Finite Group Databasebeta

← #15881202

Subgroups — #15881202

Order 3328.

1018 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
1C2×#11718825C211yesmaximal, characteristic
2C2×SD16×Dic13C211yesmaximal, characteristic
3Q8×#6101C211yesmaximal, characteristic, complemented
4D4×#6101C211yesmaximal, characteristic
5#13115579C211yesmaximal, characteristic, complemented
6#13115577C211yesmaximal, characteristic, complemented
7#13115090C211yesmaximal, characteristic, complemented
8#13115088C211yesmaximal, characteristic, complemented
9#13115083C211yesmaximal, characteristic
10#13115081C211yesmaximal, characteristic
11#13114994C211yesmaximal, characteristic, complemented
12#13114839C211yesmaximal, characteristic, complemented
13#13114834C211yesmaximal, characteristic
14#13114704C211yesmaximal, characteristic, complemented
15C2×C13×#440C211yesmaximal, characteristic
16C22×C13×SD16C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×D4×C13C2211yescharacteristic
19C2×Q8×Dic13C2211yescharacteristic
20C2×Q8×Dic13C2211yescharacteristic
21C2×D4×Dic13C2211yescharacteristic
22C2×D4×Dic13C2211yescharacteristic
23C22×#6101—21no—
24C13×#440—24no—
25C2×C13×#107C2211yescharacteristic
26C2×C13×#103C2211yescharacteristic
27C2×C13×#102C2211yescharacteristic
28C2×C13×#97C2211yescharacteristic
29C2×#88075C2211yescharacteristic
30C2×#88075C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32C2×#88072C2211yescharacteristic
33#11718871—21no—
34#11718825—24no—
35SD16×Dic13C418yescomplemented
36C2×#88058C2211yescharacteristic
37C2×#88057C2211yescharacteristic
38C2×#88055C2211yescharacteristic
39C2×#88054C2211yescharacteristic
40C2×C8×Dic13C2211yescharacteristic
41#11718637—21no—
42#11718637C2211yescharacteristic
43#11718637C2211yescharacteristic
44#11718634—21no—
45#11718634C2211yescharacteristic
46#11718634C2211yescharacteristic
47C2×#88047C2211yescharacteristic
48C2×#88047C2211yescharacteristic
49#11718607—21no—
50#11718603—21no—
51C4×#6101—21no—
52C4×#6101C2211yescharacteristic
53#11718589—21no—
54#11718589C2211yescharacteristic
55#11718589C2211yescharacteristic
56#11718143C2211yescharacteristic
57#11718143C2211yescharacteristic
58#11718142C2211yescharacteristic
59#11718121C2211yescharacteristic
60#11718121C2211yescharacteristic
61#11718116C2211yescharacteristic
62C22×Q8×C13C2311yescharacteristic
63C22×D4×C13C2311yescharacteristic
64C23×C4×C13—21no—
65C23×Dic13—21no—
66C23×Dic13—21no—
67C2×C13×SD16C2×C414yes—
68C2×C13×SD16D414yes—
69C2×C13×SD16Q814yes—
70C22×C8×C13C2311yescharacteristic
71C22×C8×C13C2311yescharacteristic
72C4×Q8×C13—22no—
73C4×Q8×C13—41no—
74C4×D4×C13—22no—
75C4×D4×C13—41no—
76C2×C13×C4⋊C4—21no—
77C2×C13×C4⋊C4C2311yescharacteristic
78C2×C13×C4⋊C4C2311yescharacteristic
79C2×C13×C22⋊C4—21no—
80C2×C42×C13—21no—
81C2×C42×C13C2311yescharacteristic
82C2×C22⋊Dic13—21no—
83C2×C22⋊Dic13—21no—
84Q8×Dic13—22no—
85Q8×Dic13—22no—
86Q8×Dic13—41no—
87Q8×Dic13C2×C414yescomplemented
88D4×Dic13—22no—
89D4×Dic13—22no—
90D4×Dic13—41no—
91D4×Dic13C2×C414yes—
92C2×C4⋊Dic13—21no—
93C2×C4⋊Dic13—21no—
94C2×C4⋊Dic13C2311yescharacteristic
95C2×C4⋊Dic13C2311yescharacteristic
96C2×C4⋊Dic13C2311yescharacteristic
97C2×C4⋊Dic13C2311yescharacteristic
98C2×#6101—21no—
99C2×#6101—21no—
100C2×#6101—21no—
101C2×#6101—21no—
102C2×#6101—21no—
103C2×#6101—21no—
104C2×#6101—21no—
105C2×#6101—21no—
106C2×#6101—44no—
107C2×#6101D411yescharacteristic
108C2×#6101D411yescharacteristic
109C2×C4×Dic13—21no—
110C2×C4×Dic13—21no—
111C2×C4×Dic13C2311yescharacteristic
112C2×C4×Dic13C2311yescharacteristic
113C22×#1055C2311yescharacteristic
114C22×#1055C2311yescharacteristic
115C13×#107—22no—
116C13×#103—22no—
117C13×#102—22no—
118C13×#97—22no—
119#88075—22no—
120#88075C2×C414yescomplemented
121#88072—22no—
122#88072C2×C414yes—
123#88071—21no—
124#88071—21no—
125#88071—21no—
126#88071—21no—
127#88071D411yescharacteristic
128#88071D411yescharacteristic
129#88058—22no—
130#88057C2×C414yescomplemented
131#88055—22no—
132#88054—22no—
133C8×Dic13C2×C414yescomplemented
134#88047—41no—
135#88047—41no—
136#22671—131nomaximal, Sylow
137C24×C13—21no—
138C2×Q8×C13—21no—
139C2×Q8×C13—21no—
140C2×Q8×C13—21no—
141C2×Q8×C13C22×C412yes—
142C2×Q8×C13C2×D412yes—
143C2×Q8×C13C2×Q812yes—
144C2×D4×C13—21no—
145C2×D4×C13—21no—
146C2×D4×C13—21no—
147C2×D4×C13C22×C412yes—
148C2×D4×C13C2×D412yes—
149C2×D4×C13C2×Q812yes—
150C22×C4×C13—21no—
151C22×C4×C13—21no—
152C22×C4×C13—21no—
153C22×C4×C13—42no—
154C22×C4×C13—42no—
155C22×C4×C13—81no—
156C22×C4×C13C2×D411yescharacteristic
157C22×C4×C13C2×D411yescharacteristic
158C22×C4×C13C2411yescharacteristic
159C22×Dic13—21no—
160C22×Dic13—21no—
161C22×Dic13—24no—
162C22×Dic13—21no—
163C22×Dic13—21no—
164C22×Dic13—42no—
165C22×Dic13—42no—
166C22×Dic13—42no—
167C22×Dic13—42no—
168C22×Dic13—81no—
169C22×Dic13C2×D411yescharacteristic
170C22×Dic13C2×D411yescharacteristic
171C22×Dic13C2×D411yescharacteristic
172C22×Dic13C2×D411yescharacteristic
173C13×SD16C4⋊C4116yescomplemented
174C2×C8×C13—21no—
175C2×C8×C13—21no—
176C2×C8×C13—21no—
177C2×C8×C13C22×C412yes—
178C2×C8×C13C2×D412yes—
179C2×C8×C13C2×Q812yes—
180C13×C4⋊C4—21no—
181C13×C4⋊C4—21no—
182C13×C4⋊C4—21no—
183C13×C4⋊C4—21no—
184C13×C4⋊C4—42no—
185C13×C22⋊C4—42no—
186C42×C13—21no—
187C42×C13—21no—
188C42×C13—42no—
189C22⋊Dic13—24no—
190C22⋊Dic13—42no—
191C4⋊Dic13—21no—
192C4⋊Dic13—21no—
193C4⋊Dic13—24no—
194C4⋊Dic13—21no—
195C4⋊Dic13—21no—
196C4⋊Dic13—21no—
197C4⋊Dic13—21no—
198C4⋊Dic13—42no—
199C4⋊Dic13C22×C412yes—
200C4⋊Dic13C22×C412yes—
201#6101—22no—
202#6101—22no—
203#6101—42no—
204#6101—42no—
205#6101—42no—
206#6101—42no—
207#6101—81no—
208#6101—81no—
209#6101—81no—
210#6101—81no—
211C4×Dic13—21no—
212C4×Dic13—21no—
213C4×Dic13—24no—
214C4×Dic13—42no—
215C4×Dic13C22×C412yes—
216C4×Dic13#4112yes—
217C2×#1055—21no—
218C2×#1055—21no—
219C2×#1055—21no—
220C2×#1055—21no—
221C2×#1055—21no—
222C2×#1055C22×C412yes—
223C2×#440—131no—
224C2×#440D1311yescharacteristic
225C2×C4×SD16—131no—
226Q8×C4⋊C4—131no—
227D4×C4⋊C4—131no—
228#1944—131no—
229#1944—131no—
230#1863—131no—
231#1863—131no—
232#1862—131no—
233#1862—131no—
234#1845—131no—
235#1803—131no—
236#1802—131no—
237#1770—131no—
238C23×C13—22no—
239C23×C13—22no—
240C23×C13—22no—
241C23×C13—41no—
242C23×C13—41no—
243C23×C13—41no—
244C23×C13—41no—
245C23×C13C22×D411yescharacteristic
246Q8×C13—22no—
247Q8×C13—22no—
248Q8×C13—22no—
249Q8×C13C2×C4⋊C414yescomplemented
250D4×C13—22no—
251D4×C13—22no—
252D4×C13—22no—
253D4×C13C2×C4⋊C414yes—
254C2×C4×C13—22no—
255C2×C4×C13—22no—
256C2×C4×C13—22no—
257C2×C4×C13—21no—
258C2×C4×C13—21no—
259C2×C4×C13—21no—
260C2×C4×C13—21no—
261C2×C4×C13—21no—
262C2×C4×C13—21no—
263C2×C4×C13—41no—
264C2×C4×C13—41no—
265C2×C4×C13—41no—
266C2×C4×C13—41no—
267C2×C4×C13—41no—
268C2×C4×C13—41no—
269C2×C4×C13—84no—
270C2×C4×C13C23×C411yescharacteristic
271C2×C4×C13C22×D411yescharacteristic
272C2×C4×C13C22×Q811yescharacteristic
273C2×C4×C13C2×#4111yescharacteristic
274C2×C4×C13H(5,𝔽2)11yescharacteristic
275C2×C4×C132−1+411yescharacteristic
276C2×Dic13—21no—
277C2×Dic13—21no—
278C2×Dic13—22no—
279C2×Dic13—22no—
280C2×Dic13—21no—
281C2×Dic13—21no—
282C2×Dic13—21no—
283C2×Dic13—21no—
284C2×Dic13—21no—
285C2×Dic13—21no—
286C2×Dic13—41no—
287C2×Dic13—41no—
288C2×Dic13—41no—
289C2×Dic13—41no—
290C2×Dic13—48no—
291C2×Dic13—41no—
292C2×Dic13—41no—
293C2×Dic13—41no—
294C2×Dic13—41no—
295C2×Dic13—41no—
296C2×Dic13—41no—
297C2×Dic13—84no—
298C2×Dic13C4×D412yes—
299C2×Dic13C4×D412yes—
300C2×Dic13AGL(1,ℤ/8)12yes—
301C2×Dic13#13712yes—
302C8×C13—41no—
303C8×C13C2×C4⋊C414yescomplemented
304#1055—22no—
305#1055—41no—
306C22×SD16C2×D1311yescharacteristic
307C2×C4×Q8—131no—
308C2×C4×Q8—131no—
309C2×C4×Q8C2×D1311yescharacteristic
310C2×C4×D4—131no—
311C2×C4×D4—131no—
312C2×C4×D4C2×D1311yescharacteristic
313C22×C4⋊C4—261no—
314#440—24no—
315#440—264no—
316C4×SD16—138no—
317C2×#107—131no—
318C2×#107—131no—
319C2×#107—131no—
320C2×#107C2×D1311yescharacteristic
321C2×#106—131no—
322C2×C4⋊C8—131no—
323C2×#103—131no—
324C2×#103—131no—
325C2×#103C2×D1311yescharacteristic
326C2×#102—131no—
327C2×#102—131no—
328C2×#102C2×D1311yescharacteristic
329C2×#97—131no—
330C2×#97C2×D1311yescharacteristic
331C2×C4×C8—131no—
332#389—131no—
333#389—131no—
334#389—131no—
335#389—131no—
336#389—261no—
337#389—261no—
338#385—261no—
339#385—261no—
340#384—131no—
341#384—131no—
342#384—261no—
343#380—261no—
344C4×C4⋊C4—131no—
345C4×C4⋊C4—261no—
346#340—131no—
347#340—131no—
348#340—131no—
349#340—131no—
350#336—131no—
351#336—131no—
352C22×C13—24no—
353C22×C13—42no—
354C22×C13—42no—
355C22×C13—42no—
356C22×C13—42no—
357C22×C13—42no—
358C22×C13—42no—
359C22×C13C2×C4×D411yescharacteristic
360C22×C13D4211yescharacteristic
361C22×C13D4×Q811yescharacteristic
362C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
363C22×C13C2×#13711yescharacteristic
364C22×C13#57611yescharacteristic
365C22×C13#57811yescharacteristic
366C4×C13—24no—
367C4×C13—41no—
368C4×C13—41no—
369C4×C13—81no—
370C4×C13—81no—
371C4×C13C22×C4⋊C411yescharacteristic
372C4×C13#52011yescharacteristic
373C4×C13#54711yescharacteristic
374C4×C13#55411yescharacteristic
375Dic13—22no—
376Dic13—42no—
377Dic13—42no—
378Dic13—41no—
379Dic13—41no—
380Dic13—81no—
381Dic13—81no—
382Dic13#44014yescomplemented
383C22×Q8C22×D1311yescharacteristic
384C22×D4C22×D1311yescharacteristic
385C23×C4—21no—
386C23×C4—261no—
387C23×C4—261no—
388C2×SD16Dic2614yes—
389C2×SD16C4×D1314yes—
390C2×SD16C22⋊D1314yes—
391C22×C8—131no—
392C22×C8—131no—
393C22×C8C22×D1311yescharacteristic
394C22×C8C22×D1311yescharacteristic
395C4×Q8—22no—
396C4×Q8—41no—
397C4×Q8—134no—
398C4×Q8—262no—
399C4×Q8—262no—
400C4×Q8—521no—
401C4×D4—22no—
402C4×D4—41no—
403C4×D4—134no—
404C4×D4—262no—
405C4×D4—262no—
406C4×D4—521no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—131no—
409C2×C4⋊C4—131no—
410C2×C4⋊C4—131no—
411C2×C4⋊C4—131no—
412C2×C4⋊C4—131no—
413C2×C4⋊C4—131no—
414C2×C4⋊C4—261no—
415C2×C4⋊C4—261no—
416C2×C4⋊C4—261no—
417C2×C4⋊C4—261no—
418C2×C4⋊C4—261no—
419C2×C4⋊C4—261no—
420C2×C4⋊C4—261no—
421C2×C4⋊C4—261no—
422C2×C4⋊C4—261no—
423C2×C4⋊C4—261no—
424C2×C4⋊C4—524no—
425C2×C4⋊C4C22×D1311yescharacteristic
426C2×C4⋊C4C22×D1311yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—261no—
429C2×C22⋊C4—261no—
430C2×C42—21no—
431C2×C42—131no—
432C2×C42—131no—
433C2×C42—261no—
434C2×C42—261no—
435C2×C42C22×D1311yescharacteristic
436#107—22no—
437#107—262no—
438#107—521no—
439#107—521no—
440#106—134no—
441C4⋊C8—262no—
442#103—22no—
443#103—134no—
444#103—262no—
445#102—22no—
446#102—134no—
447#102—262no—
448#97—22no—
449#97—262no—
450C4×C8—134no—
451#95—131no—
452#95—131no—
453#95—261no—
454#95—261no—
455#95—261no—
456#95—261no—
457C2×C13—44no—
458C2×C13D4×C4⋊C411yescharacteristic
459C2×C13C2×#44011yescharacteristic
460C2×C13#295111yescharacteristic
461C2×C13#327411yescharacteristic
462C2×C13#327611yescharacteristic
463C2×C13#338511yescharacteristic
464C2×C13#338611yescharacteristic
465C24—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8C2×Dic2612yes—
470C2×Q8C2×C4×D1312yes—
471C2×Q8C2×C22⋊D1312yes—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4C2×Dic2612yes—
476C2×D4C2×C4×D1312yes—
477C2×D4C2×C22⋊D1312yes—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—81no—
484C22×C4—131no—
485C22×C4—131no—
486C22×C4—131no—
487C22×C4—131no—
488C22×C4—264no—
489C22×C4—261no—
490C22×C4—261no—
491C22×C4—261no—
492C22×C4—261no—
493C22×C4—522no—
494C22×C4—522no—
495C22×C4—522no—
496C22×C4—522no—
497C22×C4—1041no—
498C22×C4D4×D1311yescharacteristic
499C22×C4D4×D1311yescharacteristic
500C22×C4C23×D1311yescharacteristic
501SD16#6101116yescomplemented
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—132no—
506C2×C8—261no—
507C2×C8—261no—
508C2×C8—261no—
509C2×C8—261no—
510C2×C8—261no—
511C2×C8C2×Dic2612yes—
512C2×C8C2×C4×D1312yes—
513C2×C8C2×C22⋊D1312yes—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—42no—
519C4⋊C4—132no—
520C4⋊C4—132no—
521C4⋊C4—262no—
522C4⋊C4—262no—
523C4⋊C4—264no—
524C4⋊C4—261no—
525C4⋊C4—261no—
526C4⋊C4—261no—
527C4⋊C4—261no—
528C4⋊C4—261no—
529C4⋊C4—261no—
530C4⋊C4—522no—
531C4⋊C4—522no—
532C4⋊C4—522no—
533C4⋊C4—522no—
534C4⋊C4—522no—
535C4⋊C4—1041no—
536C4⋊C4—1041no—
537C4⋊C4—1041no—
538C4⋊C4—1041no—
539C22⋊C4—42no—
540C22⋊C4—264no—
541C22⋊C4—522no—
542C42—21no—
543C42—21no—
544C42—42no—
545C42—132no—
546C42—132no—
547C42—264no—
548C42—261no—
549C42—261no—
550C42—522no—
551C13#2267111yescharacteristic, complemented, Sylow
552C23—22no—
553C23—22no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×D4×D1311yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8C2×#610114yescomplemented
564D4—22no—
565D4—22no—
566D4—22no—
567D4C2×#610114yes—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—84no—
584C2×C4—132no—
585C2×C4—132no—
586C2×C4—132no—
587C2×C4—132no—
588C2×C4—261no—
589C2×C4—261no—
590C2×C4—261no—
591C2×C4—261no—
592C2×C4—261no—
593C2×C4—261no—
594C2×C4—262no—
595C2×C4—262no—
596C2×C4—261no—
597C2×C4—261no—
598C2×C4—528no—
599C2×C4—521no—
600C2×C4—521no—
601C2×C4—521no—
602C2×C4—521no—
603C2×C4—521no—
604C2×C4—521no—
605C2×C4—521no—
606C2×C4—521no—
607C2×C4—521no—
608C2×C4—521no—
609C2×C4—1044no—
610C2×C4C22×Dic2611yescharacteristic
611C2×C4C22×C4×D1311yescharacteristic
612C2×C4#8825111yescharacteristic
613C2×C4#41×D1311yescharacteristic
614C2×C4#8825711yescharacteristic
615C2×C4C22×C22⋊D1311yescharacteristic
616C8—41no—
617C8—262no—
618C8—521no—
619C8C2×#610114yescomplemented
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22D4×Dic2611yescharacteristic
628C22C4×D4×D1311yescharacteristic
629C22D13×AGL(1,ℤ/8)11yescharacteristic
630C22#1171947911yescharacteristic
631C22D13×#13711yescharacteristic
632C22#1171948411yescharacteristic
633C22D4×C22⋊D1311yescharacteristic
634C4—24no—
635C4—41no—
636C4—41no—
637C4—81no—
638C4—81no—
639C4—134no—
640C4—262no—
641C4—522no—
642C4—521no—
643C4—521no—
644C4—522no—
645C4—1041no—
646C4—1041no—
647C4#1171924411yescharacteristic
648C4#1171925111yescharacteristic
649C4C22×#610111yescharacteristic
650C4#1171952911yescharacteristic
651C2—44no—
652C2D4×#610111yescharacteristic
653C2D13×#44011yescharacteristic
654C2#1312370811yescharacteristic
655C2#1312645011yescharacteristic
656C2#1312647611yescharacteristic
657C2#1312774711yescharacteristic
658C2#1312775811yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#440C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
cosocleC22×C4×C13C24
hypercenterC2×#440D13
hypocenterC13#22671
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
3. Q8×#6101C2semidirect
5. #13115579C2semidirect
6. #13115577C2semidirect
7. #13115090C2semidirect
8. #13115088C2semidirect
11. #13114994C2semidirect
12. #13114839C2semidirect
14. #13114704C2semidirect
35. SD16×Dic13C4semidirect
87. Q8×Dic13C2×C4semidirect
120. #88075C2×C4semidirect
130. #88057C2×C4semidirect
133. C8×Dic13C2×C4semidirect
173. C13×SD16C4⋊C4semidirect
249. Q8×C13C2×C4⋊C4semidirect
303. C8×C13C2×C4⋊C4semidirect
382. Dic13#440semidirect
501. SD16#6101semidirect
551. C13#22671semidirect
563. Q8C2×#6101semidirect
619. C8C2×#6101semidirect

Maximal subgroups

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

subgroupindex
C2×#117188252
C2×SD16×Dic132
Q8×#61012
D4×#61012
#131155792
#131155772
#131150902
#131150882
#131150832
#131150812
#131149942
#131148392
#131148342
#131147042
C2×C13×#4402
#2267113