Finite Group Databasebeta

← #15923522

Subgroups — #15923522

Order 3328.

874 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Dic13C211yesmaximal, characteristic
2#13116394C211yesmaximal, characteristic, complemented
3#13116273C211yesmaximal, characteristic, complemented
4#13116244C211yesmaximal, characteristic, complemented
5#13116029C211yesmaximal, characteristic, complemented
6#13115467C211yesmaximal, characteristic, complemented
7#13115418C211yesmaximal, characteristic, complemented
8#13115260C211yesmaximal, characteristic, complemented
9#13115144C211yesmaximal, characteristic, complemented
10#13115144C211yesmaximal, characteristic, complemented
11#13115129C211yesmaximal, characteristic
12#13115029C211yesmaximal, characteristic, complemented
13#13114920C211yesmaximal, characteristic, complemented
14C2×#11718121C211yesmaximal, characteristic, complemented
15C13×#2417C211yesmaximal, characteristic
16C2×C13×C22⋊Q8C2211yescharacteristic
17C2×C13×C22⋊Q8C2211yescharacteristic
18C2×C4×D4×C13C2211yescharacteristic
19C22×C13×C4⋊C4C2211yescharacteristic
20C2×D4×Dic13—21no—
21C22×C4⋊Dic13C2211yescharacteristic
22C22×C4⋊Dic13C2211yescharacteristic
23C23×#1055C2211yescharacteristic
24C13×#398—21no—
25C13×#396—21no—
26C13×#391C2211yescharacteristic
27C13×#389C2211yescharacteristic
28C13×#386—21no—
29C13×#380—21no—
30C13×#380C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33C2×#88075C2211yescharacteristic
34C2×#88075C2211yescharacteristic
35#11718873C2211yescharacteristic
36#11718873C2211yescharacteristic
37C2×C22⋊#1055C2211yescharacteristic
38C2×C22⋊#1055C2211yescharacteristic
39C2×C22⋊#1055C2211yescharacteristic
40#11718646—21no—
41C2×#88048C2211yescharacteristic
42C2×#88048C2211yescharacteristic
43#11718611—21no—
44#11718611—21no—
45#11718596C2211yescharacteristic
46C4×C4⋊Dic13C2211yescharacteristic
47#11718121C2211yescharacteristic
48#11718121C2211yescharacteristic
49#11718121C2211yescharacteristic
50#11718121C2211yescharacteristic
51#11718121C2211yescharacteristic
52#11718121C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718116C2211yescharacteristic
58#11718116C2211yescharacteristic
59C22×Q8×C13C2311yescharacteristic
60C22×D4×C13D411yescharacteristic
61C23×C4×C13—21no—
62C23×C4×C13C2311yescharacteristic
63C23×C4×C13C2311yescharacteristic
64C23×Dic13—21no—
65C23×Dic13—21no—
66C13×C22⋊Q8—22no—
67C13×C22⋊Q8—22no—
68C13×C22⋊Q8—22no—
69C13×C22⋊Q8—41no—
70C4×D4×C13—22no—
71C4×D4×C13—22no—
72C2×C13×C4⋊C4—22no—
73C2×C13×C4⋊C4—21no—
74C2×C13×C4⋊C4—21no—
75C2×C13×C4⋊C4—21no—
76C2×C13×C4⋊C4—21no—
77C2×C13×C4⋊C4—41no—
78C2×C13×C4⋊C4D411yescharacteristic
79C2×C13×C4⋊C4C2311yescharacteristic
80C2×C13×C4⋊C4C2311yescharacteristic
81C2×C13×C4⋊C4C2311yescharacteristic
82C2×C13×C22⋊C4—21no—
83C2×C13×C22⋊C4—21no—
84C2×C13×C22⋊C4—21no—
85C2×C13×C22⋊C4D411yescharacteristic
86C2×C13×C22⋊C4D411yescharacteristic
87C2×C42×C13C2311yescharacteristic
88C2×C22⋊Dic13—21no—
89C2×C22⋊Dic13—21no—
90D4×Dic13—44no—
91C2×C4⋊Dic13—22no—
92C2×C4⋊Dic13—22no—
93C2×C4⋊Dic13—21no—
94C2×C4⋊Dic13—21no—
95C2×C4⋊Dic13—21no—
96C2×C4⋊Dic13—21no—
97C2×C4⋊Dic13—41no—
98C2×C4⋊Dic13—41no—
99C2×C4⋊Dic13C2311yescharacteristic
100C2×C4⋊Dic13C2311yescharacteristic
101C2×C4⋊Dic13C2311yescharacteristic
102C2×C4⋊Dic13C2311yescharacteristic
103C2×C4×Dic13—21no—
104C22×#1055—22no—
105C22×#1055—41no—
106C22×#1055—41no—
107C22×#1055C2311yescharacteristic
108C22×#1055C2311yescharacteristic
109C22×#1055C2311yescharacteristic
110C22×#1055C2311yescharacteristic
111C13×#95—21no—
112C13×#95—21no—
113C13×#95—21no—
114C13×#95—21no—
115#88075—22no—
116#88075—41no—
117#88075—41no—
118#88075—41no—
119#88071—21no—
120#88071—21no—
121#88071—21no—
122C22⋊#1055—22no—
123C22⋊#1055—41no—
124C22⋊#1055—41no—
125#88048—22no—
126#88048—41no—
127#26200—131nomaximal, Sylow
128C24×C13C2×D411yescharacteristic
129C24×C13C2×D411yescharacteristic
130C2×Q8×C13—21no—
131C2×Q8×C13—21no—
132C2×Q8×C13—21no—
133C2×Q8×C13—21no—
134C2×Q8×C13—21no—
135C2×Q8×C13—21no—
136C2×D4×C13—22no—
137C2×D4×C13—22no—
138C2×D4×C13SD1614yes—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—21no—
142C22×C4×C13—21no—
143C22×C4×C13—21no—
144C22×C4×C13—21no—
145C22×C4×C13—21no—
146C22×C4×C13—21no—
147C22×C4×C13—21no—
148C22×C4×C13—21no—
149C22×C4×C13—21no—
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×C13C2×D411yescharacteristic
159C22×C4×C13C2×D411yescharacteristic
160C22×C4×C13#4112yes—
161C22×C4×C13#4112yes—
162C22×C4×C13C2411yescharacteristic
163C22×Dic13—21no—
164C22×Dic13—21no—
165C22×Dic13—21no—
166C22×Dic13—21no—
167C22×Dic13—42no—
168C22×Dic13—42no—
169C22×Dic13—42no—
170C22×Dic13—42no—
171C22×Dic13—81no—
172C22×Dic13—81no—
173C13×C4⋊C4—22no—
174C13×C4⋊C4—21no—
175C13×C4⋊C4—21no—
176C13×C4⋊C4—21no—
177C13×C4⋊C4—21no—
178C13×C4⋊C4—21no—
179C13×C4⋊C4—42no—
180C13×C4⋊C4—42no—
181C13×C4⋊C4—42no—
182C13×C4⋊C4—81no—
183C13×C4⋊C4#4112yes—
184C13×C22⋊C4—22no—
185C13×C22⋊C4—22no—
186C13×C22⋊C4—42no—
187C13×C22⋊C4—42no—
188C13×C22⋊C4—81no—
189C42×C13—21no—
190C42×C13—21no—
191C22⋊Dic13—42no—
192C22⋊Dic13—42no—
193C4⋊Dic13—21no—
194C4⋊Dic13—21no—
195C4⋊Dic13—21no—
196C4⋊Dic13—21no—
197C4⋊Dic13—21no—
198C4⋊Dic13—21no—
199C4⋊Dic13—21no—
200C4⋊Dic13—42no—
201C4⋊Dic13—42no—
202C4⋊Dic13—42no—
203C4⋊Dic13—81no—
204C4⋊Dic13#4112yes—
205C4×Dic13—42no—
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×#1055—21no—
216C2×#1055—21no—
217C2×#1055—21no—
218C2×#1055—44no—
219#2417D1311yescharacteristic
220D4×C4⋊C4—131no—
221#2099—131no—
222#2077—131no—
223#2072—131no—
224#2035—131no—
225#1927—131no—
226#1920—131no—
227#1892—131no—
228#1872—131no—
229#1872—131no—
230#1869—131no—
231#1853—131no—
232#1823—131no—
233C2×#340—131no—
234C23×C13—21no—
235C23×C13—21no—
236C23×C13—21no—
237C23×C13—21no—
238C23×C13—21no—
239C23×C13—21no—
240C23×C13—21no—
241C23×C13—21no—
242C23×C13—21no—
243C23×C13—21no—
244C23×C13#12312yes—
245C23×C13C2×SD1612yes—
246C23×C13C2×SD1612yes—
247C23×C13#13712yes—
248C23×C13C22×D411yescharacteristic
249Q8×C13—41no—
250Q8×C13—41no—
251Q8×C13—41no—
252Q8×C13—41no—
253D4×C13—28no—
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—22no—
259C2×C4×C13—22no—
260C2×C4×C13—22no—
261C2×C4×C13—22no—
262C2×C4×C13—21no—
263C2×C4×C13—21no—
264C2×C4×C13—21no—
265C2×C4×C13—22no—
266C2×C4×C13—21no—
267C2×C4×C13—21no—
268C2×C4×C13—21no—
269C2×C4×C13—41no—
270C2×C4×C13—41no—
271C2×C4×C13—41no—
272C2×C4×C13—41no—
273C2×C4×C13—41no—
274C2×C4×C13—41no—
275C2×C4×C13—41no—
276C2×C4×C13—41no—
277C2×C4×C13—41no—
278C2×C4×C13—41no—
279C2×C4×C13—41no—
280C2×C4×C13—41no—
281C2×C4×C13—41no—
282C2×C4×C13—41no—
283C2×C4×C13—41no—
284C2×C4×C13—84no—
285C2×C4×C13#12312yes—
286C2×C4×C13C2×SD1612yes—
287C2×C4×C13C2×#4111yescharacteristic
288C2×C4×C13C2×#4111yescharacteristic
289C2×C4×C13C2×#4111yescharacteristic
290C2×C4×C13C2×#4111yescharacteristic
291C2×C4×C13H(5,𝔽2)11yescharacteristic
292C2×C4×C13H(5,𝔽2)11yescharacteristic
293C2×Dic13—22no—
294C2×Dic13—41no—
295C2×Dic13—41no—
296C2×Dic13—41no—
297C2×Dic13—41no—
298C2×Dic13—41no—
299C2×Dic13—41no—
300C2×Dic13—41no—
301C2×Dic13—41no—
302C2×Dic13—41no—
303C2×Dic13—41no—
304C2×Dic13—41no—
305C2×Dic13—84no—
306C2×Dic13—84no—
307#1055—41no—
308#1055—41no—
309#1055—41no—
310#1055—41no—
311C23×C8—131no—
312C2×C22⋊Q8C2×D1311yescharacteristic
313C2×C22⋊Q8C2×D1311yescharacteristic
314C2×C4×D4—261no—
315C2×C4×D4C2×D1311yescharacteristic
316C22×C4⋊C4—131no—
317C22×C4⋊C4—131no—
318C22×C4⋊C4C2×D1311yescharacteristic
319C2×#106—131no—
320C2×#106—131no—
321C2×#103—131no—
322C2×#103—131no—
323C2×#103—131no—
324C2×#103—131no—
325C2×C22⋊C8—131no—
326C2×C22⋊C8—131no—
327C2×C22⋊C8—131no—
328#398—21no—
329#396—21no—
330#391C2×D1311yescharacteristic
331#389—261no—
332#389C2×D1311yescharacteristic
333#386—21no—
334#385—261no—
335#385—261no—
336#384—131no—
337#380—21no—
338#380—131no—
339#380—131no—
340#380C2×D1311yescharacteristic
341C4×C4⋊C4—131no—
342#340—131no—
343#340—131no—
344#340—131no—
345#340—131no—
346#340—131no—
347#340—131no—
348#340—131no—
349#340—131no—
350#340—131no—
351#340—131no—
352#336—131no—
353#336—131no—
354C22×C13—22no—
355C22×C13—22no—
356C22×C13—22no—
357C22×C13—22no—
358C22×C13—22no—
359C22×C13—22no—
360C22×C13—22no—
361C22×C13—22no—
362C22×C13—22no—
363C22×C13—41no—
364C22×C13—41no—
365C22×C13—41no—
366C22×C13—41no—
367C22×C13#48314yes—
368C22×C13C2×#12311yescharacteristic
369C22×C13#54611yescharacteristic
370C22×C13#54611yescharacteristic
371C22×C13C22×SD1611yescharacteristic
372C22×C13C2×#13711yescharacteristic
373C22×C13#57611yescharacteristic
374C22×C13#57811yescharacteristic
375C4×C13—22no—
376C4×C13—22no—
377C4×C13—41no—
378C4×C13—41no—
379C4×C13—42no—
380C4×C13—81no—
381C4×C13—81no—
382C4×C13—81no—
383C4×C13#52911yescharacteristic
384C4×C13#53811yescharacteristic
385C4×C13#55311yescharacteristic
386C4×C13#55311yescharacteristic
387Dic13—42no—
388Dic13—81no—
389Dic13—81no—
390Dic13—81no—
391C22×Q8C22×D1311yescharacteristic
392C22×D4C22⋊D1311yescharacteristic
393C23×C4—21no—
394C23×C4—261no—
395C23×C4—261no—
396C23×C4C22×D1311yescharacteristic
397C23×C4C22×D1311yescharacteristic
398C22×C8—131no—
399C22×C8—131no—
400C22×C8—131no—
401C22×C8—131no—
402C22×C8—262no—
403C22×C8—521no—
404C22×C8—521no—
405C22⋊Q8—22no—
406C22⋊Q8—22no—
407C22⋊Q8—22no—
408C22⋊Q8—41no—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4—524no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—41no—
418C2×C4⋊C4—131no—
419C2×C4⋊C4—131no—
420C2×C4⋊C4—131no—
421C2×C4⋊C4—131no—
422C2×C4⋊C4—262no—
423C2×C4⋊C4—262no—
424C2×C4⋊C4—261no—
425C2×C4⋊C4—261no—
426C2×C4⋊C4—261no—
427C2×C4⋊C4—261no—
428C2×C4⋊C4—521no—
429C2×C4⋊C4—521no—
430C2×C4⋊C4C22⋊D1311yescharacteristic
431C2×C4⋊C4C22×D1311yescharacteristic
432C2×C4⋊C4C22×D1311yescharacteristic
433C2×C4⋊C4C22×D1311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—261no—
438C2×C22⋊C4—261no—
439C2×C22⋊C4C22⋊D1311yescharacteristic
440C2×C22⋊C4C22⋊D1311yescharacteristic
441C2×C42—261no—
442C2×C42C22×D1311yescharacteristic
443#106—262no—
444#106—521no—
445#103—262no—
446#103—521no—
447#103—521no—
448#103—521no—
449C22⋊C8—262no—
450C22⋊C8—521no—
451C22⋊C8—521no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—261no—
457#95—261no—
458#95—261no—
459C2×C13—24no—
460C2×C13—42no—
461C2×C13#244411yescharacteristic
462C2×C13C2×#48311yescharacteristic
463C2×C13#309811yescharacteristic
464C2×C13#329511yescharacteristic
465C2×C13#329811yescharacteristic
466C2×C13#331211yescharacteristic
467C2×C13#331411yescharacteristic
468C24C2×C22⋊D1311yescharacteristic
469C24C2×C22⋊D1311yescharacteristic
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4#610614yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—81no—
496C22×C4—261no—
497C22×C4—261no—
498C22×C4—261no—
499C22×C4—261no—
500C22×C4—522no—
501C22×C4—522no—
502C22×C4—522no—
503C22×C4—522no—
504C22×C4—1041no—
505C22×C4—1041no—
506C22×C4#612912yes—
507C22×C4#612912yes—
508C22×C4C2×C22⋊D1311yescharacteristic
509C22×C4C2×C22⋊D1311yescharacteristic
510C22×C4C2×C22⋊D1311yescharacteristic
511C22×C4C2×C22⋊D1311yescharacteristic
512C22×C4C23×D1311yescharacteristic
513C2×C8—261no—
514C2×C8—261no—
515C2×C8—261no—
516C2×C8—261no—
517C2×C8—261no—
518C2×C8—261no—
519C2×C8—261no—
520C2×C8—261no—
521C2×C8—261no—
522C2×C8—261no—
523C2×C8—261no—
524C2×C8—261no—
525C2×C8—524no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—81no—
536C4⋊C4—132no—
537C4⋊C4—261no—
538C4⋊C4—261no—
539C4⋊C4—261no—
540C4⋊C4—261no—
541C4⋊C4—261no—
542C4⋊C4—261no—
543C4⋊C4—261no—
544C4⋊C4—522no—
545C4⋊C4—522no—
546C4⋊C4—522no—
547C4⋊C4—1041no—
548C4⋊C4#612912yes—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—81no—
554C22⋊C4—522no—
555C22⋊C4—522no—
556C42—21no—
557C42—21no—
558C42—522no—
559C13#2620011yescharacteristic, complemented, Sylow
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23#8818912yes—
571C23C2×#610612yes—
572C23C2×#610612yes—
573C23#8820512yes—
574C23C22×C22⋊D1311yescharacteristic
575Q8—41no—
576Q8—41no—
577Q8—41no—
578Q8—41no—
579D4—28no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—84no—
611C2×C4—262no—
612C2×C4—521no—
613C2×C4—521no—
614C2×C4—521no—
615C2×C4—521no—
616C2×C4—521no—
617C2×C4—521no—
618C2×C4—521no—
619C2×C4—521no—
620C2×C4—521no—
621C2×C4—521no—
622C2×C4—521no—
623C2×C4—1044no—
624C2×C4—1044no—
625C2×C4#8818912yes—
626C2×C4C2×#610612yes—
627C2×C4C2×#612911yescharacteristic
628C2×C4C2×#612911yescharacteristic
629C2×C4C2×#612911yescharacteristic
630C2×C4#8825111yescharacteristic
631C2×C4#41×D1311yescharacteristic
632C2×C4#8825611yescharacteristic
633C8—521no—
634C8—521no—
635C8—521no—
636C8—521no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22#1171870414yes—
651C22C2×#8818911yescharacteristic
652C22#1171951111yescharacteristic
653C22C22×#610611yescharacteristic
654C22C2×#8820511yescharacteristic
655C22#1171953411yescharacteristic
656C22#1171954011yescharacteristic
657C22#1171954311yescharacteristic
658C4—22no—
659C4—22no—
660C4—42no—
661C4—41no—
662C4—41no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—522no—
667C4—1041no—
668C4—1041no—
669C4—1041no—
670C4#1171930511yescharacteristic
671C4#1171930611yescharacteristic
672C4#1171932211yescharacteristic
673C4#1171932311yescharacteristic
674C2—24no—
675C2—42no—
676C2#1311858211yescharacteristic
677C2C2×#1171870411yescharacteristic
678C2#1312481011yescharacteristic
679C2#1312665411yescharacteristic
680C2#1312670111yescharacteristic
681C2#1312688111yescharacteristic
682C2#1312691711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2417C2
socleC23×C13C22×D4
prime radicalC22×D4×C13D4
hypercenter#2417D13
hypocenterC13#26200
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2417

Complements

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

NH ≅ G/Ntype
2. #13116394C2semidirect
3. #13116273C2semidirect
4. #13116244C2semidirect
5. #13116029C2semidirect
6. #13115467C2semidirect
7. #13115418C2semidirect
8. #13115260C2semidirect
9. #13115144C2semidirect
10. #13115144C2semidirect
12. #13115029C2semidirect
13. #13114920C2semidirect
14. C2×#11718121C2semidirect
559. C13#26200semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Dic132
#131163942
#131162732
#131162442
#131160292
#131154672
#131154182
#131152602
#131151442
#131151292
#131150292
#131149202
C2×#117181212
C13×#24172
#2620013