Finite Group Databasebeta

← #16799400

Subgroups — #16799400

Order 3328.

1338 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131315C211yesmaximal, characteristic
2#13131272C211yesmaximal, characteristic
3#13131027C212yesmaximal, complemented
4#13131027C212yesmaximal, complemented
5#13131026C212yesmaximal, complemented
6#13131025C212yesmaximal, complemented
7#13130533C212yesmaximal
8C2×#11718289C212yesmaximal
9C2×#11719339C211yesmaximal, characteristic
10C22×C22⋊#261C2212yes—
11C22×C22⋊#261C2212yes—
12C2×C13×C22⋊Q8C411yescharacteristic, complemented
13C2×#88200C411yescharacteristic, complemented
14C2×#88200C2211yescharacteristic
15C22×#6103C2211yescharacteristic
16C2×#88178C411yescharacteristic, complemented
17#11719339—24no—
18#11719339—24no—
19C2×#88151C411yescharacteristic, complemented
20C2×#88151C2211yescharacteristic
21C2×#88150C411yescharacteristic, complemented
22C2×#88150C411yescharacteristic, complemented
23C2×#88150C2211yescharacteristic
24C2×#88150C2211yescharacteristic
25C2×C22⋊C4×D13C2211yescharacteristic
26C2×C22⋊C4×D13C2211yescharacteristic
27C2×#88132C411yescharacteristic, complemented
28C2×#88132C411yescharacteristic, complemented
29#11719136C2212yes—
30#11719136C2212yes—
31C4×C22⋊#261C2212yes—
32C4×C22⋊#261C2212yes—
33#11719070—24no—
34C22⋊C4×#261—24no—
35#11718289C2214yes—
36#11718289C2214yes—
37#11718289C2214yes—
38#11718289C2214yes—
39C24×D13C2311yescharacteristic
40C23×#261—22no—
41C23×#261—22no—
42C22×Q8×C13C2×C411yescharacteristic, complemented
43C23×C4×C13C2×C411yescharacteristic
44C23×Dic13C2×C411yescharacteristic
45C22×C4×D13C2×C411yescharacteristic
46C22×C4×D13C2×C411yescharacteristic
47C22×C4×D13C2311yescharacteristic
48C22×C4×D13C2311yescharacteristic
49C22×Dic26C2×C411yescharacteristic, complemented
50C22×Dic26C2×C411yescharacteristic, complemented
51C22×Dic26C2×C411yescharacteristic, complemented
52C2×C22⋊#261—22no—
53C2×C22⋊#261—24no—
54C2×C22⋊#261—28no—
55C2×C22⋊#261—22no—
56C2×C22⋊#261Q818yescomplemented
57C2×C22⋊#261C2314yes—
58C2×C22⋊#261C2314yes—
59C2×C4⋊#261—22no—
60C2×C4⋊#261—22no—
61C2×C4×#261—22no—
62C2×C4×#261—22no—
63C13×C22⋊Q8—24no—
64C2×C13×C4⋊C4C2×C411yescharacteristic, complemented
65C2×C13×C4⋊C4C2×C411yescharacteristic, complemented
66C2×C13×C4⋊C4C2×C411yescharacteristic, complemented
67C2×C13×C22⋊C4C2×C411yescharacteristic
68C2×C13×C22⋊C4C2×C411yescharacteristic
69C2×C22⋊Dic13C2×C411yescharacteristic
70C2×C22⋊Dic13C2×C411yescharacteristic
71#88200—22no—
72#88200—22no—
73#88200—24no—
74C2×#6103—22no—
75C2×#6103—22no—
76C2×#6103C2×C411yescharacteristic
77C2×#6103C2×C411yescharacteristic
78C2×#6103C2×C411yescharacteristic
79C2×#6103C2×C411yescharacteristic
80C2×#6103C2311yescharacteristic
81C2×#6103C2311yescharacteristic
82C2×#6103C2311yescharacteristic
83C2×#6103C2311yescharacteristic
84C2×C4⋊Dic13C2×C411yescharacteristic, complemented
85C2×C4⋊Dic13C2×C411yescharacteristic, complemented
86C2×C4⋊Dic13C2×C411yescharacteristic, complemented
87#88178—24no—
88C2×#6101C2×C411yescharacteristic, complemented
89C2×#6101C2×C411yescharacteristic, complemented
90C2×#6101C2×C411yescharacteristic, complemented
91C2×#6101C2×C411yescharacteristic, complemented
92C2×#6101C2×C411yescharacteristic, complemented
93C2×#6101C2×C411yescharacteristic, complemented
94#88151—22no—
95#88151—22no—
96#88151—24no—
97#88150—22no—
98#88150—22no—
99#88150—22no—
100#88150—22no—
101#88150—24no—
102#88150—24no—
103C22⋊C4×D13—22no—
104C22⋊C4×D13—22no—
105C22⋊C4×D13D414yes—
106C22⋊C4×D13D414yes—
107#88132—24no—
108#88132—24no—
109(C2×#261)⋊C4—22no—
110(C2×#261)⋊C4—22no—
111(C2×#261)⋊C4—22no—
112(C2×#261)⋊C4—22no—
113#21333—131nomaximal, Sylow
114C24×C13C22×C411yescharacteristic
115C23×D13—81no—
116C23×D13—81no—
117C23×D13C22×C411yescharacteristic
118C23×D13C2×D412yes—
119C23×D13C2×D412yes—
120C23×D13C2×Q814yes—
121C23×D13#4114yes—
122C23×D13C2411yescharacteristic
123C22×#261—24no—
124C22×#261—28no—
125C22×#261—24no—
126C22×#261—44no—
127C22×#261—82no—
128C22×#261—82no—
129C22×#261—82no—
130C22×#261—82no—
131C2×Q8×C13—22no—
132C2×Q8×C13—81no—
133C22×C4×C13—22no—
134C22×C4×C13—81no—
135C22×C4×C13C22×C411yescharacteristic
136C22×C4×C13C22×C411yescharacteristic
137C22×C4×C13C22×C411yescharacteristic
138C22×C4×C13C22×C411yescharacteristic
139C22×C4×C13C22×C411yescharacteristic
140C22×C4×C13C22×C411yescharacteristic
141C22×Dic13—22no—
142C22×Dic13—81no—
143C22×Dic13C22×C411yescharacteristic
144C22×Dic13C22×C411yescharacteristic
145C22×Dic13C22×C411yescharacteristic
146C22×Dic13C22×C411yescharacteristic
147C22×Dic13C22×C411yescharacteristic
148C22×Dic13C22×C411yescharacteristic
149C2×C4×D13—22no—
150C2×C4×D13—22no—
151C2×C4×D13—81no—
152C2×C4×D13—81no—
153C2×C4×D13—81no—
154C2×C4×D13—81no—
155C2×C4×D13C2×D412yes—
156C2×C4×D13C2×D412yes—
157C2×C4×D13C2×D412yes—
158C2×C4×D13C2×D412yes—
159C2×Dic26—22no—
160C2×Dic26—22no—
161C2×Dic26—22no—
162C2×Dic26—81no—
163C2×Dic26—81no—
164C2×Dic26—81no—
165C22⋊#261—28no—
166C22⋊#261—24no—
167C22⋊#261—28no—
168C22⋊#261—44no—
169C22⋊#261C2×Q818yes—
170C4⋊#261—82no—
171C4⋊#261—82no—
172C4×#261—82no—
173C4×#261—82no—
174C13×C4⋊C4—22no—
175C13×C4⋊C4—22no—
176C13×C4⋊C4—22no—
177C13×C22⋊C4—22no—
178C13×C22⋊C4C22⋊C414yes—
179C22⋊Dic13—22no—
180C22⋊Dic13C22⋊C414yes—
181#6103—22no—
182#6103—22no—
183#6103—21no—
184#6103—21no—
185#6103—21no—
186#6103—21no—
187#6103—21no—
188#6103—21no—
189#6103—21no—
190#6103—21no—
191#6103C22⋊C414yes—
192#6103C22⋊C414yes—
193C4⋊Dic13—22no—
194C4⋊Dic13—22no—
195C4⋊Dic13—22no—
196#6101—22no—
197#6101—22no—
198#6101—22no—
199#6101—22no—
200#6101—22no—
201#6101—22no—
202C2×#543—131no—
203#2315—131no—
204#2273—131no—
205#1996—132no—
206#1996—132no—
207#1996—132no—
208#1996—132no—
209C2×#342—132no—
210C2×#342—132no—
211C23×C13—41no—
212C23×C13—41no—
213C23×C13C2×C22⋊C412yes—
214C23×C13C4×Q814yes—
215C23×C13C23×C411yescharacteristic
216C22×D13—41no—
217C22×D13—41no—
218C22×D13—41no—
219C22×D13—41no—
220C22×D13—42no—
221C22×D13—42no—
222C22×D13—42no—
223C22×D13—42no—
224C22×D13—84no—
225C22×D13—84no—
226C22×D13C2×C22⋊C412yes—
227C22×D13C4×Q814yes—
228C22×D13C22⋊Q814yes—
229C22×D13C22⋊Q814yes—
230C22×D13#12414yes—
231C22×D13C4⋊Q814yes—
232C22×D13C22×D411yescharacteristic
233C22×D13C22×D411yescharacteristic
234C22×D13C22×Q812yes—
235C22×D13C2×#4112yes—
236C2×#261—28no—
237C2×#261—44no—
238C2×#261—44no—
239C2×#261—44no—
240C2×#261—44no—
241C2×#261—44no—
242C2×#261—88no—
243C2×#261—88no—
244Q8×C13—82no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—41no—
250C2×C4×C13—41no—
251C2×C4×C13—41no—
252C2×C4×C13—41no—
253C2×C4×C13—41no—
254C2×C4×C13—41no—
255C2×C4×C13—82no—
256C2×C4×C13C2×C22⋊C412yes—
257C2×C4×C13C2×C22⋊C412yes—
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—21no—
262C2×Dic13—41no—
263C2×Dic13—41no—
264C2×Dic13—41no—
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13—41no—
268C2×Dic13—82no—
269C2×Dic13C2×C22⋊C412yes—
270C2×Dic13C2×C22⋊C412yes—
271C4×D13—42no—
272C4×D13—42no—
273C4×D13—42no—
274C4×D13—42no—
275C4×D13—82no—
276C4×D13—82no—
277Dic26—82no—
278Dic26—82no—
279Dic26—82no—
280#543—264no—
281#543—264no—
282C2×C22⋊Q8—131no—
283C2×C22⋊Q8—131no—
284C2×C22⋊Q8—131no—
285C2×C22⋊Q8—131no—
286C2×C22⋊Q8—131no—
287C2×C22⋊Q8—131no—
288C2×C22⋊Q8—131no—
289C2×C22⋊Q8—131no—
290C2×C22⋊Q8—131no—
291C2×C22⋊Q8—131no—
292C2×C22⋊Q8—131no—
293C2×C22⋊Q8#26111yescharacteristic, complemented
294C22×C22⋊C4—131no—
295C22×C22⋊C4—132no—
296C22×C22⋊C4—131no—
297C22×C22⋊C4—131no—
298C22×C22⋊C4—132no—
299#385—264no—
300#380—132no—
301#380—132no—
302C4×C22⋊C4—132no—
303C4×C22⋊C4—132no—
304C4×C22⋊C4—264no—
305#342—134no—
306#342—134no—
307#342—134no—
308#342—134no—
309C22×C13—21no—
310C22×C13—21no—
311C22×C13—22no—
312C22×C13—22no—
313C22×C13—44no—
314C22×C13#39114yes—
315C22×C13C22×C22⋊C411yescharacteristic
316C22×C13C2×C4×Q812yes—
317C2×D13—21no—
318C2×D13—21no—
319C2×D13—21no—
320C2×D13—21no—
321C2×D13—22no—
322C2×D13—22no—
323C2×D13—44no—
324C2×D13—44no—
325C2×D13—44no—
326C2×D13—44no—
327C2×D13—44no—
328C2×D13—42no—
329C2×D13—42no—
330C2×D13#39114yes—
331C2×D13C2×C22⋊Q812yes—
332C2×D13C2×#12411yescharacteristic
333C2×D13C2×C4⋊Q811yescharacteristic
334#261—48no—
335#261—84no—
336C4×C13—22no—
337C4×C13—22no—
338C4×C13—41no—
339C4×C13—41no—
340C4×C13—41no—
341C4×C13—41no—
342Dic13—22no—
343Dic13—22no—
344Dic13—41no—
345Dic13—41no—
346Dic13—41no—
347Dic13—41no—
348C25—131no—
349C22×Q8—131no—
350C22×Q8—131no—
351C22×Q8—131no—
352C22×Q8C2×#26111yescharacteristic, complemented
353C23×C4—131no—
354C23×C4—131no—
355C23×C4—131no—
356C23×C4—131no—
357C23×C4—131no—
358C23×C4—262no—
359C23×C4—262no—
360C23×C4C2×#26111yescharacteristic
361C22⋊Q8—24no—
362C22⋊Q8—264no—
363C22⋊Q8—264no—
364C22⋊Q8—264no—
365C22⋊Q8—264no—
366C22⋊Q8—264no—
367C22⋊Q8—264no—
368C22⋊Q8—262no—
369C22⋊Q8—262no—
370C22⋊Q8—262no—
371C22⋊Q8—262no—
372C22⋊Q8—262no—
373C22⋊Q8—262no—
374C22⋊Q8—262no—
375C22⋊Q8—262no—
376C22⋊Q8—264no—
377C2×C4⋊C4—131no—
378C2×C4⋊C4—131no—
379C2×C4⋊C4—131no—
380C2×C4⋊C4—131no—
381C2×C4⋊C4—131no—
382C2×C4⋊C4—131no—
383C2×C4⋊C4—131no—
384C2×C4⋊C4—131no—
385C2×C4⋊C4—131no—
386C2×C4⋊C4—262no—
387C2×C4⋊C4—262no—
388C2×C4⋊C4C2×#26111yescharacteristic, complemented
389C2×C4⋊C4C2×#26111yescharacteristic, complemented
390C2×C4⋊C4C2×#26111yescharacteristic, complemented
391C2×C22⋊C4—131no—
392C2×C22⋊C4—131no—
393C2×C22⋊C4—131no—
394C2×C22⋊C4—134no—
395C2×C22⋊C4—134no—
396C2×C22⋊C4—134no—
397C2×C22⋊C4—138no—
398C2×C22⋊C4—131no—
399C2×C22⋊C4—131no—
400C2×C22⋊C4—131no—
401C2×C22⋊C4—131no—
402C2×C22⋊C4—131no—
403C2×C22⋊C4—131no—
404C2×C22⋊C4—131no—
405C2×C22⋊C4—134no—
406C2×C22⋊C4—262no—
407C2×C22⋊C4—262no—
408C2×C22⋊C4—262no—
409C2×C22⋊C4—262no—
410C2×C22⋊C4—262no—
411C2×C22⋊C4—264no—
412C2×C22⋊C4—268no—
413C2×C22⋊C4—262no—
414C2×C22⋊C4C2×#26111yescharacteristic
415C2×C22⋊C4C2×#26111yescharacteristic
416C2×C42—262no—
417C2×C42—262no—
418#95—262no—
419#95—262no—
420#95—262no—
421#95—262no—
422C2×C13—22no—
423C2×C13—24no—
424C2×C13C2×#39111yescharacteristic
425C2×C13#288311yescharacteristic
426C2×C13#288311yescharacteristic
427D13—22no—
428D13—21no—
429D13—21no—
430D13—24no—
431C24—132no—
432C24—132no—
433C24—131no—
434C24—131no—
435C24—134no—
436C24—134no—
437C24—1041no—
438C24—1041no—
439C24C22×#26111yescharacteristic
440C2×Q8—22no—
441C2×Q8—81no—
442C2×Q8—262no—
443C2×Q8—262no—
444C2×Q8—262no—
445C2×Q8—1041no—
446C2×Q8—1041no—
447C2×Q8—1041no—
448C22×C4—22no—
449C22×C4—81no—
450C22×C4—131no—
451C22×C4—131no—
452C22×C4—132no—
453C22×C4—132no—
454C22×C4—132no—
455C22×C4—132no—
456C22×C4—131no—
457C22×C4—131no—
458C22×C4—131no—
459C22×C4—131no—
460C22×C4—262no—
461C22×C4—262no—
462C22×C4—264no—
463C22×C4—268no—
464C22×C4—262no—
465C22×C4—264no—
466C22×C4—524no—
467C22×C4—1041no—
468C22×C4—1041no—
469C22×C4—1041no—
470C22×C4—1041no—
471C22×C4—1042no—
472C22×C4—1042no—
473C22×C4—1042no—
474C22×C4—1042no—
475C22×C4—1041no—
476C22×C4C22×#26111yescharacteristic
477C22×C4C22×#26111yescharacteristic
478C22×C4C22×#26111yescharacteristic
479C22×C4C22×#26111yescharacteristic
480C22×C4C22×#26111yescharacteristic
481C22×C4C22×#26111yescharacteristic
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—262no—
486C4⋊C4—262no—
487C4⋊C4—262no—
488C4⋊C4—262no—
489C4⋊C4—262no—
490C4⋊C4—262no—
491C4⋊C4—262no—
492C4⋊C4—262no—
493C4⋊C4—262no—
494C4⋊C4—1042no—
495C4⋊C4—1042no—
496C22⋊C4—22no—
497C22⋊C4—134no—
498C22⋊C4—134no—
499C22⋊C4—134no—
500C22⋊C4—138no—
501C22⋊C4—261no—
502C22⋊C4—261no—
503C22⋊C4—261no—
504C22⋊C4—261no—
505C22⋊C4—262no—
506C22⋊C4—261no—
507C22⋊C4—261no—
508C22⋊C4—262no—
509C22⋊C4—261no—
510C22⋊C4—261no—
511C22⋊C4—264no—
512C22⋊C4—268no—
513C22⋊C4—262no—
514C22⋊C4—268no—
515C22⋊C4—524no—
516C22⋊C4C22⋊#26114yes—
517C42—1042no—
518C42—1042no—
519C13#2133311yescharacteristic, complemented, Sylow
520C23—41no—
521C23—41no—
522C23—132no—
523C23—131no—
524C23—131no—
525C23—132no—
526C23—132no—
527C23—134no—
528C23—134no—
529C23—134no—
530C23—134no—
531C23—134no—
532C23—521no—
533C23—521no—
534C23—522no—
535C23—522no—
536C23—521no—
537C23—521no—
538C23—522no—
539C23—522no—
540C23—1044no—
541C23—1044no—
542C23Q8×#26114yes—
543C23C2×C22⋊#26112yes—
544C23C23×#26111yescharacteristic
545Q8—82no—
546Q8—1042no—
547Q8—1042no—
548Q8—1042no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—82no—
560C2×C4—132no—
561C2×C4—132no—
562C2×C4—268no—
563C2×C4—261no—
564C2×C4—261no—
565C2×C4—261no—
566C2×C4—261no—
567C2×C4—522no—
568C2×C4—522no—
569C2×C4—522no—
570C2×C4—522no—
571C2×C4—521no—
572C2×C4—521no—
573C2×C4—524no—
574C2×C4—524no—
575C2×C4—521no—
576C2×C4—521no—
577C2×C4—521no—
578C2×C4—521no—
579C2×C4—524no—
580C2×C4—524no—
581C2×C4—524no—
582C2×C4—1042no—
583C2×C4—1042no—
584C2×C4—1042no—
585C2×C4—1048no—
586C2×C4—1048no—
587C2×C4C2×C22⋊#26112yes—
588C2×C4C2×C22⋊#26112yes—
589C22—21no—
590C22—22no—
591C22—21no—
592C22—22no—
593C22—44no—
594C22—132no—
595C22—131no—
596C22—131no—
597C22—134no—
598C22—261no—
599C22—261no—
600C22—262no—
601C22—261no—
602C22—261no—
603C22—262no—
604C22—524no—
605C22—524no—
606C22—522no—
607C22—522no—
608C22—524no—
609C22—524no—
610C22—524no—
611C22(C2×#261)⋊Q814yes—
612C22C2×Q8×#26112yes—
613C22C22×C22⋊#26111yescharacteristic
614C4—22no—
615C4—22no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—262no—
621C4—262no—
622C4—528no—
623C4—521no—
624C4—521no—
625C4—521no—
626C4—521no—
627C4—1044no—
628C2—22no—
629C2—24no—
630C2—262no—
631C2—261no—
632C2—261no—
633C2—264no—
634C2C2×(C2×#261)⋊Q811yescharacteristic
635C2#1313143511yescharacteristic
636C2#1313143611yescharacteristic
subgroupG/H
centerC22C22×C22⋊#261
commutatorC23×C13C23×C4
FrattiniC23C23×#261
FittingC2×C13×C22⋊Q8C4
socleC22×C13C22×C22⋊C4
prime radicalC24×D13C23
cosocleC23×D13C24
hypercenterC2×C22⋊Q8#261
hypocenterC13#21333
LCSG ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #13131027C2semidirect
4. #13131027C2semidirect
5. #13131026C2semidirect
6. #13131025C2semidirect
12. C2×C13×C22⋊Q8C4semidirect
13. C2×#88200C4semidirect
16. C2×#88178C4semidirect
19. C2×#88151C4semidirect
21. C2×#88150C4semidirect
22. C2×#88150C4semidirect
27. C2×#88132C4semidirect
28. C2×#88132C4semidirect
42. C22×Q8×C13C2×C4semidirect
49. C22×Dic26C2×C4semidirect
50. C22×Dic26C2×C4semidirect
51. C22×Dic26C2×C4semidirect
56. C2×C22⋊#261Q8semidirect
64. C2×C13×C4⋊C4C2×C4semidirect
65. C2×C13×C4⋊C4C2×C4semidirect
66. C2×C13×C4⋊C4C2×C4semidirect
84. C2×C4⋊Dic13C2×C4semidirect
85. C2×C4⋊Dic13C2×C4semidirect
86. C2×C4⋊Dic13C2×C4semidirect
88. C2×#6101C2×C4semidirect
89. C2×#6101C2×C4semidirect
90. C2×#6101C2×C4semidirect
91. C2×#6101C2×C4semidirect
92. C2×#6101C2×C4semidirect
93. C2×#6101C2×C4semidirect
293. C2×C22⋊Q8#261semidirect
352. C22×Q8C2×#261semidirect
388. C2×C4⋊C4C2×#261semidirect
389. C2×C4⋊C4C2×#261semidirect
390. C2×C4⋊C4C2×#261semidirect
519. C13#21333semidirect

Maximal subgroups

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

subgroupindex
#131313152
#131312722
#131310272
#131310262
#131310252
#131305332
C2×#117182892
C2×#117193392
#2133313