Q8×C13—22no—
16D4×Q8×C13—22no—
17D4×Q8×C13C2214yes—
18D4×Q8×C13C2214yes—
19C2×C13×C4⋊Q8—21no—
20C2×C13×C4⋊Q8C2211yescharacteristic
21C2×C13×C22⋊Q8—22no—
22C2×C13×C22⋊Q8C2212yes—
23C2×C4×Q8×C13C2211yescharacteristic
24C2×C4×D4×C13—21no—
25C2×C4×D4×C13C2211yescharacteristic
26C2×#88196C2218yescomplemented
27#11719510—21no—
28#11719496C2214yes—
29#11719268C2212yes—
30#11719264—22no—
31#11719263—22no—
32#11719258C2212yes—
33#11719256C2212yes—
34#11719247C2212yes—
35#11719209—21no—
36#11719199—22no—
37#11719182C2211yescharacteristic
38#11719175C2211yescharacteristic
39#11718889C2212yes—
40#11718889C2214yes—
41#11718889C2218yescomplemented
42#11718888C2212yes—
43#11718888C2214yes—
44#11718888C2218yescomplemented
45#11718886C2218yescomplemented
46#11718868C2214yes—
47#11718753C2212yescomplemented
48#11718751C2214yescomplemented
49#11718746C2212yescomplemented
50#11718744C2214yescomplemented
51#11718742C2212yescomplemented
52#11718742C2212yescomplemented
53#11718713C2218yescomplemented
54#11718710C2218yescomplemented
55#11718708C2218yescomplemented
56#11718706C2218yescomplemented
57#11718692C2212yescomplemented
58C4×#6107C2214yescomplemented
59#11718688C2212yescomplemented
60C4×#6106C2214yescomplemented
61#11718682C2212yescomplemented
62#11718681C2212yescomplemented
63#11718673C2214yes—
64#11718673C2214yes—
65#11718631C2212yes—
66#11718629C2212yes—
67#11718586C2211yescharacteristic
68#11718586C2211yescharacteristic
69C22×Q8×C13—22no—
70C22×Q8×C13—24no—
71C22×Q8×C13—48no—
72C22×Q8×C13C2311yescharacteristic
73C22×Q8×C13C2318yes—
74C22×Q8×C13C2312yes—
75C22×D4×C13D411yescharacteristic
76C23×C4×C13—22no—
77C23×C4×C13C2312yes—
78C2×#6127—21no—
79C2×#6127C2314yes—
80C13×C4⋊Q8—24no—
81C13×C4⋊Q8—24no—
82C13×C4⋊Q8—21no—
83C13×C4⋊Q8—21no—
84C13×C4⋊Q8C2312yes—
85C13×C4⋊Q8C2312yes—
86C13×C22⋊Q8—24no—
87C13×C22⋊Q8—28no—
88C13×C22⋊Q8—28no—
89C13×C22⋊Q8C2318yes—
90C4×Q8×C13—21no—
91C4×Q8×C13—21no—
92C4×Q8×C13C2312yes—
93C4×Q8×C13C2312yes—
94C4×D4×C13—24no—
95C4×D4×C13—24no—
96C4×D4×C13D412yes—
97C4×D4×C13D412yes—
98C4×D4×C13C2312yes—
99C4×D4×C13C2312yes—
100C2×C13×C4⋊C4—21no—
101C2×C13×C4⋊C4—21no—
102C2×C13×C4⋊C4—21no—
103C2×C13×C4⋊C4—22no—
104C2×C13×C4⋊C4D411yescharacteristic
105C2×C13×C4⋊C4C2311yescharacteristic
106C2×C13×C22⋊C4—22no—
107C2×C13×C22⋊C4D412yes—
108C2×C42×C13—21no—
109C2×C42×C13C2311yescharacteristic
110#88207—24no—
111C2×#6107C2314yes—
112C2×#6107C2318yes—
113#88196—416no—
114C2×#6106C2314yes—
115C2×#6106C2318yes—
116#88194—22no—
117#88192—22no—
118#88192—22no—
119#88190—22no—
120#88184—21no—
121#88184C2312yes—
122#88184C2314yes—
123#88183—24no—
124C4×C22⋊D13—24no—
125C4×C22⋊D13—24no—
126#88180—21no—
127#88180C2314yes—
128#88178—21no—
129#88178C2312yes—
130#88178C2314yes—
131C2×#6099C2318yes—
132#88147—22no—
133#88146—22no—
134C4⋊C4×D13—22no—
135#88141—22no—
136#88139—24no—
137#88138—24no—
138#88137—24no—
139#88133—24no—
140#88129—21no—
141#88128C2311yescharacteristic
142C4×D52—21no—
143C4×D52C2312yes—
144#88124—21no—
145#88123C2311yescharacteristic
146#88123C2311yescharacteristic
147#88123C2311yescharacteristic
148C4×Dic26—21no—
149C4×Dic26C2312yes—
150#88075C2314yes—
151#88075C2314yes—
152#88075C2318yes—
153#88075C2318yes—
154C22⋊#1055C2318yes—
155C22⋊#1055C2312yes—
156C22⋊#1055C2312yes—
157#88050C2314yes—
158#88049C2314yes—
159#88048C2314yes—
160#88047C2314yes—
161C4⋊#1055C2312yes—
162C4⋊#1055C2312yes—
163C4×#1055C2312yes—
164C4×#1055C2312yes—
165#61584—131nomaximal, Sylow
166C24×C13C2×D412yes—
167C2×Q8×C13—24no—
168C2×Q8×C13—21no—
169C2×Q8×C13—28no—
170C2×Q8×C13—22no—
171C2×Q8×C13—216no—
172C2×Q8×C13—48no—
173C2×Q8×C13—44no—
174C2×Q8×C13—48no—
175C2×Q8×C13—416no—
176C2×Q8×C13C2414yes—
177C2×Q8×C13C2412yes—
178C2×D4×C13—24no—
179C2×D4×C13C2×D412yes—
180C2×D4×C13C2×D412yes—
181C22×C4×C13—21no—
182C22×C4×C13—21no—
183C22×C4×C13—22no—
184C22×C4×C13—28no—
185C22×C4×C13—44no—
186C22×C4×C13—48no—
187C22×C4×C13C2×D411yescharacteristic
188C22×C4×C13C2×D412yes—
189C22×C4×C13C2×D414yes—
190C22×C4×C13C2×D411yescharacteristic
191C22×C4×C13C2411yescharacteristic
192C22×C4×C13C2414yes—
193C2×C22⋊D13—22no—
194C2×C22⋊D13—24no—
195C2×C22⋊D13—24no—
196#6127—48no—
197#6127—44no—
198C2×D52—21no—
199C2×D52—21no—
200C2×D52C2412yes—
201C2×C4×D13—22no—
202C2×C4×D13—24no—
203C2×Dic26—21no—
204C2×Dic26—21no—
205C2×Dic26C2412yes—
206C13×C4⋊C4—22no—
207C13×C4⋊C4—22no—
208C13×C4⋊C4—22no—
209C13×C4⋊C4—22no—
210C13×C4⋊C4—22no—
211C13×C4⋊C4—22no—
212C13×C4⋊C4—28no—
213C13×C4⋊C4—21no—
214C13×C4⋊C4C2×D412yes—
215C13×C4⋊C4C2×D412yes—
216C13×C4⋊C4C2412yes—
217C13×C22⋊C4—28no—
218C13×C22⋊C4C2×D414yes—
219C13×C22⋊C4C2×D414yes—
220C42×C13—22no—
221C42×C13—22no—
222C42×C13C2×D411yescharacteristic
223C42×C13C2×D411yescharacteristic
224C42×C13C2411yescharacteristic
225C42×C13C2411yescharacteristic
226C22⋊Dic13—24no—
227C22⋊Dic13—24no—
228C22⋊Dic13—22no—
229#6107—28no—
230#6107—48no—
231#6106—28no—
232#6106—48no—
233#6103—22no—
234#6103—22no—
235#6103—24no—
236C4⋊Dic13—21no—
237C4⋊Dic13—21no—
238C4⋊Dic13—21no—
239C4⋊Dic13—21no—
240C4⋊Dic13C2412yes—
241C4⋊Dic13C2412yes—
242#6101—22no—
243#6101—22no—
244#6101—24no—
245C4×Dic13—22no—
246C4×Dic13—24no—
247#6099—48no—
248C2×#1055C2414yes—
249C2×#1055C2414yes—
250#3506—131no—
251C2×D4×Q8D1311yescharacteristic, complemented
252#3329—134no—
253#3328—134no—
254#3278—134no—
255#3276—134no—
256#3209—131no—
257#3206—131no—
258#3196—134no—
259#3104—131no—
260#3103—131no—
261#3004—134no—
262#2975—131no—
263C23×C13—24no—
264C23×C13—42no—
265C23×C13—42no—
266C23×C13C22×D414yes—
267C23×C13C22×D411yescharacteristic
268C22×D13—22no—
269Q8×C13—24no—
270Q8×C13—22no—
271Q8×C13—28no—
272Q8×C13—416no—
273Q8×C13—44no—
274Q8×C13H(5,𝔽2)14yes—
275D4×C13—24no—
276D4×C13—24no—
277C2×C4×C13—24no—
278C2×C4×C13—22no—
279C2×C4×C13—28no—
280C2×C4×C13—22no—
281C2×C4×C13—22no—
282C2×C4×C13—24no—
283C2×C4×C13—42no—
284C2×C4×C13—44no—
285C2×C4×C13—41no—
286C2×C4×C13—416no—
287C2×C4×C13—42no—
288C2×C4×C13—44no—
289C2×C4×C13#13714yescomplemented
290C2×C4×C13C22×D414yes—
291C2×C4×C13C22×D411yescharacteristic
292C2×C4×C13C22×D411yescharacteristic
293C2×C4×C13C22×D411yescharacteristic
294C2×C4×C13C22×D411yescharacteristic
295C2×C4×C13C22×D411yescharacteristic
296C2×C4×C13C2511yescharacteristic
297C22⋊D13—44no—
298C22⋊D13—84no—
299C22⋊D13—84no—
300C2×Dic13—22no—
301C2×Dic13—22no—
302C2×Dic13—22no—
303D52—24no—
304D52—42no—
305D52—81no—
306C4×D13—44no—
307C4×D13—84no—
308Dic26—24no—
309Dic26—42no—
310Dic26—81no—
311#1055—24no—
312#1055—24no—
313C23×Q8C2×D1312yes—
314C2×#137—138no—
315#559—262no—
316#556—132no—
317#555—132no—
318#553—262no—
319#550—132no—
320D4×Q8—22no—
321D4×Q8—22no—
322D4×Q8C2×D1314yes—
323D4×Q8C2×D1314yes—
324#547—132no—
325#538—262no—
326#535—261no—
327#533—131no—
328#532—131no—
329C2×C4⋊Q8—21no—
330C2×C4⋊Q8C2×D1311yescharacteristic
331#525—134no—
332C2×C22⋊Q8—22no—
333C2×C22⋊Q8C2×D1312yes—
334#520—261no—
335C2×C4×Q8C2×D1311yescharacteristic
336C2×C4×D4—21no—
337C2×C4×D4C2×D1311yescharacteristic
338#494—132no—
339#492—132no—
340#487—132no—
341#487—132no—
342#485—138no—
343#484—132no—
344#481—132no—
345#477—132no—
346#475—132no—
347#470—138no—
348#468—138no—
349#462—134no—
350#462—132no—
351#461—134no—
352#460—132no—
353#452—138no—
354#451—138no—
355#451—132no—
356#451—134no—
357#448—138no—
358#448—132no—
359#448—134no—
360C4×Q16—134no—
361C4×SD16—134no—
362#436—134no—
363#436—134no—
364#431—131no—
365#431—131no—
366#419—138no—
367#407—134no—
368C22×C13—21no—
369C22×C13—21no—
370C22×C13—24no—
371C22×C13—24no—
372C22×C13—24no—
373C22×C13—48no—
374C22×C13C2×#13712yes—
375C22×C13C23×D411yescharacteristic
376C2×D13—42no—
377C2×D13—42no—
378C2×D13—42no—
379C4×C13—21no—
380C4×C13—21no—
381C4×C13—24no—
382C4×C13—24no—
383C4×C13—21no—
384C4×C13—42no—
385C4×C13—44no—
386C4×C13C2×#13712yes—
387C4×C13C2×#13712yes—
388C4×C13C2×H(5,𝔽2)12yes—
389Dic13—42no—
390Dic13—42no—
391Dic13—42no—
392C2×#41—134no—
393C2×#41—261no—
394C22×Q8—22no—
395C22×Q8—24no—
396C22×Q8—48no—
397C22×Q8C22×D1311yescharacteristic
398C22×Q8C22×D1318yes—
399C22×Q8C22×D1312yes—
400C22×D4C22⋊D1311yescharacteristic
401C23×C4—22no—
402C23×C4C22×D1312yes—
403#137—5216no—
404C2×Q16—134no—
405C2×Q16—138no—
406C2×SD16—134no—
407C2×SD16—138no—
408C2×M16—138no—
409C4⋊Q8—24no—
410C4⋊Q8—24no—
411C4⋊Q8—21no—
412C4⋊Q8—21no—
413C4⋊Q8—131no—
414C4⋊Q8—131no—
415C4⋊Q8—131no—
416C4⋊Q8C22×D1312yes—
417C4⋊Q8C22×D1312yes—
418#127—131no—
419#127—262no—
420#126—264no—
421#125—261no—
422#124—262no—
423#124—264no—
424#124—261no—
425#123—264no—
426#123—264no—
427C22⋊Q8—24no—
428C22⋊Q8—28no—
429C22⋊Q8—28no—
430C22⋊Q8—132no—
431C22⋊Q8—134no—
432C22⋊Q8—261no—
433C22⋊Q8C22×D1318yes—
434#121—132no—
435#121—134no—
436#121—262no—
437#121—262no—
438#121—261no—
439#121—264no—
440#120—264no—
441C4×Q8—21no—
442C4×Q8—21no—
443C4×Q8—132no—
444C4×Q8—262no—
445C4×Q8—261no—
446C4×Q8C22×D1312yes—
447C4×Q8C22×D1312yes—
448C4×D4—24no—
449C4×D4—24no—
450C4×D4—132no—
451C4×D4—264no—
452C4×D4—264no—
453C4×D4—262no—
454C4×D4—261no—
455C4×D4C22⋊D1312yes—
456C4×D4C22⋊D1312yes—
457C4×D4C22×D1312yes—
458C4×D4C22×D1312yes—
459#117—134no—
460#117—261no—
461#117—262no—
462C2×C4⋊C4—21no—
463C2×C4⋊C4—21no—
464C2×C4⋊C4—21no—
465C2×C4⋊C4—22no—
466C2×C4⋊C4—262no—
467C2×C4⋊C4C22⋊D1311yescharacteristic
468C2×C4⋊C4C22×D1311yescharacteristic
469C2×C22⋊C4—22no—
470C2×C22⋊C4C22⋊D1312yes—
471C2×C42—21no—
472C2×C42C22×D1311yescharacteristic
473#107—134no—
474#106—134no—
475C4⋊C8—132no—
476C4⋊C8—132no—
477#103—134no—
478#103—134no—
479#103—138no—
480#103—138no—
481#103—134no—
482#102—134no—
483C22⋊C8—132no—
484C22⋊C8—132no—
485C22⋊C8—138no—
486C4×C8—132no—
487C4×C8—132no—
488C2×C13—22no—
489C2×C13—44no—
490C2×C13#348411yescharacteristic
491C2×C13C22×#13711yescharacteristic
492C2×C13#358611yescharacteristic
493D13—82no—
494C24C2×C22⋊D1312yes—
495#41—528no—
496#41—524no—
497C2×Q8—24no—
498C2×Q8—21no—
499C2×Q8—28no—
500C2×Q8—22no—
501C2×Q8—216no—
502C2×Q8—48no—
503C2×Q8—44no—
504C2×Q8—48no—
505C2×Q8—416no—
506C2×Q8—132no—
507C2×Q8—261no—
508C2×Q8—261no—
509C2×Q8C23×D1314yes—
510C2×Q8C23×D1312yes—
511C2×D4—24no—
512C2×D4—132no—
513C2×D4—262no—
514C2×D4—264no—
515C2×D4—264no—
516C2×D4—261no—
517C2×D4—261no—
518C2×D4C2×C22⋊D1312yes—
519C2×D4C2×C22⋊D1312yes—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—22no—
523C22×C4—28no—
524C22×C4—44no—
525C22×C4—48no—
526C22×C4—262no—
527C22×C4—264no—
528C22×C4C2×C22⋊D1311yescharacteristic
529C22×C4C2×C22⋊D1312yes—
530C22×C4C2×C22⋊D1314yes—
531C22×C4C2×C22⋊D1311yescharacteristic
532C22×C4C23×D1311yescharacteristic
533C22×C4C23×D1314yes—
534Q16—268no—
535Q16—528no—
536SD16—268no—
537SD16—528no—
538M16—528no—
539C2×C8—134no—
540C2×C8—134no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—28no—
548C4⋊C4—21no—
549C4⋊C4—132no—
550C4⋊C4—132no—
551C4⋊C4—261no—
552C4⋊C4—264no—
553C4⋊C4—261no—
554C4⋊C4—262no—
555C4⋊C4—261no—
556C4⋊C4—262no—
557C4⋊C4—261no—
558C4⋊C4C2×C22⋊D1312yes—
559C4⋊C4C2×C22⋊D1312yes—
560C4⋊C4C23×D1312yes—
561C22⋊C4—28no—
562C22⋊C4—264no—
563C22⋊C4—262no—
564C22⋊C4—264no—
565C22⋊C4—264no—
566C22⋊C4—262no—
567C22⋊C4—262no—
568C22⋊C4C2×C22⋊D1314yes—
569C22⋊C4C2×C22⋊D1314yes—
570C42—22no—
571C42—22no—
572C42—264no—
573C42—262no—
574C42C2×C22⋊D1311yescharacteristic
575C42C2×C22⋊D1311yescharacteristic
576C42C23×D1311yescharacteristic
577C42C23×D1311yescharacteristic
578C13#6158411yescharacteristic, complemented, Sylow
579C23—24no—
580C23—42no—
581C23—42no—
582C23—262no—
583C23C22×C22⋊D1314yes—
584C23C22×C22⋊D1311yescharacteristic
585Q8—24no—
586Q8—22no—
587Q8—28no—
588Q8—416no—
589Q8—44no—
590Q8—264no—
591Q8—522no—
592Q8—1041no—
593Q8#8825114yes—
594D4—24no—
595D4—24no—
596D4—264no—
597D4—524no—
598D4—522no—
599D4—1041no—
600D4—1044no—
601D4—1044no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—416no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—262no—
615C2×C4—262no—
616C2×C4—262no—
617C2×C4—524no—
618C2×C4—1044no—
619C2×C4#8819614yescomplemented
620C2×C4C22×C22⋊D1314yes—
621C2×C4C22×C22⋊D1311yescharacteristic
622C2×C4C22×C22⋊D1311yescharacteristic
623C2×C4C22×C22⋊D1311yescharacteristic
624C2×C4C22×C22⋊D1311yescharacteristic
625C2×C4C22×C22⋊D1311yescharacteristic
626C2×C4C24×D1311yescharacteristic
627C8—264no—
628C8—264no—
629C22—21no—
630C22—21no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—48no—
635C22—522no—
636C22—522no—
637C22—522no—
638C22C2×#8819612yes—
639C22C23×C22⋊D1311yescharacteristic
640C4—24no—
641C4—21no—
642C4—24no—
643C4—21no—
644C4—21no—
645C4—44no—
646C4—42no—
647C4—522no—
648C4—522no—
649C4—522no—
650C4C2×#8819612yes—
651C4C2×#8819612yes—
652C4C2×#8825112yes—
653C2—22no—
654C2—44no—
655C2—1042no—
656C2#1312847111yescharacteristic
657C2C22×#8819611yescharacteristic
658C2#1312962311yescharacteristic
subgroupG/H
centerC22C23×C22⋊D13
commutator, cosocleC2×C4×C13C25
FrattiniC2×C4C24×D13
FittingC2×D4×Q8×C13C2
socleC22×C13C23×D4
prime radical#13128688C2
hypercenterC2×D4×Q8D13
hypocenterC13#61584
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
2. #13127130C2semidirect
3. #13127119C2semidirect
4. #13126498C2semidirect
5. #13126482C2semidirect
6. #13125784C2semidirect
7. #13125761C2semidirect
8. #13125666C2semidirect
9. #13124875C2semidirect
10. #13124867C2semidirect
11. #13124070C2semidirect
12. #13123877C2semidirect
13. C2×D4×Q8×C13C2semidirect
26. C2×#88196C22semidirect
41. #11718889C22semidirect
44. #11718888C22semidirect
45. #11718886C22semidirect
47. #11718753C22semidirect
48. #11718751C22semidirect
49. #11718746C22semidirect
50. #11718744C22semidirect
51. #11718742C22semidirect
52. #11718742C22semidirect
53. #11718713C22semidirect
54. #11718710C22semidirect
55. #11718708C22semidirect
56. #11718706C22semidirect
57. #11718692C22semidirect
58. C4×#6107C22semidirect
59. #11718688C22semidirect
60. C4×#6106C22semidirect
61. #11718682C22semidirect
62. #11718681C22semidirect
251. C2×D4×Q8D13semidirect
289. C2×C4×C13#137semidirect
578. C13#61584semidirect
619. C2×C4#88196semidirect

Maximal subgroups

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

subgroupindex
#131286882
#131271302
#131271192
#131264982
#131264822
#131257842
#131257612
#131256662
#131248752
#131248672
#131240702
#131238772
C2×D4×Q8×C132
#6158413