Finite Group Databasebeta

← #15894017

Subgroups — #15894017

Order 3328.

1734 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#6107C211yesmaximal, characteristic, complemented
2C2×#11718889C212yesmaximal, complemented
3C2×#11718711C211yesmaximal, characteristic, complemented
4#13118201C211yesmaximal, characteristic
5#13115936C212yesmaximal, complemented
6#13115844C212yesmaximal, complemented
7#13115260C211yesmaximal, characteristic, complemented
8#13115177C212yesmaximal, complemented
9#13115129C211yesmaximal, characteristic
10#13114771C211yesmaximal, characteristic, complemented
11C2×D4×Q8×C13C211yesmaximal, characteristic
12C23×Q8×C13C2211yescharacteristic
13C23×Q8×C13C2211yescharacteristic
14C23×Dic26C2211yescharacteristic
15D4×Q8×C13—24no—
16D4×Q8×C13—24no—
17C2×C13×C4⋊Q8—21no—
18C2×C13×C4⋊Q8C2211yescharacteristic
19C2×C13×C22⋊Q8—21no—
20C2×C13×C22⋊Q8—21no—
21C2×C13×C22⋊Q8C2211yescharacteristic
22C2×C13×C22⋊Q8C2211yescharacteristic
23C2×C4×Q8×C13C2211yescharacteristic
24C2×C4×D4×C13—21no—
25C2×C4×D4×C13C2211yescharacteristic
26C22×#6107—24no—
27C22×#6107—28no—
28C22×#6107C2212yes—
29C22×#6107C2212yes—
30C22×#6107C2212yes—
31C2×#88178C2212yes—
32C23×#1055C2211yescharacteristic
33#11718889—28no—
34C2×#88075C2212yes—
35C2×#88075C2212yes—
36C2×#88075C2212yes—
37#11718881—21no—
38#11718873C2211yescharacteristic
39C2×C22⋊#1055C2211yescharacteristic
40C2×C22⋊#1055C2212yes—
41#11718711—24no—
42#11718640—21no—
43C2×#88050C2212yes—
44#11718608—21no—
45#11718607—21no—
46#11718590C2212yes—
47#11718121C2212yes—
48#11718116C2212yes—
49C22×Q8×C13—22no—
50C22×Q8×C13—22no—
51C22×Q8×C13—21no—
52C22×Q8×C13—21no—
53C22×Q8×C13—22no—
54C22×Q8×C13—24no—
55C22×Q8×C13—24no—
56C22×Q8×C13—42no—
57C22×Q8×C13—42no—
58C22×Q8×C13D414yes—
59C22×Q8×C13C2311yescharacteristic
60C22×Q8×C13C2311yescharacteristic
61C22×Q8×C13C2311yescharacteristic
62C22×D4×C13D411yescharacteristic
63C23×C4×C13—21no—
64C23×C4×C13—21no—
65C23×C4×C13C2311yescharacteristic
66C23×C4×C13C2311yescharacteristic
67C23×Dic13—21no—
68C22×Dic26—22no—
69C22×Dic26—24no—
70C22×Dic26—22no—
71C22×Dic26—21no—
72C22×Dic26—42no—
73C22×Dic26C2312yes—
74C13×C4⋊Q8—28no—
75C13×C4⋊Q8—22no—
76C13×C4⋊Q8—22no—
77C13×C22⋊Q8—28no—
78C13×C22⋊Q8—28no—
79C13×C22⋊Q8—22no—
80C13×C22⋊Q8—22no—
81C13×C22⋊Q8—22no—
82C13×C22⋊Q8D414yes—
83C4×Q8×C13—22no—
84C4×Q8×C13—22no—
85C4×D4×C13—28no—
86C4×D4×C13—22no—
87C4×D4×C13—22no—
88C2×C13×C4⋊C4—21no—
89C2×C13×C4⋊C4—21no—
90C2×C13×C4⋊C4—21no—
91C2×C13×C4⋊C4—21no—
92C2×C13×C4⋊C4—21no—
93C2×C13×C4⋊C4D411yescharacteristic
94C2×C13×C4⋊C4C2311yescharacteristic
95C2×C13×C22⋊C4—21no—
96C2×C13×C22⋊C4—21no—
97C2×C13×C22⋊C4D411yescharacteristic
98C2×C13×C22⋊C4D411yescharacteristic
99C2×C42×C13—21no—
100C2×C42×C13C2311yescharacteristic
101C2×C22⋊Dic13—22no—
102C2×C22⋊Dic13—21no—
103C2×#6107—232no—
104C2×#6107—24no—
105C2×#6107—24no—
106C2×#6107—24no—
107C2×#6107—24no—
108C2×#6107—24no—
109C2×#6107—24no—
110C2×#6107—24no—
111C2×#6107—42no—
112C2×#6107D418yescomplemented
113C2×C4⋊Dic13C2312yes—
114#88178—24no—
115#88178—42no—
116C2×#6101—22no—
117C22×#1055—22no—
118C22×#1055—24no—
119C22×#1055C2312yes—
120C22×#1055C2312yes—
121#88075—24no—
122#88075—24no—
123#88075—42no—
124#88071—22no—
125#88071—22no—
126#88071—21no—
127C22⋊#1055—22no—
128C22⋊#1055—24no—
129#88050—24no—
130#23856—131nomaximal, Sylow
131C24×C13C2×D411yescharacteristic
132C24×C13C2×D411yescharacteristic
133C2×Q8×C13—24no—
134C2×Q8×C13—24no—
135C2×Q8×C13—21no—
136C2×Q8×C13—21no—
137C2×Q8×C13—21no—
138C2×Q8×C13—22no—
139C2×Q8×C13—28no—
140C2×Q8×C13—22no—
141C2×Q8×C13—22no—
142C2×Q8×C13—22no—
143C2×Q8×C13—22no—
144C2×Q8×C13—44no—
145C2×Q8×C13—44no—
146C2×Q8×C13—44no—
147C2×Q8×C13—48no—
148C2×Q8×C13—48no—
149C2×Q8×C13—44no—
150C2×Q8×C13—44no—
151C2×Q8×C13C2×D414yes—
152C2×Q8×C13C2×D412yes—
153C2×Q8×C13C2×D412yes—
154C2×Q8×C13C2×D412yes—
155C2×D4×C13—22no—
156C2×D4×C13—22no—
157C2×D4×C13Q1614yescomplemented
158C22×C4×C13—21no—
159C22×C4×C13—22no—
160C22×C4×C13—22no—
161C22×C4×C13—24no—
162C22×C4×C13—24no—
163C22×C4×C13—21no—
164C22×C4×C13—21no—
165C22×C4×C13—21no—
166C22×C4×C13—22no—
167C22×C4×C13—21no—
168C22×C4×C13—21no—
169C22×C4×C13—22no—
170C22×C4×C13—44no—
171C22×C4×C13—44no—
172C22×C4×C13C2×D411yescharacteristic
173C22×C4×C13C2×D411yescharacteristic
174C22×C4×C13C2×D411yescharacteristic
175C22×C4×C13C2×D412yes—
176C22×C4×C13C2×D412yes—
177C22×C4×C13C2×D411yescharacteristic
178C22×C4×C13C2411yescharacteristic
179C22×Dic13—22no—
180C22×Dic13—22no—
181C22×Dic13—44no—
182C22×Dic13—42no—
183C2×Dic26—28no—
184C2×Dic26—22no—
185C2×Dic26—22no—
186C2×Dic26—22no—
187C2×Dic26—22no—
188C2×Dic26—22no—
189C2×Dic26—48no—
190C2×Dic26—44no—
191C2×Dic26—44no—
192C2×Dic26—42no—
193C2×Dic26C2×D414yes—
194C13×C4⋊C4—24no—
195C13×C4⋊C4—24no—
196C13×C4⋊C4—24no—
197C13×C4⋊C4—24no—
198C13×C4⋊C4—24no—
199C13×C4⋊C4—22no—
200C13×C4⋊C4—21no—
201C13×C4⋊C4C2×D412yes—
202C13×C22⋊C4—24no—
203C13×C22⋊C4—24no—
204C13×C22⋊C4—22no—
205C13×C22⋊C4—22no—
206C42×C13—24no—
207C42×C13—21no—
208C42×C13—21no—
209C22⋊Dic13—44no—
210C22⋊Dic13—42no—
211#6107—216no—
212#6107—216no—
213#6107—48no—
214C4⋊Dic13—22no—
215C4⋊Dic13—22no—
216#6101—44no—
217C2×#1055—28no—
218C2×#1055—22no—
219C2×#1055—22no—
220C2×#1055—22no—
221C2×#1055—22no—
222C2×#1055—22no—
223C2×#1055C2×D414yes—
224C23×Q16—131no—
225C2×D4×Q8D1311yescharacteristic
226C2×#451—132no—
227C2×#451—131no—
228#2409—131no—
229#2016—132no—
230#2001—132no—
231#1892—131no—
232#1878—132no—
233#1869—131no—
234#1787—131no—
235C23×C13—22no—
236C23×C13—22no—
237C23×C13—21no—
238C23×C13—22no—
239C23×C13—21no—
240C23×C13—22no—
241C23×C13#12012yes—
242C23×C13C2×Q1612yes—
243C23×C13C2×Q1612yescomplemented
244C23×C13#13712yes—
245C23×C13C22×D411yescharacteristic
246Q8×C13—22no—
247Q8×C13—24no—
248Q8×C13—22no—
249Q8×C13—22no—
250Q8×C13—24no—
251Q8×C13—48no—
252Q8×C13—48no—
253Q8×C13—41no—
254Q8×C13—41no—
255Q8×C13—42no—
256Q8×C13#12014yes—
257D4×C13—28no—
258C2×C4×C13—21no—
259C2×C4×C13—22no—
260C2×C4×C13—22no—
261C2×C4×C13—24no—
262C2×C4×C13—21no—
263C2×C4×C13—22no—
264C2×C4×C13—22no—
265C2×C4×C13—24no—
266C2×C4×C13—21no—
267C2×C4×C13—22no—
268C2×C4×C13—22no—
269C2×C4×C13—22no—
270C2×C4×C13—21no—
271C2×C4×C13—22no—
272C2×C4×C13—22no—
273C2×C4×C13—48no—
274C2×C4×C13—48no—
275C2×C4×C13—42no—
276C2×C4×C13—42no—
277C2×C4×C13—42no—
278C2×C4×C13—41no—
279C2×C4×C13—41no—
280C2×C4×C13—42no—
281C2×C4×C13#12012yes—
282C2×C4×C13C2×Q1612yescomplemented
283C2×C4×C13C22×D411yescharacteristic
284C2×C4×C13C22×D412yes—
285C2×C4×C13C22×D411yescharacteristic
286C2×C4×C13H(5,𝔽2)12yes—
287C2×Dic13—24no—
288C2×Dic13—48no—
289C2×Dic13—42no—
290C2×Dic13—42no—
291C2×Dic13—42no—
292C2×Dic13—42no—
293C2×Dic13—42no—
294Dic26—24no—
295Dic26—24no—
296Dic26—48no—
297Dic26—42no—
298Dic26—42no—
299#1055—24no—
300#1055—42no—
301C23×Q8—131no—
302C23×Q8C2×D1311yescharacteristic
303C23×Q8C2×D1311yescharacteristic
304C22×Q16—132no—
305C22×Q16—132no—
306C22×Q16—132no—
307C22×Q16—268no—
308C22×Q16—264no—
309C23×C8—131no—
310D4×Q8—24no—
311D4×Q8—24no—
312C2×C4⋊Q8—21no—
313C2×C4⋊Q8C2×D1311yescharacteristic
314C2×C22⋊Q8—21no—
315C2×C22⋊Q8—21no—
316C2×C22⋊Q8—132no—
317C2×C22⋊Q8C2×D1311yescharacteristic
318C2×C22⋊Q8C2×D1311yescharacteristic
319C2×C4×Q8C2×D1311yescharacteristic
320C2×C4×D4—21no—
321C2×C4×D4C2×D1311yescharacteristic
322#451—268no—
323#451—264no—
324C2×#103—132no—
325C2×#103—132no—
326C2×#103—132no—
327C2×#103—132no—
328C2×C22⋊C8—132no—
329C2×C22⋊C8—131no—
330#395—261no—
331#393—261no—
332#391—132no—
333#386—261no—
334#385—261no—
335#380—131no—
336#340—132no—
337#336—132no—
338C22×C13—22no—
339C22×C13—22no—
340C22×C13—24no—
341C22×C13—24no—
342C22×C13—22no—
343C22×C13—24no—
344C22×C13—41no—
345C22×C13—41no—
346C22×C13—42no—
347C22×C13#45114yescomplemented
348C22×C13C2×#12011yescharacteristic
349C22×C13D4212yes—
350C22×C13C22×Q1611yescharacteristic
351C22×C13C2×#13711yescharacteristic
352C22×C13#57812yes—
353C4×C13—22no—
354C4×C13—24no—
355C4×C13—22no—
356C4×C13—41no—
357C4×C13—42no—
358C4×C13—42no—
359C4×C13—41no—
360C4×C13—42no—
361C4×C13C2×#12011yescharacteristic
362C4×C13#53411yescharacteristic
363C4×C13D4212yes—
364Dic13—44no—
365Dic13—82no—
366C22×Q8—22no—
367C22×Q8—22no—
368C22×Q8—21no—
369C22×Q8—21no—
370C22×Q8—22no—
371C22×Q8—24no—
372C22×Q8—24no—
373C22×Q8—42no—
374C22×Q8—42no—
375C22×Q8—132no—
376C22×Q8—264no—
377C22×Q8—261no—
378C22×Q8—262no—
379C22×Q8—262no—
380C22×Q8—522no—
381C22×Q8C22⋊D1314yes—
382C22×Q8C22×D1311yescharacteristic
383C22×Q8C22×D1311yescharacteristic
384C22×Q8C22×D1311yescharacteristic
385C22×D4C22⋊D1311yescharacteristic
386C23×C4—21no—
387C23×C4—21no—
388C23×C4—261no—
389C23×C4C22×D1311yescharacteristic
390C23×C4C22×D1311yescharacteristic
391C2×Q16—138no—
392C2×Q16—264no—
393C2×Q16—264no—
394C2×Q16—2632no—
395C2×Q16—264no—
396C2×Q16—264no—
397C2×Q16—264no—
398C2×Q16—264no—
399C2×Q16—264no—
400C2×Q16—522no—
401C22×C8—132no—
402C22×C8—132no—
403C22×C8—264no—
404C22×C8—262no—
405C4⋊Q8—28no—
406C4⋊Q8—22no—
407C4⋊Q8—22no—
408C22⋊Q8—28no—
409C22⋊Q8—28no—
410C22⋊Q8—22no—
411C22⋊Q8—22no—
412C22⋊Q8—22no—
413C22⋊Q8—264no—
414C22⋊Q8—522no—
415C22⋊Q8C22⋊D1314yes—
416C4×Q8—22no—
417C4×Q8—22no—
418C4×D4—28no—
419C4×D4—22no—
420C4×D4—22no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—132no—
427C2×C4⋊C4—262no—
428C2×C4⋊C4C22⋊D1311yescharacteristic
429C2×C4⋊C4C22×D1311yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—262no—
433C2×C22⋊C4—261no—
434C2×C22⋊C4C22⋊D1311yescharacteristic
435C2×C22⋊C4C22⋊D1311yescharacteristic
436C2×C42—21no—
437C2×C42C22×D1311yescharacteristic
438#103—264no—
439#103—264no—
440#103—264no—
441#103—522no—
442C22⋊C8—264no—
443C22⋊C8—262no—
444#95—261no—
445#95—262no—
446#95—262no—
447C2×C13—24no—
448C2×C13—42no—
449C2×C13#240311yescharacteristic
450C2×C13C2×#45111yescharacteristic
451C2×C13#301111yescharacteristic
452C2×C13#327012yes—
453C2×C13D4×Q1612yes—
454C24C2×C22⋊D1311yescharacteristic
455C24C2×C22⋊D1311yescharacteristic
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8—22no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—28no—
463C2×Q8—22no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—44no—
468C2×Q8—44no—
469C2×Q8—44no—
470C2×Q8—48no—
471C2×Q8—48no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—134no—
475C2×Q8—268no—
476C2×Q8—262no—
477C2×Q8—262no—
478C2×Q8—262no—
479C2×Q8—262no—
480C2×Q8—262no—
481C2×Q8—524no—
482C2×Q8—528no—
483C2×Q8—524no—
484C2×Q8—522no—
485C2×Q8D4×D1314yes—
486C2×Q8C2×C22⋊D1312yes—
487C2×Q8C2×C22⋊D1312yes—
488C2×Q8C2×C22⋊D1312yes—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4#610714yescomplemented
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—262no—
507C22×C4—262no—
508C22×C4—524no—
509C22×C4—522no—
510C22×C4C2×C22⋊D1311yescharacteristic
511C22×C4C2×C22⋊D1311yescharacteristic
512C22×C4C2×C22⋊D1311yescharacteristic
513C22×C4C2×C22⋊D1311yescharacteristic
514C22×C4C2×C22⋊D1312yes—
515C22×C4C2×C22⋊D1312yes—
516C22×C4C23×D1311yescharacteristic
517Q16—2616no—
518Q16—2616no—
519Q16—528no—
520C2×C8—134no—
521C2×C8—268no—
522C2×C8—262no—
523C2×C8—262no—
524C2×C8—262no—
525C2×C8—262no—
526C2×C8—262no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—262no—
535C4⋊C4—262no—
536C4⋊C4—524no—
537C4⋊C4C2×C22⋊D1312yes—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—524no—
543C22⋊C4—522no—
544C42—24no—
545C42—21no—
546C42—21no—
547C13#2385611yescharacteristic, complemented, Sylow
548C23—22no—
549C23—22no—
550C23—22no—
551C23—21no—
552C23—22no—
553C23—21no—
554C23#8819612yes—
555C23C2×#610712yes—
556C23C2×#610712yescomplemented
557C23#8820712yes—
558C23C22×C22⋊D1311yescharacteristic
559Q8—24no—
560Q8—22no—
561Q8—22no—
562Q8—24no—
563Q8—22no—
564Q8—48no—
565Q8—48no—
566Q8—41no—
567Q8—42no—
568Q8—41no—
569Q8—264no—
570Q8—264no—
571Q8—528no—
572Q8—522no—
573Q8—522no—
574Q8#8819114yes—
575D4—28no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—264no—
600C2×C4—528no—
601C2×C4—522no—
602C2×C4—522no—
603C2×C4—522no—
604C2×C4—522no—
605C2×C4—522no—
606C2×C4C2×#610712yescomplemented
607C2×C4#8820712yes—
608C2×C4C2×D4×D1312yes—
609C2×C4#8825112yes—
610C2×C4C22×C22⋊D1311yescharacteristic
611C2×C4C22×C22⋊D1311yescharacteristic
612C8—264no—
613C8—522no—
614C22—24no—
615C22—22no—
616C22—24no—
617C22—22no—
618C22—24no—
619C22—22no—
620C22—42no—
621C22—41no—
622C22—41no—
623C22#1171888914yescomplemented
624C22D4×C22⋊D1312yes—
625C22C2×#8819611yescharacteristic
626C22C22×#610711yescharacteristic
627C22#1171954312yes—
628C22C2×#8820711yescharacteristic
629C4—22no—
630C4—22no—
631C4—24no—
632C4—41no—
633C4—42no—
634C4—41no—
635C4—42no—
636C4—42no—
637C4—524no—
638C4—1042no—
639C4C2×#8819111yescharacteristic
640C4D4×C22⋊D1312yes—
641C4#1171950911yescharacteristic
642C2—24no—
643C2—42no—
644C2#1311814611yescharacteristic
645C2C2×#1171888911yescharacteristic
646C2#1312413111yescharacteristic
647C2#1312640712yes—
648C2D4×#610712yes—
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC2×D4×Q8×C13C2
socleC23×C13C22×D4
prime radicalC22×D4×C13D4
hypercenterC2×D4×Q8D13
hypocenterC13#23856
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C23×#6107C2semidirect
2. C2×#11718889C2semidirect
3. C2×#11718711C2semidirect
5. #13115936C2semidirect
6. #13115844C2semidirect
7. #13115260C2semidirect
8. #13115177C2semidirect
10. #13114771C2semidirect
112. C2×#6107D4semidirect
157. C2×D4×C13Q16semidirect
243. C23×C13C2×Q16semidirect
282. C2×C4×C13C2×Q16semidirect
347. C22×C13#451semidirect
491. C2×D4#6107semidirect
547. C13#23856semidirect
556. C23C2×#6107semidirect
606. C2×C4C2×#6107semidirect
623. C22#11718889semidirect

Maximal subgroups

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

subgroupindex
C23×#61072
C2×#117188892
C2×#117187112
#131182012
#131159362
#131158442
#131152602
#131151772
#131151292
#131147712
C2×D4×Q8×C132
#2385613