Finite Group Databasebeta

← #15897275

Subgroups — #15897275

Order 3328.

1114 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718752C211yesmaximal, characteristic, complemented
2C2×#11718751C211yesmaximal, characteristic, complemented
3C4×#88178C211yesmaximal, characteristic
4#13116036C212yesmaximal
5#13115844C212yesmaximal
6#13115785C211yesmaximal, characteristic, complemented
7#13115760C211yesmaximal, characteristic, complemented
8#13115177C212yesmaximal, complemented
9#13115090C212yesmaximal, complemented
10#13114816C211yesmaximal, characteristic
11C2×C13×#533C211yesmaximal, characteristic
12C13×#533—24no—
13C13×#533—24no—
14C2×C13×C4⋊Q8C2211yescharacteristic
15C2×C13×C4⋊Q8C2211yescharacteristic
16C2×C13×#125—21no—
17C2×C13×C22⋊Q8—21no—
18C2×C13×C22⋊Q8C2212yes—
19C2×C4×Q8×C13—21no—
20C2×C4×Q8×C13C2212yes—
21C2×C13×#117—21no—
22C22×C42×C13C2211yescharacteristic
23C22×#6107C2212yes—
24C2×#88178C2211yescharacteristic
25C2×C4×Dic26C2211yescharacteristic
26C2×#88075C2212yes—
27C2×#88075C2212yes—
28C2×#88075C2212yes—
29#11718873C2211yescharacteristic
30#11718871—21no—
31C4×C22⋊Dic13—21no—
32C2×C22⋊#1055C2212yes—
33#11718752—24no—
34#11718751—24no—
35C2×#88050C2212yes—
36C2×#88047C2212yes—
37C4×C4⋊Dic13C2211yescharacteristic
38#11718594—21no—
39C4×#6101—21no—
40#11718590C2211yescharacteristic
41#11718589C2211yescharacteristic
42C2×C4⋊#1055C2211yescharacteristic
43C2×C4⋊#1055C2211yescharacteristic
44#11718121C2212yes—
45#11718121C2212yes—
46#11718116C2212yes—
47C22×Q8×C13—21no—
48C22×Q8×C13C2312yes—
49C23×C4×C13D411yescharacteristic
50C23×C4×C13D411yescharacteristic
51C23×C4×C13C2311yescharacteristic
52C22×Dic26C2311yescharacteristic
53C13×C4⋊Q8—22no—
54C13×C4⋊Q8—22no—
55C13×C4⋊Q8D414yes—
56C13×C4⋊Q8D414yes—
57C13×#125—28no—
58C13×C22⋊Q8—28no—
59C13×C22⋊Q8—24no—
60C13×C22⋊Q8—24no—
61C4×Q8×C13—28no—
62C4×Q8×C13—24no—
63C4×Q8×C13—24no—
64C13×#117—28no—
65C2×C13×C4⋊C4—21no—
66C2×C13×C4⋊C4—21no—
67C2×C13×C4⋊C4—21no—
68C2×C13×C4⋊C4—22no—
69C2×C13×C4⋊C4—22no—
70C2×C13×C4⋊C4C2312yes—
71C2×C13×C22⋊C4—22no—
72C2×C42×C13—22no—
73C2×C42×C13—22no—
74C2×C42×C13—22no—
75C2×C42×C13D411yescharacteristic
76C2×C42×C13D411yescharacteristic
77C2×C42×C13C2311yescharacteristic
78C2×C42×C13C2311yescharacteristic
79C2×C22⋊Dic13—21no—
80C2×C22⋊Dic13—21no—
81C2×#6107—24no—
82C2×#6107—24no—
83C2×#6107—42no—
84C2×C4⋊Dic13C2311yescharacteristic
85C2×C4⋊Dic13C2311yescharacteristic
86C2×C4⋊Dic13C2311yescharacteristic
87#88178—22no—
88#88178—22no—
89C2×#6101—21no—
90C2×#6101—21no—
91C2×#6101—21no—
92C2×C4×Dic13—21no—
93C2×C4×Dic13—21no—
94C22×#1055C2312yes—
95C22×#1055C2312yes—
96C4×Dic26—22no—
97C4×Dic26—22no—
98#88075—24no—
99#88075—24no—
100#88075—42no—
101#88071—21no—
102#88071—21no—
103#88071—21no—
104#88071—21no—
105#88071—21no—
106C22⋊#1055—42no—
107#88050—24no—
108#88047—24no—
109C4⋊#1055—22no—
110C4⋊#1055—22no—
111#24155—131nomaximal, Sylow
112C24×C13C2×D411yescharacteristic
113C2×Q8×C13—24no—
114C2×Q8×C13—22no—
115C2×Q8×C13—22no—
116C2×Q8×C13—22no—
117C2×Q8×C13—22no—
118C2×Q8×C13—22no—
119C2×Q8×C13—42no—
120C2×Q8×C13—42no—
121C2×Q8×C13C2×D414yes—
122C22×C4×C13—22no—
123C22×C4×C13—22no—
124C22×C4×C13—22no—
125C22×C4×C13—22no—
126C22×C4×C13—22no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—41no—
134C22×C4×C13—41no—
135C22×C4×C13Q1614yescomplemented
136C22×C4×C13C2×D411yescharacteristic
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2411yescharacteristic
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—21no—
145C22×Dic13—21no—
146C2×Dic26—21no—
147C2×Dic26—21no—
148C2×Dic26—21no—
149C2×Dic26—21no—
150C2×Dic26—21no—
151C2×Dic26#4112yes—
152C13×C4⋊C4—28no—
153C13×C4⋊C4—28no—
154C13×C4⋊C4—24no—
155C13×C4⋊C4—24no—
156C13×C4⋊C4—24no—
157C13×C4⋊C4—22no—
158C13×C4⋊C4C2×D414yes—
159C13×C22⋊C4—28no—
160C42×C13—28no—
161C42×C13—22no—
162C42×C13—22no—
163C42×C13—21no—
164C42×C13—21no—
165C42×C13C2×D412yes—
166C42×C13C2×D412yes—
167C22⋊Dic13—24no—
168C22⋊Dic13—42no—
169#6107—48no—
170C4⋊Dic13—21no—
171C4⋊Dic13—21no—
172C4⋊Dic13—21no—
173C4⋊Dic13—21no—
174C4⋊Dic13—21no—
175C4⋊Dic13#4112yes—
176#6101—24no—
177#6101—42no—
178#6101—42no—
179C4×Dic13—42no—
180C4×Dic13—42no—
181C2×#1055—22no—
182C2×#1055—22no—
183C2×#1055—22no—
184C2×#1055—22no—
185C2×#1055—22no—
186C2×#1055—22no—
187C2×#533D1311yescharacteristic
188C2×#477—131no—
189C2×#462—131no—
190C4×C22⋊Q8—131no—
191#2036—132no—
192#2001—132no—
193#1987—131no—
194#1981—131no—
195#1878—132no—
196#1863—132no—
197#1797—131no—
198C23×C13—21no—
199C23×C13—21no—
200C23×C13—21no—
201C23×C13—21no—
202C23×C13—21no—
203C23×C13—21no—
204C23×C13C2×Q1612yes—
205C23×C13C22×D411yescharacteristic
206Q8×C13—24no—
207Q8×C13—24no—
208Q8×C13—48no—
209Q8×C13—42no—
210Q8×C13—42no—
211C2×C4×C13—21no—
212C2×C4×C13—21no—
213C2×C4×C13—21no—
214C2×C4×C13—21no—
215C2×C4×C13—28no—
216C2×C4×C13—21no—
217C2×C4×C13—21no—
218C2×C4×C13—24no—
219C2×C4×C13—24no—
220C2×C4×C13—21no—
221C2×C4×C13—21no—
222C2×C4×C13—21no—
223C2×C4×C13—21no—
224C2×C4×C13—44no—
225C2×C4×C13—42no—
226C2×C4×C13—42no—
227C2×C4×C13—42no—
228C2×C4×C13—42no—
229C2×C4×C13—41no—
230C2×C4×C13—41no—
231C2×C4×C13—41no—
232C2×C4×C13—41no—
233C2×C4×C13#12012yes—
234C2×C4×C13#12012yes—
235C2×C4×C13C2×Q1612yes—
236C2×C4×C13C2×Q1612yes—
237C2×C4×C13#13712yes—
238C2×C4×C13C22×D411yescharacteristic
239C2×C4×C13C22×D411yescharacteristic
240C2×C4×C13C2×#4111yescharacteristic
241C2×C4×C13C2×#4111yescharacteristic
242C2×C4×C13H(5,𝔽2)11yescharacteristic
243C2×C4×C132−1+411yescharacteristic
244C2×Dic13—22no—
245C2×Dic13—22no—
246C2×Dic13—41no—
247C2×Dic13—41no—
248C2×Dic13—41no—
249C2×Dic13—41no—
250C2×Dic13—41no—
251C2×Dic13—41no—
252C2×Dic13—41no—
253C2×Dic13—41no—
254C2×Dic13—41no—
255C2×Dic13—41no—
256Dic26—22no—
257Dic26—22no—
258Dic26—41no—
259Dic26—41no—
260#1055—42no—
261#1055—42no—
262C22×Q16—132no—
263#533—24no—
264#533—24no—
265C2×C4⋊Q8C2×D1311yescharacteristic
266C2×C4⋊Q8C2×D1311yescharacteristic
267C2×#125—21no—
268C2×C22⋊Q8—21no—
269C2×C22⋊Q8—131no—
270C2×C22⋊Q8C2×D1312yes—
271C2×C4×Q8—21no—
272C2×C4×Q8—131no—
273C2×C4×Q8C2×D1312yes—
274C2×#117—21no—
275C22×C42C2×D1311yescharacteristic
276#477—264no—
277#462—264no—
278C2×#107—132no—
279C2×C4⋊C8—131no—
280C2×C4⋊C8—131no—
281C2×#103—132no—
282C2×#103—132no—
283C2×#103—132no—
284C2×#103—132no—
285C2×C22⋊C8—132no—
286#391—131no—
287#389—131no—
288#387—261no—
289#385—261no—
290#380—131no—
291C4×C4⋊C4—131no—
292C4×C4⋊C4—261no—
293C4×C22⋊C4—261no—
294#340—132no—
295#340—132no—
296#336—132no—
297C22×C13—22no—
298C22×C13—22no—
299C22×C13—22no—
300C22×C13—41no—
301C22×C13—41no—
302C22×C13—41no—
303C22×C13—41no—
304C22×C13C2×#12011yescharacteristic
305C22×C13#52511yescharacteristic
306C22×C13#53711yescharacteristic
307C22×C13C22×Q1611yescharacteristic
308C22×C13C2×#13711yescharacteristic
309C22×C13#57511yescharacteristic
310C22×C13#57511yescharacteristic
311C4×C13—22no—
312C4×C13—22no—
313C4×C13—41no—
314C4×C13—44no—
315C4×C13—41no—
316C4×C13—44no—
317C4×C13#45114yes—
318C4×C13#54611yescharacteristic
319C4×C13#54711yescharacteristic
320C4×C13#54811yescharacteristic
321C4×C13#55011yescharacteristic
322Dic13—42no—
323Dic13—42no—
324Dic13—81no—
325Dic13—81no—
326C22×Q8—21no—
327C22×Q8—131no—
328C22×Q8C22×D1312yes—
329C23×C4C22⋊D1311yescharacteristic
330C23×C4C22⋊D1311yescharacteristic
331C23×C4C22×D1311yescharacteristic
332C2×Q16—264no—
333C2×Q16—264no—
334C2×Q16—522no—
335C22×C8—132no—
336C22×C8—132no—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339C4⋊Q8C22⋊D1314yes—
340C4⋊Q8C22⋊D1314yes—
341#125—28no—
342C22⋊Q8—28no—
343C22⋊Q8—24no—
344C22⋊Q8—24no—
345C22⋊Q8—262no—
346C22⋊Q8—262no—
347C4×Q8—28no—
348C4×Q8—24no—
349C4×Q8—24no—
350C4×Q8—262no—
351C4×Q8—262no—
352#117—28no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—131no—
359C2×C4⋊C4—131no—
360C2×C4⋊C4—131no—
361C2×C4⋊C4—261no—
362C2×C4⋊C4—261no—
363C2×C4⋊C4—261no—
364C2×C4⋊C4C22×D1312yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—261no—
367C2×C22⋊C4—261no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—261no—
372C2×C42—261no—
373C2×C42C22⋊D1311yescharacteristic
374C2×C42C22⋊D1311yescharacteristic
375C2×C42C22×D1311yescharacteristic
376C2×C42C22×D1311yescharacteristic
377#107—264no—
378C4⋊C8—262no—
379C4⋊C8—262no—
380#103—264no—
381#103—264no—
382#103—264no—
383#103—522no—
384C22⋊C8—522no—
385#95—261no—
386#95—261no—
387#95—261no—
388#95—261no—
389#95—261no—
390C2×C13—42no—
391C2×C13#240711yescharacteristic
392C2×C13C2×#45111yescharacteristic
393C2×C13#300511yescharacteristic
394C2×C13#310411yescharacteristic
395C2×C13#311611yescharacteristic
396C2×C13#324911yescharacteristic
397C2×C13#325211yescharacteristic
398C24C2×C22⋊D1311yescharacteristic
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—132no—
408C2×Q8—261no—
409C2×Q8—261no—
410C2×Q8—261no—
411C2×Q8—261no—
412C2×Q8—261no—
413C2×Q8C2×C22⋊D1314yes—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—261no—
428C22×C4—261no—
429C22×C4—261no—
430C22×C4—261no—
431C22×C4#610714yescomplemented
432C22×C4C2×C22⋊D1311yescharacteristic
433C22×C4C2×C22⋊D1311yescharacteristic
434C22×C4C2×C22⋊D1311yescharacteristic
435C22×C4C2×C22⋊D1311yescharacteristic
436C22×C4C2×C22⋊D1311yescharacteristic
437C22×C4C23×D1311yescharacteristic
438Q16—528no—
439C2×C8—262no—
440C2×C8—262no—
441C2×C8—262no—
442C2×C8—262no—
443C2×C8—262no—
444C2×C8—262no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—132no—
452C4⋊C4—264no—
453C4⋊C4—261no—
454C4⋊C4—261no—
455C4⋊C4—261no—
456C4⋊C4—261no—
457C4⋊C4—261no—
458C4⋊C4—522no—
459C4⋊C4—522no—
460C4⋊C4C2×C22⋊D1314yes—
461C22⋊C4—28no—
462C22⋊C4—264no—
463C22⋊C4—522no—
464C42—28no—
465C42—22no—
466C42—22no—
467C42—21no—
468C42—21no—
469C42—522no—
470C42—522no—
471C42C2×C22⋊D1312yes—
472C42C2×C22⋊D1312yes—
473C13#2415511yescharacteristic, complemented, Sylow
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23C2×#610712yes—
481C23C22×C22⋊D1311yescharacteristic
482Q8—24no—
483Q8—24no—
484Q8—48no—
485Q8—42no—
486Q8—42no—
487Q8—262no—
488Q8—262no—
489Q8—521no—
490Q8—521no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—28no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—262no—
514C2×C4—262no—
515C2×C4—521no—
516C2×C4—521no—
517C2×C4—521no—
518C2×C4—521no—
519C2×C4—521no—
520C2×C4—521no—
521C2×C4—521no—
522C2×C4—521no—
523C2×C4—521no—
524C2×C4—521no—
525C2×C4#8819612yes—
526C2×C4C2×#610712yes—
527C2×C4C2×#610712yes—
528C2×C4#8820712yes—
529C2×C4#8820712yes—
530C2×C4#41×D1311yescharacteristic
531C2×C4#41×D1311yescharacteristic
532C2×C4#8825611yescharacteristic
533C2×C4#8825711yescharacteristic
534C2×C4C22×C22⋊D1311yescharacteristic
535C2×C4C22×C22⋊D1311yescharacteristic
536C8—522no—
537C8—522no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22C2×#8819611yescharacteristic
546C22C22×#610711yescharacteristic
547C22#1171952711yescharacteristic
548C22#1171952711yescharacteristic
549C22#1171953311yescharacteristic
550C22#1171953811yescharacteristic
551C22C2×#8820711yescharacteristic
552C4—22no—
553C4—22no—
554C4—41no—
555C4—44no—
556C4—41no—
557C4—44no—
558C4—522no—
559C4—522no—
560C4—1041no—
561C4—1041no—
562C4#1171888914yes—
563C4#1171952911yescharacteristic
564C4#1171953411yescharacteristic
565C4#1171953611yescharacteristic
566C4#1171953911yescharacteristic
567C2—42no—
568C2#1311817511yescharacteristic
569C2C2×#1171888911yescharacteristic
570C2#1312407811yescharacteristic
571C2#1312487111yescharacteristic
572C2#1312499211yescharacteristic
573C2#1312617811yescharacteristic
574C2#1312620311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC2×C13×#533C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenterC2×#533D13
hypocenterC13#24155
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×#11718752C2semidirect
2. C2×#11718751C2semidirect
6. #13115785C2semidirect
7. #13115760C2semidirect
8. #13115177C2semidirect
9. #13115090C2semidirect
135. C22×C4×C13Q16semidirect
431. C22×C4#6107semidirect
473. C13#24155semidirect

Maximal subgroups

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

subgroupindex
C2×#117187522
C2×#117187512
C4×#881782
#131160362
#131158442
#131157852
#131157602
#131151772
#131150902
#131148162
C2×C13×#5332
#2415513