Finite Group Databasebeta

← #15898331

Subgroups — #15898331

Order 3328.

1406 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718751C211yesmaximal, characteristic, complemented
2C2×#11718889C212yesmaximal, complemented
3C2×C4×#6107C211yesmaximal, characteristic, complemented
4#13119758C211yesmaximal, characteristic
5#13116036C212yesmaximal, complemented
6#13115844C212yesmaximal, complemented
7#13115639C211yesmaximal, characteristic, complemented
8#13115631C211yesmaximal, characteristic
9#13114841C211yesmaximal, characteristic, complemented
10#13114771C212yesmaximal, complemented
11C2×D4×Q8×C13C211yesmaximal, characteristic
12C23×Q8×C13C2212yes—
13D4×Q8×C13—24no—
14D4×Q8×C13—24no—
15C2×C13×C4⋊Q8—21no—
16C2×C13×C4⋊Q8C2211yescharacteristic
17C2×C13×C22⋊Q8—22no—
18C2×C13×C22⋊Q8C2212yes—
19C2×C4×Q8×C13C2211yescharacteristic
20C2×C4×D4×C13—21no—
21C2×C4×D4×C13C2211yescharacteristic
22C22×#6107C2212yes—
23C22×#6107C2211yescharacteristic
24C2×#88178C2212yes—
25C2×C4×Dic26C2211yescharacteristic
26#11718889—28no—
27C2×#88075C2211yescharacteristic
28C2×#88075C2212yes—
29C2×#88075C2212yes—
30C2×#88075C2212yes—
31C2×#88075C2212yes—
32#11718882—21no—
33#11718873C2212yes—
34C2×C22⋊#1055C2212yes—
35C2×C22⋊#1055C2212yes—
36#11718751—24no—
37C4×#6107—24no—
38#11718634—21no—
39C2×#88050C2211yescharacteristic
40C2×#88047C2211yescharacteristic
41#11718609—22no—
42#11718596C2211yescharacteristic
43C2×C4⋊#1055C2211yescharacteristic
44C2×C4×#1055C2211yescharacteristic
45#11718121C2212yes—
46C22×Q8×C13—24no—
47C22×Q8×C13—24no—
48C22×Q8×C13—22no—
49C22×Q8×C13—24no—
50C22×Q8×C13—24no—
51C22×Q8×C13—42no—
52C22×Q8×C13—42no—
53C22×Q8×C13C2312yes—
54C22×Q8×C13C2311yescharacteristic
55C22×D4×C13D411yescharacteristic
56C23×C4×C13—22no—
57C23×C4×C13C2312yes—
58C22×Dic26C2311yescharacteristic
59C13×C4⋊Q8—28no—
60C13×C4⋊Q8—22no—
61C13×C4⋊Q8D414yes—
62C13×C22⋊Q8—216no—
63C13×C22⋊Q8—24no—
64C13×C22⋊Q8—24no—
65C4×Q8×C13—22no—
66C4×Q8×C13D414yes—
67C4×D4×C13—28no—
68C4×D4×C13—22no—
69C4×D4×C13—22no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4—22no—
74C2×C13×C4⋊C4D411yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C22⋊C4—22no—
77C2×C13×C22⋊C4D412yes—
78C2×C42×C13—21no—
79C2×C42×C13C2311yescharacteristic
80C2×C22⋊Dic13—22no—
81C2×C22⋊Dic13—22no—
82C2×#6107—22no—
83C2×#6107—22no—
84C2×#6107—24no—
85C2×#6107—24no—
86C2×#6107—42no—
87C2×#6107D414yescomplemented
88C2×C4⋊Dic13—21no—
89C2×C4⋊Dic13—21no—
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2312yes—
92#88178—24no—
93#88178—42no—
94C2×#6101—21no—
95C2×#6101—22no—
96C2×C4×Dic13—21no—
97C22×#1055C2312yes—
98C22×#1055C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C4×Dic26—22no—
101C4×Dic26—22no—
102#88075—22no—
103#88075—24no—
104#88075—24no—
105#88075—24no—
106#88075—42no—
107#88071—22no—
108C22⋊#1055—24no—
109C22⋊#1055—42no—
110#88050—22no—
111#88047—22no—
112C4⋊#1055—22no—
113C4×#1055—22no—
114#24243—131nomaximal, Sylow
115C24×C13C2×D412yes—
116C2×Q8×C13—28no—
117C2×Q8×C13—21no—
118C2×Q8×C13—22no—
119C2×Q8×C13—21no—
120C2×Q8×C13—21no—
121C2×Q8×C13—22no—
122C2×Q8×C13—22no—
123C2×Q8×C13—22no—
124C2×Q8×C13—22no—
125C2×Q8×C13—44no—
126C2×Q8×C13—44no—
127C2×Q8×C13—48no—
128C2×Q8×C13—48no—
129C2×Q8×C13—48no—
130C2×Q8×C13—48no—
131C2×Q8×C13C2×D412yes—
132C2×Q8×C13C2×D414yes—
133C2×Q8×C13C2×D412yes—
134C2×Q8×C13#4112yes—
135C2×D4×C13—22no—
136C2×D4×C13—22no—
137C2×D4×C13Q1614yescomplemented
138C22×C4×C13—21no—
139C22×C4×C13—28no—
140C22×C4×C13—21no—
141C22×C4×C13—22no—
142C22×C4×C13—22no—
143C22×C4×C13—22no—
144C22×C4×C13—22no—
145C22×C4×C13—22no—
146C22×C4×C13—22no—
147C22×C4×C13—22no—
148C22×C4×C13—44no—
149C22×C4×C13—44no—
150C22×C4×C13C2×D412yes—
151C22×C4×C13C2×D411yescharacteristic
152C22×C4×C13C2×D411yescharacteristic
153C22×C4×C13C2411yescharacteristic
154C22×Dic13—21no—
155C22×Dic13—21no—
156C22×Dic13—22no—
157C2×Dic26—21no—
158C2×Dic26—21no—
159C2×Dic26—21no—
160C2×Dic26—21no—
161C2×Dic26—21no—
162C2×Dic26C2×D412yes—
163C13×C4⋊C4—28no—
164C13×C4⋊C4—24no—
165C13×C4⋊C4—24no—
166C13×C4⋊C4—24no—
167C13×C4⋊C4—22no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4C2×D412yes—
170C13×C22⋊C4—28no—
171C13×C22⋊C4—24no—
172C42×C13—24no—
173C42×C13C2×D412yes—
174C42×C13C2×D412yes—
175C22⋊Dic13—44no—
176C22⋊Dic13—82no—
177#6107—28no—
178#6107—48no—
179C4⋊Dic13—21no—
180C4⋊Dic13—22no—
181C4⋊Dic13—21no—
182C4⋊Dic13—22no—
183C4⋊Dic13—42no—
184C4⋊Dic13—81no—
185#6101—44no—
186#6101—42no—
187C4×Dic13—42no—
188C2×#1055—21no—
189C2×#1055—21no—
190C2×#1055—22no—
191C2×#1055—22no—
192C2×#1055—21no—
193C2×#1055—21no—
194C2×#1055—22no—
195C2×#1055—21no—
196C2×#1055C2×D412yes—
197C2×D4×Q8D1311yescharacteristic
198C2×#462—131no—
199C2×#451—132no—
200C2×C4×Q16—131no—
201#2538—131no—
202#2036—132no—
203#2001—132no—
204#1956—131no—
205#1954—131no—
206#1803—131no—
207#1787—132no—
208C23×C13—22no—
209C23×C13—22no—
210C23×C13—22no—
211C23×C13—22no—
212C23×C13—22no—
213C23×C13—22no—
214C23×C13C2×Q1614yes—
215C23×C13C22×D411yescharacteristic
216Q8×C13—22no—
217Q8×C13—22no—
218Q8×C13—22no—
219Q8×C13—24no—
220Q8×C13—24no—
221Q8×C13—416no—
222Q8×C13—42no—
223Q8×C13—42no—
224Q8×C13#12114yes—
225D4×C13—28no—
226C2×C4×C13—22no—
227C2×C4×C13—22no—
228C2×C4×C13—22no—
229C2×C4×C13—21no—
230C2×C4×C13—21no—
231C2×C4×C13—22no—
232C2×C4×C13—24no—
233C2×C4×C13—22no—
234C2×C4×C13—22no—
235C2×C4×C13—22no—
236C2×C4×C13—416no—
237C2×C4×C13—42no—
238C2×C4×C13—42no—
239C2×C4×C13—41no—
240C2×C4×C13—41no—
241C2×C4×C13—42no—
242C2×C4×C13—42no—
243C2×C4×C13—42no—
244C2×C4×C13—42no—
245C2×C4×C13#12012yes—
246C2×C4×C13#12012yes—
247C2×C4×C13C2×Q1612yes—
248C2×C4×C13#13712yes—
249C2×C4×C13C22×D411yescharacteristic
250C2×C4×C13C22×D411yescharacteristic
251C2×C4×C13C22×D411yescharacteristic
252C2×C4×C13C2×#4111yescharacteristic
253C2×C4×C13H(5,𝔽2)11yescharacteristic
254C2×C4×C13H(5,𝔽2)11yescharacteristic
255C2×Dic13—22no—
256C2×Dic13—41no—
257C2×Dic13—41no—
258C2×Dic13—41no—
259C2×Dic13—42no—
260C2×Dic13—42no—
261C2×Dic13—42no—
262C2×Dic13—41no—
263C2×Dic13—41no—
264Dic26—22no—
265Dic26—22no—
266Dic26—41no—
267Dic26—41no—
268#1055—22no—
269#1055—41no—
270#1055—42no—
271C23×Q8C2×D1312yes—
272C22×Q16—131no—
273C22×Q16—132no—
274D4×Q8—24no—
275D4×Q8—24no—
276C2×C4⋊Q8—21no—
277C2×C4⋊Q8C2×D1311yescharacteristic
278C2×C22⋊Q8—22no—
279C2×C22⋊Q8—132no—
280C2×C22⋊Q8C2×D1312yes—
281C2×C4×Q8—131no—
282C2×C4×Q8C2×D1311yescharacteristic
283C2×C4×D4—21no—
284C2×C4×D4C2×D1311yescharacteristic
285#462—264no—
286#451—268no—
287C4×Q16—264no—
288C2×#107—131no—
289C2×C4⋊C8—131no—
290C2×#103—132no—
291C2×#103—131no—
292C2×#103—132no—
293C2×#103—132no—
294C2×#103—131no—
295C2×#103—132no—
296C2×C22⋊C8—132no—
297C2×C22⋊C8—132no—
298C2×C4×C8—131no—
299#396—262no—
300#390—261no—
301#389—261no—
302#384—131no—
303#380—132no—
304#340—132no—
305C22×C13—24no—
306C22×C13—24no—
307C22×C13—24no—
308C22×C13—42no—
309C22×C13—42no—
310C22×C13—42no—
311C22×C13—42no—
312C22×C13C2×#12011yescharacteristic
313C22×C13D4211yescharacteristic
314C22×C13#54611yescharacteristic
315C22×C13C22×Q1611yescharacteristic
316C22×C13C2×#13711yescharacteristic
317C22×C13#57711yescharacteristic
318C22×C13#57811yescharacteristic
319C4×C13—24no—
320C4×C13—22no—
321C4×C13—42no—
322C4×C13—44no—
323C4×C13—42no—
324C4×C13#45114yescomplemented
325C4×C13C2×#12111yescharacteristic
326C4×C13#53511yescharacteristic
327C4×C13D4211yescharacteristic
328C4×C13#54611yescharacteristic
329Dic13—42no—
330Dic13—81no—
331Dic13—82no—
332C22×Q8—24no—
333C22×Q8—24no—
334C22×Q8—22no—
335C22×Q8—24no—
336C22×Q8—24no—
337C22×Q8—42no—
338C22×Q8—42no—
339C22×Q8—131no—
340C22×Q8C22×D1312yes—
341C22×Q8C22×D1311yescharacteristic
342C22×D4C22⋊D1311yescharacteristic
343C23×C4—22no—
344C23×C4C22×D1312yes—
345C2×Q16—134no—
346C2×Q16—262no—
347C2×Q16—262no—
348C2×Q16—264no—
349C2×Q16—264no—
350C2×Q16—522no—
351C22×C8—131no—
352C22×C8—132no—
353C22×C8—131no—
354C4⋊Q8—28no—
355C4⋊Q8—22no—
356C4⋊Q8C22⋊D1314yes—
357C22⋊Q8—216no—
358C22⋊Q8—24no—
359C22⋊Q8—24no—
360C22⋊Q8—264no—
361C22⋊Q8—522no—
362C4×Q8—22no—
363C4×Q8—262no—
364C4×Q8—262no—
365C4×Q8C22⋊D1314yes—
366C4×D4—28no—
367C4×D4—22no—
368C4×D4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—131no—
374C2×C4⋊C4—132no—
375C2×C4⋊C4—261no—
376C2×C4⋊C4—262no—
377C2×C4⋊C4—261no—
378C2×C4⋊C4—261no—
379C2×C4⋊C4C22⋊D1311yescharacteristic
380C2×C4⋊C4C22×D1311yescharacteristic
381C2×C22⋊C4—22no—
382C2×C22⋊C4—262no—
383C2×C22⋊C4—262no—
384C2×C22⋊C4C22⋊D1312yes—
385C2×C42—21no—
386C2×C42—261no—
387C2×C42C22×D1311yescharacteristic
388#107—262no—
389C4⋊C8—262no—
390#103—264no—
391#103—264no—
392#103—262no—
393#103—264no—
394#103—262no—
395#103—522no—
396C22⋊C8—264no—
397C22⋊C8—522no—
398C4×C8—262no—
399#95—262no—
400C2×C13—44no—
401C2×C13#240811yescharacteristic
402C2×C13C2×#45111yescharacteristic
403C2×C13#301311yescharacteristic
404C2×C13#327811yescharacteristic
405C2×C13D4×Q1611yescharacteristic
406C2×C13#329911yescharacteristic
407C2×C13#332811yescharacteristic
408C24C2×C22⋊D1312yes—
409C2×Q8—28no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—48no—
421C2×Q8—48no—
422C2×Q8—48no—
423C2×Q8—48no—
424C2×Q8—132no—
425C2×Q8—261no—
426C2×Q8—261no—
427C2×Q8—261no—
428C2×Q8—261no—
429C2×Q8—261no—
430C2×Q8D4×D1312yes—
431C2×Q8#612912yes—
432C2×Q8C2×C22⋊D1314yes—
433C2×Q8C2×C22⋊D1312yes—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4#610714yescomplemented
437C22×C4—21no—
438C22×C4—28no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—261no—
450C22×C4—261no—
451C22×C4—262no—
452C22×C4C2×C22⋊D1312yes—
453C22×C4C2×C22⋊D1311yescharacteristic
454C22×C4C2×C22⋊D1311yescharacteristic
455C22×C4C23×D1311yescharacteristic
456Q16—268no—
457Q16—528no—
458C2×C8—132no—
459C2×C8—261no—
460C2×C8—261no—
461C2×C8—262no—
462C2×C8—261no—
463C2×C8—262no—
464C2×C8—261no—
465C2×C8—261no—
466C2×C8—262no—
467C4⋊C4—28no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—262no—
474C4⋊C4—262no—
475C4⋊C4—261no—
476C4⋊C4—261no—
477C4⋊C4—524no—
478C4⋊C4—522no—
479C4⋊C4—522no—
480C4⋊C4—1041no—
481C4⋊C4C2×C22⋊D1312yes—
482C22⋊C4—28no—
483C22⋊C4—24no—
484C22⋊C4—524no—
485C22⋊C4—1042no—
486C42—24no—
487C42—522no—
488C42C2×C22⋊D1312yes—
489C42C2×C22⋊D1312yes—
490C13#2424311yescharacteristic, complemented, Sylow
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23C2×#610714yes—
498C23C22×C22⋊D1311yescharacteristic
499Q8—22no—
500Q8—22no—
501Q8—22no—
502Q8—24no—
503Q8—24no—
504Q8—416no—
505Q8—42no—
506Q8—42no—
507Q8—262no—
508Q8—262no—
509Q8—521no—
510Q8—521no—
511Q8#8819214yes—
512D4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—416no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—262no—
533C2×C4—521no—
534C2×C4—521no—
535C2×C4—522no—
536C2×C4—522no—
537C2×C4—521no—
538C2×C4—522no—
539C2×C4—521no—
540C2×C4—521no—
541C2×C4#8819612yes—
542C2×C4C2×#610712yes—
543C2×C4#8820712yes—
544C2×C4#8820712yes—
545C2×C4C2×D4×D1311yescharacteristic
546C2×C4C2×#612911yescharacteristic
547C2×C4#8825111yescharacteristic
548C2×C4#8825111yescharacteristic
549C2×C4C22×C22⋊D1311yescharacteristic
550C2×C4C22×C22⋊D1311yescharacteristic
551C8—262no—
552C8—522no—
553C8—521no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22D4×C22⋊D1311yescharacteristic
562C22#1171951111yescharacteristic
563C22C2×#8819611yescharacteristic
564C22C22×#610711yescharacteristic
565C22#1171954211yescharacteristic
566C22#1171954311yescharacteristic
567C22C2×#8820711yescharacteristic
568C4—22no—
569C4—24no—
570C4—42no—
571C4—42no—
572C4—44no—
573C4—522no—
574C4—1041no—
575C4—1042no—
576C4#1171888914yescomplemented
577C4C2×#8819211yescharacteristic
578C4D4×C22⋊D1311yescharacteristic
579C4#1171951011yescharacteristic
580C4#1171951111yescharacteristic
581C2—44no—
582C2#1311818611yescharacteristic
583C2C2×#1171888911yescharacteristic
584C2#1312415111yescharacteristic
585C2#1312649811yescharacteristic
586C2D4×#610711yescharacteristic
587C2#1312671211yescharacteristic
588C2#1312711911yescharacteristic
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#24243
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. C2×#11718751C2semidirect
2. C2×#11718889C2semidirect
3. C2×C4×#6107C2semidirect
5. #13116036C2semidirect
6. #13115844C2semidirect
7. #13115639C2semidirect
9. #13114841C2semidirect
10. #13114771C2semidirect
87. C2×#6107D4semidirect
137. C2×D4×C13Q16semidirect
324. C4×C13#451semidirect
436. C2×D4#6107semidirect
490. C13#24243semidirect
576. C4#11718889semidirect

Maximal subgroups

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

subgroupindex
C2×#117187512
C2×#117188892
C2×C4×#61072
#131197582
#131160362
#131158442
#131156392
#131156312
#131148412
#131147712
C2×D4×Q8×C132
#2424313