Finite Group Databasebeta

← #15870392

Subgroups — #15870392

Order 3328.

886 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×#11718362C211yesmaximal, characteristic, complemented
2C4×#88178C211yesmaximal, characteristic
3#13115776C211yesmaximal, characteristic, complemented
4#13115559C211yesmaximal, characteristic, complemented
5#13115393C211yesmaximal, characteristic, complemented
6#13115393C211yesmaximal, characteristic, complemented
7#13115019C211yesmaximal, characteristic
8#13115019C211yesmaximal, characteristic
9#13114816C211yesmaximal, characteristic
10#13114710C211yesmaximal, characteristic, complemented
11#13114708C211yesmaximal, characteristic, complemented
12C8×#6101C211yesmaximal, characteristic, complemented
13#13114617C211yesmaximal, characteristic
14#13114617C211yesmaximal, characteristic
15C2×C4×C13×M16C211yesmaximal, characteristic
16C22×C13×M16C2211yescharacteristic
17C22×C13×M16C2211yescharacteristic
18C22×C42×C13C2211yescharacteristic
19C2×#88178C411yescharacteristic
20C2×C4×Dic26C2211yescharacteristic
21C4×C13×M16—24no—
22C2×C13×#97C2211yescharacteristic
23C2×C13×#97C2211yescharacteristic
24C2×C4×C8×C13C2211yescharacteristic
25C2×C4×C8×C13C2211yescharacteristic
26#11718873C411yescharacteristic
27#11718871C411yescharacteristic
28#11718871C411yescharacteristic
29C4×C22⋊Dic13C2211yescharacteristic
30C4×C22⋊Dic13C2211yescharacteristic
31C2×C22⋊#1055C2211yescharacteristic
32C2×C22⋊#1055C2211yescharacteristic
33C2×#88055C2211yescharacteristic
34C2×#88055C2211yescharacteristic
35C2×#88054C2211yescharacteristic
36C2×#88054C2211yescharacteristic
37C2×#88054C2211yescharacteristic
38C2×#88054C2211yescharacteristic
39C2×C8×Dic13C2211yescharacteristic
40C2×C8×Dic13C2211yescharacteristic
41C4×C4⋊Dic13C2211yescharacteristic
42#11718594C411yescharacteristic
43#11718594C411yescharacteristic
44C4×#6101C2211yescharacteristic
45C4×#6101C2211yescharacteristic
46#11718590C411yescharacteristic
47#11718589C411yescharacteristic
48C2×C4⋊#1055C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50#11718362—24no—
51#11718142C2211yescharacteristic
52#11718142C2211yescharacteristic
53#11718142C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718142C2211yescharacteristic
58#11718142C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C23×C4×C13C2×C411yescharacteristic
62C23×C4×C13C2×C411yescharacteristic
63C23×C4×C13C2311yescharacteristic
64C22×Dic26C2×C411yescharacteristic
65C2×C13×M16—22no—
66C2×C13×M16—22no—
67C2×C13×M16—22no—
68C2×C13×M16—22no—
69C2×C13×M16—22no—
70C2×C13×M16Q814yescomplemented
71C22×C8×C13C2311yescharacteristic
72C22×C8×C13C2311yescharacteristic
73C22×C8×C13C2311yescharacteristic
74C22×C8×C13C2311yescharacteristic
75C2×C42×C13—22no—
76C2×C42×C13—22no—
77C2×C42×C13C2×C411yescharacteristic
78C2×C42×C13C2×C411yescharacteristic
79C2×C42×C13C2311yescharacteristic
80C2×C42×C13C2311yescharacteristic
81C2×C22⋊Dic13C2×C411yescharacteristic
82C2×C22⋊Dic13C2×C411yescharacteristic
83C2×C22⋊Dic13C2×C411yescharacteristic
84C2×C22⋊Dic13C2×C411yescharacteristic
85C2×C4⋊Dic13C2×C411yescharacteristic
86C2×C4⋊Dic13C2×C411yescharacteristic
87C2×C4⋊Dic13C2×C411yescharacteristic
88#88178—24no—
89C2×#6101C2×C411yescharacteristic
90C2×#6101C2×C411yescharacteristic
91C2×#6101C2×C411yescharacteristic
92C2×#6101C2×C411yescharacteristic
93C2×#6101C2×C411yescharacteristic
94C2×#6101C2×C411yescharacteristic
95C2×C4×Dic13C2311yescharacteristic
96C2×C4×Dic13C2311yescharacteristic
97C2×C4×Dic13C2311yescharacteristic
98C2×C4×Dic13C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C22×#1055C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102C22×#1055C2311yescharacteristic
103C4×Dic26—22no—
104C4×Dic26—22no—
105C13×#97—22no—
106C13×#97D414yes—
107C4×C8×C13—22no—
108C4×C8×C13—22no—
109#88071C2×C411yescharacteristic
110#88071C2×C411yescharacteristic
111#88071C2×C411yescharacteristic
112#88071C2×C411yescharacteristic
113#88071C2×C411yescharacteristic
114#88071C2×C411yescharacteristic
115#88071C2×C411yescharacteristic
116#88071C2×C411yescharacteristic
117#88071C2×C411yescharacteristic
118#88071C2×C411yescharacteristic
119C22⋊#1055—22no—
120C22⋊#1055—22no—
121#88055—22no—
122#88055—22no—
123#88054—22no—
124#88054—22no—
125#88054—22no—
126#88054—22no—
127C8×Dic13—22no—
128C8×Dic13—22no—
129C4⋊#1055—22no—
130C4⋊#1055—22no—
131#21600—131nomaximal, Sylow
132C24×C13C22×C411yescharacteristic
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—22no—
138C22×C4×C13—22no—
139C22×C4×C13—22no—
140C22×C4×C13—22no—
141C22×C4×C13—41no—
142C22×C4×C13—41no—
143C22×C4×C13C22×C411yescharacteristic
144C22×C4×C13C22×C411yescharacteristic
145C22×C4×C13C22×C411yescharacteristic
146C22×C4×C13C22×C411yescharacteristic
147C22×C4×C13C22×C411yescharacteristic
148C22×C4×C13C2×Q812yes—
149C22×C4×C13#4112yes—
150C22×C4×C13C2411yescharacteristic
151C22×Dic13C22×C411yescharacteristic
152C22×Dic13C22×C411yescharacteristic
153C22×Dic13C22×C411yescharacteristic
154C22×Dic13C22×C411yescharacteristic
155C22×Dic13C22×C411yescharacteristic
156C22×Dic13C22×C411yescharacteristic
157C22×Dic13C22×C411yescharacteristic
158C22×Dic13C22×C411yescharacteristic
159C2×Dic26—22no—
160C2×Dic26—22no—
161C2×Dic26—41no—
162C13×M16—28no—
163C13×M16—28no—
164C2×C8×C13—21no—
165C2×C8×C13—21no—
166C2×C8×C13—21no—
167C2×C8×C13—21no—
168C2×C8×C13—21no—
169C2×C8×C13—21no—
170C2×C8×C13C2×D412yes—
171C2×C8×C13C2×D412yes—
172C2×C8×C13C2×Q812yes—
173C2×C8×C13C2×Q812yes—
174C2×C8×C13#4112yes—
175C2×C8×C13#4112yes—
176C42×C13—22no—
177C42×C13—22no—
178C42×C13—21no—
179C42×C13—21no—
180C42×C13C2×D412yes—
181C42×C13#4112yes—
182C22⋊Dic13—22no—
183C22⋊Dic13—22no—
184C22⋊Dic13—41no—
185C22⋊Dic13—41no—
186C4⋊Dic13—22no—
187C4⋊Dic13—22no—
188C4⋊Dic13—41no—
189#6101—22no—
190#6101—22no—
191#6101—22no—
192#6101—22no—
193#6101—22no—
194#6101M1614yescomplemented
195C4×Dic13—21no—
196C4×Dic13—21no—
197C4×Dic13—21no—
198C4×Dic13—21no—
199C4×Dic13—21no—
200C4×Dic13—21no—
201C4×Dic13—21no—
202C4×Dic13—21no—
203C2×#1055—21no—
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×#1055—21no—
215C2×#446—131no—
216C2×C4×M16D1311yescharacteristic
217C4×C22⋊Q8—131no—
218#1985—131no—
219#1940—131no—
220#1916—131no—
221#1916—131no—
222#1851—131no—
223#1851—131no—
224#1797—131no—
225#1771—131no—
226#1771—131no—
227C8×C4⋊C4—131no—
228#1753—131no—
229#1753—131no—
230C23×C13—21no—
231C23×C13—21no—
232C23×C13—21no—
233C23×C13—21no—
234C23×C13—21no—
235C23×C13—21no—
236C23×C13C4×Q812yes—
237C23×C13C23×C411yescharacteristic
238C2×C4×C13—21no—
239C2×C4×C13—21no—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—21no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13—22no—
249C2×C4×C13—22no—
250C2×C4×C13—22no—
251C2×C4×C13—22no—
252C2×C4×C13—44no—
253C2×C4×C13—44no—
254C2×C4×C13C4×D412yes—
255C2×C4×C13C4×D412yes—
256C2×C4×C13C4×Q812yes—
257C2×C4×C13C2×M1612yes—
258C2×C4×C13#13112yes—
259C2×C4×C13C22×D411yescharacteristic
260C2×C4×C13C22×Q811yescharacteristic
261C2×C4×C13C2×#4111yescharacteristic
262C2×C4×C13C2×#4111yescharacteristic
263C2×C4×C13C2×#4111yescharacteristic
264C2×C4×C13C2×#4111yescharacteristic
265C2×Dic13—21no—
266C2×Dic13—21no—
267C2×Dic13—21no—
268C2×Dic13—21no—
269C2×Dic13—21no—
270C2×Dic13—21no—
271C2×Dic13—21no—
272C2×Dic13—21no—
273C2×Dic13—21no—
274C2×Dic13—21no—
275C2×Dic13—21no—
276C2×Dic13—21no—
277C2×Dic13—21no—
278C2×Dic13—21no—
279C2×Dic13—21no—
280C2×Dic13—21no—
281C2×Dic13—21no—
282C2×Dic13—21no—
283C2×Dic13—21no—
284C2×Dic13—21no—
285C2×Dic13—21no—
286C2×Dic13—21no—
287C2×Dic13C2×M1612yes—
288C2×Dic13C2×M1612yes—
289Dic26—44no—
290C8×C13—22no—
291C8×C13—22no—
292C8×C13—22no—
293C8×C13C22⋊Q814yescomplemented
294#1055—41no—
295#1055—41no—
296#1055—41no—
297#1055—41no—
298C22×M16C2×D1311yescharacteristic
299C22×M16C2×D1311yescharacteristic
300C2×C22⋊Q8—131no—
301C2×C4×Q8—131no—
302C22×C42C2×D1311yescharacteristic
303#446—264no—
304C2×C4⋊C8—131no—
305C2×C4⋊C8—131no—
306C2×C4⋊C8—131no—
307C2×C4⋊C8—131no—
308C2×C4⋊C8—131no—
309C2×C4⋊C8—131no—
310C2×C22⋊C8—131no—
311C2×C22⋊C8—131no—
312C4×M16—24no—
313C2×#97—131no—
314C2×#97—131no—
315C2×#97C2×D1311yescharacteristic
316C2×#97C2×D1311yescharacteristic
317C2×C4×C8—131no—
318C2×C4×C8—131no—
319C2×C4×C8C2×D1311yescharacteristic
320C2×C4×C8C2×D1311yescharacteristic
321#391—131no—
322#389—131no—
323#387—131no—
324#387—131no—
325#385—131no—
326#385—131no—
327#380—131no—
328C4×C4⋊C4—131no—
329C4×C4⋊C4—131no—
330C4×C4⋊C4—131no—
331C4×C22⋊C4—131no—
332C4×C22⋊C4—131no—
333#336—131no—
334#336—131no—
335#336—131no—
336#336—131no—
337#336—131no—
338#336—131no—
339#336—131no—
340#336—131no—
341#336—131no—
342#336—131no—
343C22×C13—22no—
344C22×C13—22no—
345C22×C13—22no—
346C22×C13—41no—
347C22×C13—41no—
348C22×C13—41no—
349C22×C13—41no—
350C22×C13C2×C4×D411yescharacteristic
351C22×C13C2×C4×Q811yescharacteristic
352C22×C13C4×#4111yescharacteristic
353C22×C13C22×M1611yescharacteristic
354C22×C13C2×#13111yescharacteristic
355C22×C13#56811yescharacteristic
356C22×C13#56811yescharacteristic
357C4×C13—22no—
358C4×C13—22no—
359C4×C13—22no—
360C4×C13—22no—
361C4×C13—22no—
362C4×C13C2×C22⋊Q811yescharacteristic
363C4×C13#52511yescharacteristic
364C4×C13#52911yescharacteristic
365C4×C13#53311yescharacteristic
366Dic13—22no—
367Dic13—22no—
368Dic13—41no—
369Dic13—41no—
370Dic13—41no—
371Dic13—41no—
372Dic13—41no—
373Dic13—41no—
374C22×Q8—131no—
375C23×C4C4×D1311yescharacteristic
376C23×C4C4×D1311yescharacteristic
377C23×C4C22×D1311yescharacteristic
378C2×M16—22no—
379C2×M16—22no—
380C2×M16—22no—
381C2×M16—22no—
382C2×M16—22no—
383C2×M16Dic2614yescomplemented
384C22×C8—131no—
385C22×C8—131no—
386C22×C8—131no—
387C22×C8—131no—
388C22×C8C22×D1311yescharacteristic
389C22×C8C22×D1311yescharacteristic
390C22×C8C22×D1311yescharacteristic
391C22×C8C22×D1311yescharacteristic
392C22⋊Q8—264no—
393C4×Q8—262no—
394C4×Q8—262no—
395C2×C4⋊C4—131no—
396C2×C4⋊C4—131no—
397C2×C4⋊C4—131no—
398C2×C4⋊C4—131no—
399C2×C4⋊C4—131no—
400C2×C4⋊C4—131no—
401C2×C4⋊C4—131no—
402C2×C4⋊C4—131no—
403C2×C4⋊C4—131no—
404C2×C22⋊C4—131no—
405C2×C22⋊C4—131no—
406C2×C22⋊C4—131no—
407C2×C22⋊C4—131no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42—131no—
411C2×C42—131no—
412C2×C42—131no—
413C2×C42—131no—
414C2×C42C4×D1311yescharacteristic
415C2×C42C4×D1311yescharacteristic
416C2×C42C22×D1311yescharacteristic
417C2×C42C22×D1311yescharacteristic
418C4⋊C8—262no—
419C4⋊C8—262no—
420C4⋊C8—262no—
421C4⋊C8—262no—
422C4⋊C8—262no—
423C4⋊C8—262no—
424C22⋊C8—262no—
425C22⋊C8—262no—
426#97—22no—
427#97—262no—
428#97—262no—
429#97C22⋊D1314yes—
430C4×C8—22no—
431C4×C8—22no—
432C4×C8—262no—
433C4×C8—262no—
434#95—131no—
435#95—131no—
436#95—131no—
437#95—131no—
438#95—131no—
439#95—131no—
440#95—131no—
441#95—131no—
442#95—131no—
443#95—131no—
444C2×C13—42no—
445C2×C13C4×C22⋊Q811yescharacteristic
446C2×C13#293311yescharacteristic
447C2×C13D4×M1611yescharacteristic
448C2×C13#296211yescharacteristic
449C2×C13Q8×M1611yescharacteristic
450C2×C13#296511yescharacteristic
451C2×C13#296711yescharacteristic
452C24C2×C4×D1311yescharacteristic
453C2×Q8—262no—
454C2×Q8—262no—
455C2×Q8—521no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—131no—
467C22×C4—131no—
468C22×C4—131no—
469C22×C4—131no—
470C22×C4—131no—
471C22×C4—131no—
472C22×C4—131no—
473C22×C4—131no—
474C22×C4C2×Dic2612yes—
475C22×C4C2×C4×D1311yescharacteristic
476C22×C4C2×C4×D1311yescharacteristic
477C22×C4C2×C4×D1311yescharacteristic
478C22×C4C2×C4×D1311yescharacteristic
479C22×C4C2×C4×D1311yescharacteristic
480C22×C4#612712yes—
481C22×C4C23×D1311yescharacteristic
482M16—28no—
483M16—28no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—261no—
491C2×C8—261no—
492C2×C8—261no—
493C2×C8—261no—
494C2×C8—261no—
495C2×C8—261no—
496C2×C8—261no—
497C2×C8—261no—
498C2×C8—261no—
499C2×C8—261no—
500C2×C8—261no—
501C2×C8—261no—
502C2×C8C2×Dic2612yes—
503C2×C8C2×Dic2612yes—
504C2×C8#612712yes—
505C2×C8#612712yes—
506C2×C8C2×C22⋊D1312yes—
507C2×C8C2×C22⋊D1312yes—
508C4⋊C4—134no—
509C4⋊C4—262no—
510C4⋊C4—262no—
511C4⋊C4—262no—
512C4⋊C4—262no—
513C4⋊C4—262no—
514C4⋊C4—262no—
515C4⋊C4—262no—
516C4⋊C4—521no—
517C22⋊C4—262no—
518C22⋊C4—262no—
519C22⋊C4—521no—
520C22⋊C4—521no—
521C42—22no—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—261no—
526C42—261no—
527C42—261no—
528C42—261no—
529C42—261no—
530C42—261no—
531C42—261no—
532C42—261no—
533C42#612712yes—
534C42C2×C22⋊D1312yes—
535C13#2160011yescharacteristic, complemented, Sylow
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23C4×Dic2612yes—
543C23C22×C4×D1311yescharacteristic
544Q8—524no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—132no—
562C2×C4—132no—
563C2×C4—261no—
564C2×C4—261no—
565C2×C4—261no—
566C2×C4—261no—
567C2×C4—261no—
568C2×C4—261no—
569C2×C4—261no—
570C2×C4—261no—
571C2×C4—261no—
572C2×C4—261no—
573C2×C4—261no—
574C2×C4—261no—
575C2×C4—261no—
576C2×C4—261no—
577C2×C4—261no—
578C2×C4—261no—
579C2×C4—261no—
580C2×C4—261no—
581C2×C4—261no—
582C2×C4—261no—
583C2×C4—261no—
584C2×C4—261no—
585C2×C4C4×Dic2612yes—
586C2×C4M16×D1312yes—
587C2×C4#8816112yes—
588C2×C4C4×C22⋊D1312yes—
589C2×C4C4×C22⋊D1312yes—
590C2×C4C22×Dic2611yescharacteristic
591C2×C4C2×#612711yescharacteristic
592C2×C4C2×#612711yescharacteristic
593C2×C4C2×#612711yescharacteristic
594C2×C4C2×#612711yescharacteristic
595C2×C4C22×C22⋊D1311yescharacteristic
596C8—22no—
597C8—22no—
598C8—22no—
599C8—521no—
600C8—521no—
601C8—521no—
602C8—521no—
603C8#8817814yescomplemented
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22C2×C4×Dic2611yescharacteristic
612C22C4×#612711yescharacteristic
613C22C2×M16×D1311yescharacteristic
614C22C2×#8816111yescharacteristic
615C22#1171945011yescharacteristic
616C22#1171945011yescharacteristic
617C22C2×C4×C22⋊D1311yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—262no—
624C4—262no—
625C4—521no—
626C4—521no—
627C4—521no—
628C4—521no—
629C4—521no—
630C4—521no—
631C4#1171917511yescharacteristic
632C4#1171918411yescharacteristic
633C4C2×#8817811yescharacteristic
634C4#1171949611yescharacteristic
635C2—42no—
636C2C4×#8817811yescharacteristic
637C2#1312356111yescharacteristic
638C2M16×C22⋊D1311yescharacteristic
639C2#1312378311yescharacteristic
640C2M16×Dic2611yescharacteristic
641C2#1312379811yescharacteristic
642C2#1312382111yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D13
commutator, socleC23×C13C23×C4
FittingC2×C4×C13×M16C2
prime radicalC24×C13C22×C4
cosocleC22×C4×C13C24
hypercenterC2×C4×M16D13
hypocenterC13#21600
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#11718362C2semidirect
3. #13115776C2semidirect
4. #13115559C2semidirect
5. #13115393C2semidirect
6. #13115393C2semidirect
10. #13114710C2semidirect
11. #13114708C2semidirect
12. C8×#6101C2semidirect
70. C2×C13×M16Q8semidirect
194. #6101M16semidirect
293. C8×C13C22⋊Q8semidirect
383. C2×M16Dic26semidirect
535. C13#21600semidirect
603. C8#88178semidirect

Maximal subgroups

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

subgroupindex
C2×#117183622
C4×#881782
#131157762
#131155592
#131153932
#131150192
#131148162
#131147102
#131147082
C8×#61012
#131146172
C2×C4×C13×M162
#2160013