Finite Group Databasebeta

← #15925945

Subgroups — #15925945

Order 3328.

864 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718681C211yesmaximal, characteristic, complemented
2C2×C4×#6106C211yesmaximal, characteristic
3#13118686C211yesmaximal, characteristic
4#13116370C211yesmaximal, characteristic
5#13116313C211yesmaximal, characteristic, complemented
6#13116207C211yesmaximal, characteristic
7#13116169C211yesmaximal, characteristic, complemented
8#13115561C211yesmaximal, characteristic, complemented
9#13115395C211yesmaximal, characteristic, complemented
10#13115191C211yesmaximal, characteristic
11#13115191C211yesmaximal, characteristic
12#13115104C211yesmaximal, characteristic, complemented
13#13115104C211yesmaximal, characteristic, complemented
14#13115096C211yesmaximal, characteristic
15Q8×C13×C4⋊C4C211yesmaximal, characteristic, complemented
16C2×C4×Q8×C13C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×Q8×C13C2211yescharacteristic
19C22×#6106C2211yescharacteristic
20C2×#88150—21no—
21C2×C4×D52C2211yescharacteristic
22C2×#88123C2211yescharacteristic
23C13×#389—21no—
24C13×#389—21no—
25C13×#389C2211yescharacteristic
26C13×#389C2211yescharacteristic
27C13×#384—21no—
28C13×#384C2211yescharacteristic
29C4×C13×C4⋊C4—21no—
30C4×C13×C4⋊C4C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33C2×#88075C2211yescharacteristic
34C4×#6106—24no—
35#11718681—24no—
36#11718655—21no—
37C2×#88049C2211yescharacteristic
38C2×#88049C2211yescharacteristic
39C2×#88049C2211yescharacteristic
40C2×#88048C2211yescharacteristic
41C2×#88048C2211yescharacteristic
42C2×#88048C2211yescharacteristic
43C2×#88048C2211yescharacteristic
44C2×#88048C2211yescharacteristic
45#11718600C2211yescharacteristic
46#11718600C2211yescharacteristic
47#11718589C2211yescharacteristic
48#11718589C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50C2×C4×#1055C2211yescharacteristic
51#11718334—21no—
52#11718329—21no—
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718116C2211yescharacteristic
58#11718116C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C22×Q8×C13C2311yescharacteristic
62C22×D52C2311yescharacteristic
63C22×C4×D13—21no—
64C22×Dic26—21no—
65C4×Q8×C13—22no—
66C4×Q8×C13—22no—
67C4×Q8×C13—22no—
68C4×Q8×C13—22no—
69C4×Q8×C13—41no—
70C4×Q8×C13—41no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4—21no—
74C2×C13×C4⋊C4—21no—
75C2×C13×C4⋊C4—21no—
76C2×C13×C4⋊C4—21no—
77C2×C13×C4⋊C4—21no—
78C2×C13×C4⋊C4D411yescharacteristic
79C2×C13×C4⋊C4D411yescharacteristic
80C2×C13×C4⋊C4C2311yescharacteristic
81C2×C13×C4⋊C4C2311yescharacteristic
82C2×C13×C4⋊C4C2311yescharacteristic
83C2×C42×C13—21no—
84C2×C42×C13—21no—
85C2×C42×C13—21no—
86C2×C42×C13C2311yescharacteristic
87C2×C42×C13C2311yescharacteristic
88C2×C42×C13C2311yescharacteristic
89C2×#6106—22no—
90C2×#6106—22no—
91C2×#6106Q814yescomplemented
92C2×#6103—21no—
93C2×#6103—21no—
94C2×#6103—21no—
95C2×#6103—21no—
96C2×#6103—21no—
97C2×C4⋊Dic13—21no—
98C2×C4⋊Dic13C2311yescharacteristic
99C2×C4⋊Dic13C2311yescharacteristic
100C2×C4⋊Dic13C2311yescharacteristic
101C2×#6101—21no—
102C2×#6101—21no—
103C22×#1055C2311yescharacteristic
104C22×#1055C2311yescharacteristic
105C22×#1055C2311yescharacteristic
106C22×#1055C2311yescharacteristic
107#88150—44no—
108C4×D52—22no—
109C4×D52—41no—
110#88123—22no—
111#88123—41no—
112C13×#95—21no—
113C13×#95—21no—
114C13×#95D411yescharacteristic
115C13×#95D411yescharacteristic
116#88075—22no—
117#88075—22no—
118#88075D414yes—
119#88071—21no—
120#88071—21no—
121#88049—22no—
122#88049—22no—
123#88049—41no—
124#88048—22no—
125#88048—22no—
126#88048—22no—
127#88048—22no—
128#88048—41no—
129C4⋊#1055—22no—
130C4×#1055—22no—
131#26373—131nomaximal, Sylow
132C23×D13—21no—
133C2×Q8×C13—21no—
134C2×Q8×C13—21no—
135C2×Q8×C13—21no—
136C2×Q8×C13C2×D412yes—
137C2×Q8×C13C2×Q812yes—
138C2×Q8×C13#4112yes—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—21no—
142C22×C4×C13—21no—
143C22×C4×C13C2×D411yescharacteristic
144C22×C4×C13C2×D411yescharacteristic
145C22×C4×C13C2×D411yescharacteristic
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2×D411yescharacteristic
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13C2411yescharacteristic
150C22×Dic13—21no—
151C22×Dic13—21no—
152C22×Dic13—21no—
153C2×D52—21no—
154C2×D52—21no—
155C2×D52—21no—
156C2×D52—21no—
157C2×D52—21no—
158C2×D52C2×Q812yes—
159C2×C4×D13—42no—
160C2×C4×D13—42no—
161C2×C4×D13—81no—
162C2×Dic26—42no—
163C2×Dic26—42no—
164C2×Dic26—81no—
165C13×C4⋊C4—24no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—21no—
170C13×C4⋊C4—21no—
171C13×C4⋊C4—41no—
172C13×C4⋊C4—42no—
173C13×C4⋊C4—42no—
174C13×C4⋊C4—42no—
175C13×C4⋊C4—81no—
176C13×C4⋊C4—81no—
177C13×C4⋊C4—81no—
178C13×C4⋊C4SD1614yescomplemented
179C13×C4⋊C4#4112yes—
180C42×C13—24no—
181C42×C13—21no—
182C42×C13—21no—
183C42×C13—21no—
184C42×C13—21no—
185C42×C13—21no—
186C42×C13—21no—
187C42×C13—42no—
188C42×C13—42no—
189#6106—28no—
190#6103—42no—
191#6103—42no—
192#6103—42no—
193#6103—42no—
194#6103—81no—
195C4⋊Dic13—21no—
196C4⋊Dic13—21no—
197C4⋊Dic13—21no—
198C4⋊Dic13—21no—
199C4⋊Dic13—21no—
200C4⋊Dic13—42no—
201C4⋊Dic13C2×D412yes—
202#6101—42no—
203#6101—42no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055—21no—
211C2×#1055—21no—
212C2×#1055—21no—
213C2×#1055—21no—
214C2×#1055C2×D412yes—
215C2×#1055C2×Q812yes—
216C2×#475—131no—
217C2×C4×SD16—131no—
218#2452—131no—
219Q8×C4⋊C4D1311yescharacteristic, complemented
220#2094—131no—
221#2084—131no—
222#2065—131no—
223#2058—131no—
224#1941—131no—
225#1916—131no—
226#1880—131no—
227#1880—131no—
228#1865—131no—
229#1865—131no—
230#1864—131no—
231C23×C13C22×D411yescharacteristic
232C22×D13—22no—
233C22×D13—41no—
234C22×D13—41no—
235C22×D13—41no—
236C22×D13—41no—
237C22×D13—41no—
238C22×D13—41no—
239Q8×C13—22no—
240Q8×C13—22no—
241Q8×C13—22no—
242Q8×C13C22⋊Q814yescomplemented
243C2×C4×C13—21no—
244C2×C4×C13—22no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—22no—
248C2×C4×C13—22no—
249C2×C4×C13—21no—
250C2×C4×C13—21no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—22no—
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—21no—
259C2×C4×C13—21no—
260C2×C4×C13—21no—
261C2×C4×C13—21no—
262C2×C4×C13—41no—
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—41no—
270C2×C4×C13C2×SD1612yes—
271C2×C4×C13C2×SD1612yes—
272C2×C4×C13C2×SD1612yes—
273C2×C4×C13C22×D411yescharacteristic
274C2×C4×C13C22×Q811yescharacteristic
275C2×C4×C13C2×#4111yescharacteristic
276C2×C4×C13C2×#4111yescharacteristic
277C2×C4×C13H(5,𝔽2)11yescharacteristic
278C2×C4×C132−1+411yescharacteristic
279C2×Dic13—22no—
280C2×Dic13—41no—
281C2×Dic13—41no—
282C2×Dic13—41no—
283C2×Dic13—41no—
284C2×Dic13—41no—
285C2×Dic13—41no—
286C2×Dic13—41no—
287C2×Dic13—41no—
288D52—22no—
289D52—22no—
290D52—41no—
291D52—41no—
292C4×D13—84no—
293Dic26—84no—
294#1055—22no—
295#1055—22no—
296#1055—41no—
297#1055—41no—
298C22×SD16—131no—
299C2×C4⋊Q8—131no—
300C2×C22⋊Q8—261no—
301C2×C4×Q8C2×D1311yescharacteristic
302C2×C4×Q8C2×D1311yescharacteristic
303C2×C4×Q8C2×D1311yescharacteristic
304C2×C4×D4—131no—
305#475—264no—
306C4×SD16—264no—
307C2×#106—131no—
308C2×#106—131no—
309C2×#106—131no—
310C2×#106—131no—
311C2×#106—131no—
312C2×C4⋊C8—131no—
313C2×#103—131no—
314C2×#103—131no—
315C2×#103—131no—
316C2×#102—131no—
317C2×#102—131no—
318C2×#102—131no—
319C2×C4×C8—131no—
320#396—261no—
321#393—261no—
322#390—131no—
323#390—131no—
324#389—21no—
325#389—21no—
326#389—131no—
327#389—131no—
328#389C2×D1311yescharacteristic
329#389C2×D1311yescharacteristic
330#385—261no—
331#384—21no—
332#384C2×D1311yescharacteristic
333C4×C4⋊C4—21no—
334C4×C4⋊C4C2×D1311yescharacteristic
335#340—131no—
336#340—131no—
337#340—131no—
338#340—131no—
339#336—131no—
340#336—131no—
341#336—131no—
342#336—131no—
343C22×C13#52511yescharacteristic
344C22×C13D4211yescharacteristic
345C22×C13D4×Q811yescharacteristic
346C22×C13C22×SD1611yescharacteristic
347C22×C13#57511yescharacteristic
348C22×C13#57711yescharacteristic
349C22×C13#57711yescharacteristic
350C2×D13—42no—
351C2×D13—42no—
352C2×D13—42no—
353C2×D13—81no—
354C2×D13—81no—
355C2×D13—81no—
356C2×D13—81no—
357C4×C13—22no—
358C4×C13—24no—
359C4×C13—22no—
360C4×C13—22no—
361C4×C13—41no—
362C4×C13—42no—
363C4×C13—41no—
364C4×C13—41no—
365C4×C13—81no—
366C4×C13—81no—
367C4×C13C2×C22⋊Q811yescharacteristic
368C4×C13#53911yescharacteristic
369C4×C13#54711yescharacteristic
370C4×C13#55411yescharacteristic
371Dic13—42no—
372Dic13—81no—
373Dic13—81no—
374C22×Q8—261no—
375C22×Q8C22×D1311yescharacteristic
376C22×D4—131no—
377C23×C4—261no—
378C2×SD16—134no—
379C2×SD16—262no—
380C2×SD16—262no—
381C22×C8—131no—
382C22×C8—131no—
383C22×C8—131no—
384C22×C8—131no—
385C4⋊Q8—262no—
386C4⋊Q8—521no—
387C22⋊Q8—524no—
388C4×Q8—22no—
389C4×Q8—22no—
390C4×Q8—22no—
391C4×Q8—22no—
392C4×Q8—41no—
393C4×Q8—41no—
394C4×D4—262no—
395C4×D4—521no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—131no—
404C2×C4⋊C4—131no—
405C2×C4⋊C4—131no—
406C2×C4⋊C4—261no—
407C2×C4⋊C4—261no—
408C2×C4⋊C4—261no—
409C2×C4⋊C4C22⋊D1311yescharacteristic
410C2×C4⋊C4C22⋊D1311yescharacteristic
411C2×C4⋊C4C22×D1311yescharacteristic
412C2×C4⋊C4C22×D1311yescharacteristic
413C2×C4⋊C4C22×D1311yescharacteristic
414C2×C22⋊C4—261no—
415C2×C22⋊C4—261no—
416C2×C22⋊C4—261no—
417C2×C22⋊C4—261no—
418C2×C22⋊C4—261no—
419C2×C42—21no—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42C22×D1311yescharacteristic
423C2×C42C22×D1311yescharacteristic
424C2×C42C22×D1311yescharacteristic
425#106—262no—
426#106—262no—
427#106—262no—
428#106—262no—
429#106—521no—
430C4⋊C8—262no—
431#103—134no—
432#103—262no—
433#103—262no—
434#102—262no—
435#102—262no—
436#102—521no—
437C4×C8—262no—
438#95—21no—
439#95—21no—
440#95—261no—
441#95—261no—
442#95C22⋊D1311yescharacteristic
443#95C22⋊D1311yescharacteristic
444C2×C13#244911yescharacteristic
445C2×C13#310311yescharacteristic
446C2×C13#312511yescharacteristic
447C2×C13#327111yescharacteristic
448C2×C13D4×SD1611yescharacteristic
449C2×C13Q8×SD1611yescharacteristic
450C2×C13#338811yescharacteristic
451D13—82no—
452C24—261no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—522no—
457C2×Q8—522no—
458C2×Q8—1041no—
459C2×Q8#612712yes—
460C2×Q8D4×D1312yes—
461C2×Q8Q8×D1312yes—
462C2×D4—132no—
463C2×D4—261no—
464C2×D4—261no—
465C2×D4—261no—
466C2×D4—261no—
467C2×D4—261no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—261no—
473C22×C4—261no—
474C22×C4—261no—
475C22×C4—522no—
476C22×C4—522no—
477C22×C4—1041no—
478C22×C4C2×C22⋊D1311yescharacteristic
479C22×C4C2×C22⋊D1311yescharacteristic
480C22×C4C2×C22⋊D1311yescharacteristic
481C22×C4C2×C22⋊D1311yescharacteristic
482C22×C4C2×C22⋊D1311yescharacteristic
483C22×C4C2×C22⋊D1311yescharacteristic
484C22×C4C23×D1311yescharacteristic
485SD16—268no—
486C2×C8—132no—
487C2×C8—132no—
488C2×C8—261no—
489C2×C8—261no—
490C2×C8—261no—
491C2×C8—261no—
492C2×C8—261no—
493C2×C8—261no—
494C2×C8—261no—
495C2×C8—261no—
496C2×C8—261no—
497C2×C8—261no—
498C4⋊C4—24no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—41no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—132no—
512C4⋊C4—261no—
513C4⋊C4—261no—
514C4⋊C4—261no—
515C4⋊C4—261no—
516C4⋊C4—261no—
517C4⋊C4—522no—
518C4⋊C4—522no—
519C4⋊C4—522no—
520C4⋊C4#610614yescomplemented
521C4⋊C4#612712yes—
522C22⋊C4—522no—
523C22⋊C4—522no—
524C22⋊C4—522no—
525C22⋊C4—522no—
526C22⋊C4—1041no—
527C42—24no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—42no—
535C42—42no—
536C13#2637311yescharacteristic, complemented, Sylow
537C23—262no—
538C23—521no—
539C23—521no—
540C23—521no—
541C23—521no—
542C23—521no—
543C23—521no—
544C23C22×C22⋊D1311yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—1044no—
549Q8#8815014yescomplemented
550D4—262no—
551D4—262no—
552D4—521no—
553D4—521no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—262no—
582C2×C4—521no—
583C2×C4—521no—
584C2×C4—521no—
585C2×C4—521no—
586C2×C4—521no—
587C2×C4—521no—
588C2×C4—521no—
589C2×C4—521no—
590C2×C4—1044no—
591C2×C4C2×#610612yes—
592C2×C4C2×#610612yes—
593C2×C4C2×#610612yes—
594C2×C4C2×#612711yescharacteristic
595C2×C4C2×#612711yescharacteristic
596C2×C4C2×D4×D1311yescharacteristic
597C2×C4#8825111yescharacteristic
598C2×C4C2×Q8×D1311yescharacteristic
599C2×C4#8825411yescharacteristic
600C8—262no—
601C8—262no—
602C8—521no—
603C8—521no—
604C22—522no—
605C22—522no—
606C22—522no—
607C22—1041no—
608C22—1041no—
609C22—1041no—
610C22—1041no—
611C22#1171949611yescharacteristic
612C22D4×C22⋊D1311yescharacteristic
613C22C22×#610611yescharacteristic
614C22Q8×C22⋊D1311yescharacteristic
615C22#1171952711yescharacteristic
616C22#1171954111yescharacteristic
617C22#1171954211yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—24no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—42no—
626C4—81no—
627C4—81no—
628C4—522no—
629C4—1041no—
630C4—1041no—
631C4#1171921211yescharacteristic
632C4C2×#8815011yescharacteristic
633C4#1171921611yescharacteristic
634C4#1171922011yescharacteristic
635C2—1042no—
636C2#1311864511yescharacteristic
637C2#1312486711yescharacteristic
638C2#1312511311yescharacteristic
639C2#1312641711yescharacteristic
640C2D4×#610611yescharacteristic
641C2Q8×#610611yescharacteristic
642C2#1312777511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingQ8×C13×C4⋊C4C2
socleC23×C13C22×D4
prime radicalC22×D52C23
hypercenterQ8×C4⋊C4D13
hypocenterC13#26373
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#11718681C2semidirect
5. #13116313C2semidirect
7. #13116169C2semidirect
8. #13115561C2semidirect
9. #13115395C2semidirect
12. #13115104C2semidirect
13. #13115104C2semidirect
15. Q8×C13×C4⋊C4C2semidirect
91. C2×#6106Q8semidirect
178. C13×C4⋊C4SD16semidirect
219. Q8×C4⋊C4D13semidirect
242. Q8×C13C22⋊Q8semidirect
520. C4⋊C4#6106semidirect
536. C13#26373semidirect
549. Q8#88150semidirect

Maximal subgroups

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

subgroupindex
C2×#117186812
C2×C4×#61062
#131186862
#131163702
#131163132
#131162072
#131161692
#131155612
#131153952
#131151912
#131151042
#131150962
Q8×C13×C4⋊C42
#2637313