Finite Group Databasebeta

← #15870340

Subgroups — #15870340

Order 3328.

894 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×C8×Dic26C211yesmaximal, characteristic, complemented
2C2×M16×Dic13C211yesmaximal, characteristic
3C4×#88132C211yesmaximal, characteristic
4#13115770C211yesmaximal, characteristic, complemented
5#13115553C211yesmaximal, characteristic, complemented
6#13115403C211yesmaximal, characteristic, complemented
7#13115399C211yesmaximal, characteristic, complemented
8#13115021C211yesmaximal, characteristic
9#13115018C211yesmaximal, characteristic
10#13114814C211yesmaximal, characteristic
11#13114710C211yesmaximal, characteristic, complemented
12#13114708C211yesmaximal, characteristic, complemented
13#13114704C211yesmaximal, characteristic, complemented
14#13114617C211yesmaximal, characteristic
15C13×#1753C211yesmaximal, characteristic
16C22×C13×M16C2211yescharacteristic
17C22×C4×Dic13C2211yescharacteristic
18C22×#6099C2211yescharacteristic
19C2×#88132C411yescharacteristic
20C2×C4×Dic26C2211yescharacteristic
21C2×C13×C22⋊C8C2211yescharacteristic
22C2×C13×#97C2211yescharacteristic
23C2×C4×C8×C13C2211yescharacteristic
24C4×C13×C22⋊C4C2211yescharacteristic
25C4×C22⋊Dic13C2211yescharacteristic
26C2×C22⋊#1055C2211yescharacteristic
27M16×Dic13—24no—
28C2×#88055C2211yescharacteristic
29C2×#88055C2211yescharacteristic
30C2×#88055C2211yescharacteristic
31C2×#88054C2211yescharacteristic
32C2×#88054C2211yescharacteristic
33C2×#88054C2211yescharacteristic
34C2×#88054C2211yescharacteristic
35C2×C8×Dic13C2211yescharacteristic
36C2×C8×Dic13C2211yescharacteristic
37C2×C8×Dic13C2211yescharacteristic
38#11718611C411yescharacteristic
39#11718607C411yescharacteristic
40#11718603C411yescharacteristic
41C4×C4⋊Dic13C2211yescharacteristic
42C4×#6101C2211yescharacteristic
43C4×#6101C2211yescharacteristic
44C2×C4⋊#1055C2211yescharacteristic
45C2×C4⋊#1055C2211yescharacteristic
46C8×Dic26—24no—
47#11718316C411yescharacteristic
48#11718315C411yescharacteristic
49#11718308C411yescharacteristic
50#11718307C411yescharacteristic
51C13×#336C2211yescharacteristic
52C13×#336C2211yescharacteristic
53#11718142C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718142C2211yescharacteristic
58#11718142C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C23×C4×C13C2311yescharacteristic
62C23×Dic13C2×C411yescharacteristic
63C23×Dic13C2×C411yescharacteristic
64C22×Dic26C2×C411yescharacteristic
65C2×C13×M16—22no—
66C2×C13×M16—22no—
67C2×C13×M16Q814yescomplemented
68C22×C8×C13C2311yescharacteristic
69C22×C8×C13C2311yescharacteristic
70C22×C8×C13C2311yescharacteristic
71C22×C8×C13C2311yescharacteristic
72C2×C13×C22⋊C4C2×C411yescharacteristic
73C2×C13×C22⋊C4C2×C411yescharacteristic
74C2×C42×C13C2311yescharacteristic
75C2×C42×C13C2311yescharacteristic
76C2×C22⋊Dic13C2×C411yescharacteristic
77C2×C22⋊Dic13C2×C411yescharacteristic
78C2×C4⋊Dic13C2×C411yescharacteristic
79C2×C4⋊Dic13C2×C411yescharacteristic
80C2×C4⋊Dic13C2×C411yescharacteristic
81C2×#6101C2×C411yescharacteristic
82C2×#6101C2×C411yescharacteristic
83C2×#6101C2×C411yescharacteristic
84C2×#6101C2×C411yescharacteristic
85C2×#6101C2×C411yescharacteristic
86C2×#6101C2×C411yescharacteristic
87C2×C4×Dic13—22no—
88C2×C4×Dic13—22no—
89C2×C4×Dic13C2×C411yescharacteristic
90C2×C4×Dic13C2×C411yescharacteristic
91C2×C4×Dic13C2311yescharacteristic
92C2×C4×Dic13C2311yescharacteristic
93C2×C4×Dic13C2311yescharacteristic
94C2×C4×Dic13C2311yescharacteristic
95C2×#6099—22no—
96C2×#6099—22no—
97C2×#6099—22no—
98C22×#1055C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C22×#1055C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102#88132—24no—
103C4×Dic26—22no—
104C4×Dic26—22no—
105C13×C22⋊C8—22no—
106C13×#97—22no—
107C4×C8×C13—22no—
108C13×#95C2×C411yescharacteristic
109C13×#95C2×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
118C22⋊#1055—22no—
119#88055—22no—
120#88055—22no—
121#88055—22no—
122#88054—22no—
123#88054—22no—
124#88054—22no—
125#88054—22no—
126C8×Dic13—22no—
127C8×Dic13—22no—
128C8×Dic13D414yes—
129C4⋊#1055—22no—
130C4⋊#1055—22no—
131#21596—131nomaximal, Sylow
132C24×C13C22×C411yescharacteristic
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13C22×C411yescharacteristic
138C22×C4×C13C22×C411yescharacteristic
139C22×C4×C13C22×C411yescharacteristic
140C22×C4×C13C22×C411yescharacteristic
141C22×C4×C13C22×C411yescharacteristic
142C22×C4×C13C2×Q812yes—
143C22×C4×C13#4112yes—
144C22×C4×C13C2411yescharacteristic
145C22×Dic13—22no—
146C22×Dic13—22no—
147C22×Dic13—22no—
148C22×Dic13—22no—
149C22×Dic13—41no—
150C22×Dic13—41no—
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×Dic26M1614yescomplemented
162C13×M16—28no—
163C2×C8×C13—21no—
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×C13—21no—
171C2×C8×C13C2×D412yes—
172C2×C8×C13C2×Q812yes—
173C2×C8×C13C2×Q812yes—
174C2×C8×C13#4112yes—
175C13×C22⋊C4—22no—
176C13×C22⋊C4—41no—
177C42×C13—21no—
178C42×C13—21no—
179C42×C13—21no—
180C42×C13—21no—
181C22⋊Dic13—22no—
182C22⋊Dic13—41no—
183C4⋊Dic13—22no—
184C4⋊Dic13—22no—
185C4⋊Dic13—41no—
186#6101—22no—
187#6101—22no—
188#6101—22no—
189#6101—22no—
190#6101—22no—
191#6101—41no—
192C4×Dic13—21no—
193C4×Dic13—21no—
194C4×Dic13—21no—
195C4×Dic13—21no—
196C4×Dic13—21no—
197C4×Dic13—21no—
198C4×Dic13—22no—
199C4×Dic13—22no—
200C4×Dic13C2×D412yes—
201C4×Dic13#4112yes—
202#6099—28no—
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×#1055C2×D412yes—
214C2×#1055#4112yes—
215C2×C8×Q8—131no—
216C2×C4×M16—131no—
217C4×C22⋊Q8—131no—
218#1984—131no—
219#1939—131no—
220#1917—131no—
221#1917—131no—
222#1851—131no—
223#1851—131no—
224#1797—131no—
225#1771—131no—
226#1771—131no—
227#1770—131no—
228#1753—131no—
229#1753D1311yescharacteristic
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—22no—
243C2×C4×C13—22no—
244C2×C4×C13—22no—
245C2×C4×C13—22no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—21no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—21no—
254C2×C4×C13—21no—
255C2×C4×C13C4×Q812yes—
256C2×C4×C13C2×M1612yes—
257C2×C4×C13C22×D411yescharacteristic
258C2×C4×C13C22×Q811yescharacteristic
259C2×C4×C13C2×#4111yescharacteristic
260C2×C4×C13C2×#4111yescharacteristic
261C2×C4×C13C2×#4111yescharacteristic
262C2×C4×C13C2×#4111yescharacteristic
263C2×Dic13—21no—
264C2×Dic13—21no—
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—44no—
283C2×Dic13—44no—
284C2×Dic13C4×D412yes—
285C2×Dic13C4×D412yes—
286C2×Dic13C2×M1612yes—
287C2×Dic13C2×M1612yes—
288C2×Dic13#13112yes—
289Dic26—28no—
290C8×C13—22no—
291C8×C13—41no—
292C8×C13—41no—
293C8×C13C22⋊Q814yescomplemented
294#1055—22no—
295#1055—22no—
296#1055—41no—
297#1055—41no—
298C22×M16—131no—
299C22×M16C2×D1311yescharacteristic
300C2×C22⋊Q8—131no—
301C2×C4×Q8—131no—
302C22×C42—131no—
303C8×Q8—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⋊C8C2×D1311yescharacteristic
312C4×M16—264no—
313C2×#97—131no—
314C2×#97—131no—
315C2×#97—131no—
316C2×#97C2×D1311yescharacteristic
317C2×C4×C8—131no—
318C2×C4×C8—131no—
319C2×C4×C8—131no—
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⋊C4C2×D1311yescharacteristic
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#336C2×D1311yescharacteristic
342#336C2×D1311yescharacteristic
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—41no—
360C4×C13—41no—
361C4×C13—41no—
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#43514yes—
374C22×Q8—131no—
375C23×C4—131no—
376C23×C4—131no—
377C23×C4C22×D1311yescharacteristic
378C2×M16—22no—
379C2×M16—22no—
380C2×M16—262no—
381C2×M16—262no—
382C2×M16—262no—
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⋊C4C4×D1311yescharacteristic
407C2×C22⋊C4C4×D1311yescharacteristic
408C2×C42—131no—
409C2×C42—131no—
410C2×C42—131no—
411C2×C42—131no—
412C2×C42—131no—
413C2×C42—131no—
414C2×C42—262no—
415C2×C42—262no—
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—22no—
425C22⋊C8—262no—
426#97—22no—
427#97—262no—
428#97—262no—
429#97—262no—
430C4×C8—22no—
431C4×C8—134no—
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#95C4×D1311yescharacteristic
443#95C4×D1311yescharacteristic
444C2×C13—42no—
445C2×C13C4×C22⋊Q811yescharacteristic
446C2×C13C2×#43511yescharacteristic
447C2×C13#293011yescharacteristic
448C2×C13#296211yescharacteristic
449C2×C13Q8×M1611yescharacteristic
450C2×C13#296911yescharacteristic
451C2×C13#297111yescharacteristic
452C24C2×C4×D1311yescharacteristic
453C2×Q8—134no—
454C2×Q8—262no—
455C2×Q8—262no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—131no—
461C22×C4—131no—
462C22×C4—131no—
463C22×C4—131no—
464C22×C4—131no—
465C22×C4—131no—
466C22×C4—131no—
467C22×C4—131no—
468C22×C4—262no—
469C22×C4—262no—
470C22×C4—262no—
471C22×C4—262no—
472C22×C4—521no—
473C22×C4—521no—
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—268no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—132no—
493C2×C8—132no—
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×C8—261no—
503C2×C8—261no—
504C2×C8C2×Dic2612yes—
505C2×C8C2×Dic2612yes—
506C2×C8D4×D1312yes—
507C2×C8#612912yes—
508C4⋊C4—262no—
509C4⋊C4—262no—
510C4⋊C4—262no—
511C4⋊C4—262no—
512C4⋊C4—262no—
513C4⋊C4—262no—
514C4⋊C4—262no—
515C4⋊C4—521no—
516C4⋊C4—521no—
517C22⋊C4—22no—
518C22⋊C4—41no—
519C22⋊C4—262no—
520C22⋊C4—521no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—132no—
526C42—132no—
527C42—262no—
528C42—262no—
529C42—261no—
530C42—261no—
531C42—261no—
532C42—261no—
533C42—261no—
534C42—261no—
535C13#2159611yescharacteristic, complemented, Sylow
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23C4×Dic2612yes—
543C23C22×C4×D1311yescharacteristic
544Q8—268no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—132no—
563C2×C4—132no—
564C2×C4—132no—
565C2×C4—132no—
566C2×C4—132no—
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×C4—261no—
586C2×C4—524no—
587C2×C4—524no—
588C2×C4C4×Dic2612yes—
589C2×C4M16×D1312yes—
590C2×C4C22×Dic2611yescharacteristic
591C2×C4C2×#612711yescharacteristic
592C2×C4C2×D4×D1311yescharacteristic
593C2×C4C2×#612911yescharacteristic
594C2×C4#41×D1311yescharacteristic
595C2×C4#41×D1311yescharacteristic
596C8—22no—
597C8—41no—
598C8—41no—
599C8—262no—
600C8—262no—
601C8—521no—
602C8—521no—
603C8#8813214yescomplemented
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22C2×C4×Dic2611yescharacteristic
612C22C4×#612911yescharacteristic
613C22C4×D4×D1311yescharacteristic
614C22C2×M16×D1311yescharacteristic
615C22#1171945011yescharacteristic
616C22D13×#13111yescharacteristic
617C22#1171945511yescharacteristic
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—134no—
624C4—262no—
625C4—262no—
626C4—521no—
627C4—521no—
628C4—521no—
629C4—521no—
630C4—521no—
631C4C2×#8813211yescharacteristic
632C4#1171922211yescharacteristic
633C4#1171924311yescharacteristic
634C4#1171925211yescharacteristic
635C2—42no—
636C2C4×#8813211yescharacteristic
637C2D13×#43511yescharacteristic
638C2#1312353411yescharacteristic
639C2#1312378011yescharacteristic
640C2M16×Dic2611yescharacteristic
641C2#1312383511yescharacteristic
642C2#1312385211yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D13
commutator, socleC23×C13C23×C4
FittingC13×#1753C2
prime radicalC24×C13C22×C4
cosocleC22×C4×C13C24
hypercenter#1753D13
hypocenterC13#21596
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
1. C2×C8×Dic26C2semidirect
4. #13115770C2semidirect
5. #13115553C2semidirect
6. #13115403C2semidirect
7. #13115399C2semidirect
11. #13114710C2semidirect
12. #13114708C2semidirect
13. #13114704C2semidirect
67. C2×C13×M16Q8semidirect
161. C2×Dic26M16semidirect
293. C8×C13C22⋊Q8semidirect
383. C2×M16Dic26semidirect
535. C13#21596semidirect
603. C8#88132semidirect

Maximal subgroups

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

subgroupindex
C2×C8×Dic262
C2×M16×Dic132
C4×#881322
#131157702
#131155532
#131154032
#131153992
#131150212
#131150182
#131148142
#131147102
#131147082
#131147042
#131146172
C13×#17532
#2159613