Finite Group Databasebeta

← #15880697

Subgroups — #15880697

Order 3328.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718625C211yesmaximal, characteristic
2#13117719C211yesmaximal, characteristic
3#13115576C211yesmaximal, characteristic, complemented
4#13115561C211yesmaximal, characteristic, complemented
5#13115425C211yesmaximal, characteristic, complemented
6#13115421C211yesmaximal, characteristic, complemented
7#13115414C211yesmaximal, characteristic, complemented
8#13115404C211yesmaximal, characteristic, complemented
9#13114928C211yesmaximal, characteristic
10#13114926C211yesmaximal, characteristic
11#13114867C211yesmaximal, characteristic
12#13114734C211yesmaximal, characteristic, complemented
13C4×#88047C211yesmaximal, characteristic, complemented
14#13114620C211yesmaximal, characteristic
15C13×#2344C211yesmaximal, characteristic
16C2×C13×#117—21no—
17C22×C13×C4⋊C4C2211yescharacteristic
18C2×#88180C2211yescharacteristic
19C22×#6099C2211yescharacteristic
20C13×#389C2211yescharacteristic
21C13×#389C2211yescharacteristic
22C13×#387—21no—
23C13×#387—21no—
24C13×#381—21no—
25C13×#380C2211yescharacteristic
26C4×C13×C4⋊C4C2211yescharacteristic
27C4×C13×C4⋊C4C2211yescharacteristic
28C4×C13×C22⋊C4C2211yescharacteristic
29#11718873C2211yescharacteristic
30C2×C22⋊#1055C2211yescharacteristic
31#11718625C418yescomplemented
32C2×#88048C2211yescharacteristic
33C2×#88048C2211yescharacteristic
34C2×#88048C2211yescharacteristic
35C2×#88047C2211yescharacteristic
36C2×#88047C2211yescharacteristic
37C2×#88047C2211yescharacteristic
38#11718605—21no—
39#11718605—21no—
40C4×C4⋊Dic13C2211yescharacteristic
41C4×C4⋊Dic13C2211yescharacteristic
42#11718589C2211yescharacteristic
43#11718589C2211yescharacteristic
44C2×#88043C2211yescharacteristic
45C2×C4×#1055C2211yescharacteristic
46#11718311—21no—
47#11718309—21no—
48#11718121C2211yescharacteristic
49#11718121C2211yescharacteristic
50#11718121C2211yescharacteristic
51#11718121C2211yescharacteristic
52#11718121C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718116C2211yescharacteristic
59#11718116C2211yescharacteristic
60C23×C4×C13—21no—
61C23×C4×C13C2311yescharacteristic
62C13×#117—44no—
63C2×C13×C4⋊C4—21no—
64C2×C13×C4⋊C4—22no—
65C2×C13×C4⋊C4—21no—
66C2×C13×C4⋊C4—21no—
67C2×C13×C4⋊C4C2×C414yes—
68C2×C13×C4⋊C4C2311yescharacteristic
69C2×C13×C4⋊C4C2311yescharacteristic
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C22⋊C4—21no—
73C2×C13×C22⋊C4D411yescharacteristic
74C2×C13×C22⋊C4D411yescharacteristic
75C2×C42×C13—21no—
76C2×C42×C13—21no—
77C2×C42×C13C2311yescharacteristic
78C2×C42×C13C2311yescharacteristic
79C2×C22⋊Dic13—21no—
80C2×C22⋊Dic13—21no—
81#88180—22no—
82#88180C2×C414yes—
83C2×C4⋊Dic13C2311yescharacteristic
84C2×C4⋊Dic13C2311yescharacteristic
85C2×C4⋊Dic13C2311yescharacteristic
86C2×C4⋊Dic13C2311yescharacteristic
87C2×#6101—21no—
88C2×#6101—21no—
89C2×C4×Dic13—21no—
90C2×C4×Dic13—21no—
91C2×C4×Dic13—21no—
92C2×#6099—22no—
93C2×#6099—22no—
94C2×#6099C2×C414yes—
95C22×#1055C2311yescharacteristic
96C22×#1055C2311yescharacteristic
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C13×#95—21no—
100C13×#95—21no—
101C13×#95—21no—
102C13×#95—21no—
103C13×#95—21no—
104C13×#95D411yescharacteristic
105C13×#95D411yescharacteristic
106#88071—21no—
107#88071—21no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111C22⋊#1055—22no—
112#88048—22no—
113#88048C2×C414yes—
114#88048C2×C414yes—
115#88047—22no—
116#88047C2×C414yes—
117#88047C2×C414yes—
118#88043—22no—
119C4×#1055—22no—
120#22624—131nomaximal, Sylow
121C24×C13C2×D411yescharacteristic
122C22×C4×C13—21no—
123C22×C4×C13—21no—
124C22×C4×C13—21no—
125C22×C4×C13—21no—
126C22×C4×C13—24no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—42no—
132C22×C4×C13—42no—
133C22×C4×C13C22×C412yes—
134C22×C4×C13C2×D411yescharacteristic
135C22×C4×C13C2×D411yescharacteristic
136C22×C4×C13C2×D411yescharacteristic
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13#4112yes—
140C22×C4×C13C2411yescharacteristic
141C22×Dic13—21no—
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—21no—
145C13×C4⋊C4—24no—
146C13×C4⋊C4—21no—
147C13×C4⋊C4—21no—
148C13×C4⋊C4—21no—
149C13×C4⋊C4—21no—
150C13×C4⋊C4—21no—
151C13×C4⋊C4—21no—
152C13×C4⋊C4—42no—
153C13×C4⋊C4—42no—
154C13×C4⋊C4C22×C412yes—
155C13×C4⋊C4C22×C412yes—
156C13×C22⋊C4—22no—
157C13×C22⋊C4—41no—
158C13×C22⋊C4—42no—
159C42×C13—21no—
160C42×C13—21no—
161C42×C13—21no—
162C42×C13—21no—
163C42×C13—42no—
164C42×C13—42no—
165C22⋊Dic13—24no—
166C22⋊Dic13—81no—
167C4⋊Dic13—21no—
168C4⋊Dic13—21no—
169C4⋊Dic13—21no—
170C4⋊Dic13—21no—
171C4⋊Dic13—21no—
172C4⋊Dic13—21no—
173C4⋊Dic13C22×C412yes—
174C4⋊Dic13C22×C412yes—
175#6101—81no—
176#6101—81no—
177C4×Dic13—24no—
178C4×Dic13—42no—
179C4×Dic13—42no—
180#6099—28no—
181C2×#1055—21no—
182C2×#1055—21no—
183C2×#1055—21no—
184C2×#1055—21no—
185C2×#1055—21no—
186C2×#1055—21no—
187C2×#1055—21no—
188C2×#1055—21no—
189C2×#1055C22×C412yes—
190C2×#1055C22×C412yes—
191C2×#1055#4112yes—
192C2×#1055#4112yes—
193C2×#428—131no—
194#2346—131no—
195#2344D1311yescharacteristic
196#1944—131no—
197#1941—131no—
198#1921—131no—
199#1920—131no—
200#1919—131no—
201#1918—131no—
202#1825—131no—
203#1824—131no—
204#1809—131no—
205#1776—131no—
206C4×#107—131no—
207#1753—131no—
208C23×C13—21no—
209C23×C13—21no—
210C23×C13—21no—
211C23×C13—21no—
212C23×C13—21no—
213C23×C13—21no—
214C23×C13C4×D412yes—
215C23×C13C22×D411yescharacteristic
216C2×C4×C13—21no—
217C2×C4×C13—21no—
218C2×C4×C13—21no—
219C2×C4×C13—21no—
220C2×C4×C13—22no—
221C2×C4×C13—22no—
222C2×C4×C13—22no—
223C2×C4×C13—22no—
224C2×C4×C13—22no—
225C2×C4×C13—22no—
226C2×C4×C13—21no—
227C2×C4×C13—21no—
228C2×C4×C13—21no—
229C2×C4×C13—21no—
230C2×C4×C13—21no—
231C2×C4×C13—21no—
232C2×C4×C13—21no—
233C2×C4×C13—21no—
234C2×C4×C13—21no—
235C2×C4×C13—48no—
236C2×C4×C13—41no—
237C2×C4×C13—41no—
238C2×C4×C13—41no—
239C2×C4×C13—41no—
240C2×C4×C13—41no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13—41no—
244C2×C4×C13—41no—
245C2×C4×C13—41no—
246C2×C4×C13C4×D412yes—
247C2×C4×C13AGL(1,ℤ/8)12yes—
248C2×C4×C13C23×C411yescharacteristic
249C2×C4×C13C2×#4111yescharacteristic
250C2×C4×C13C2×#4111yescharacteristic
251C2×C4×C13C2×#4111yescharacteristic
252C2×C4×C132−1+411yescharacteristic
253C2×C4×C132−1+411yescharacteristic
254C2×Dic13—22no—
255C2×Dic13—22no—
256C2×Dic13—41no—
257C2×Dic13—41no—
258C2×Dic13—41no—
259C2×Dic13—41no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13—41no—
263C2×Dic13—41no—
264C2×Dic13—41no—
265C2×Dic13—41no—
266#1055—22no—
267#1055—41no—
268#1055—41no—
269#1055#11714yescomplemented
270C22×M16—131no—
271C2×#117—21no—
272C2×#117—131no—
273C22×C4⋊C4C2×D1311yescharacteristic
274#428—138no—
275C2×#107—131no—
276C2×#107—131no—
277C2×#107—131no—
278C2×#106—131no—
279C2×#106—131no—
280C2×#106—131no—
281C2×C22⋊C8—131no—
282C2×#97—131no—
283C2×C4×C8—131no—
284#389—131no—
285#389—131no—
286#389C2×D1311yescharacteristic
287#389C2×D1311yescharacteristic
288#388—261no—
289#388—261no—
290#387—21no—
291#387—21no—
292#382—261no—
293#381—21no—
294#380—131no—
295#380C2×D1311yescharacteristic
296C4×C4⋊C4—131no—
297C4×C4⋊C4—131no—
298C4×C4⋊C4C2×D1311yescharacteristic
299C4×C4⋊C4C2×D1311yescharacteristic
300C4×C22⋊C4C2×D1311yescharacteristic
301#376—261no—
302#340—131no—
303#340—131no—
304#340—131no—
305#340—131no—
306#340—131no—
307#340—131no—
308#340—131no—
309#340—131no—
310#340—131no—
311#340—131no—
312#336—131no—
313#336—131no—
314C22×C13—22no—
315C22×C13—22no—
316C22×C13—22no—
317C22×C13—41no—
318C22×C13—41no—
319C22×C13—41no—
320C22×C13—41no—
321C22×C13C2×C4×D411yescharacteristic
322C22×C13#54711yescharacteristic
323C22×C13#54711yescharacteristic
324C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
325C22×C13#57511yescharacteristic
326C22×C13#57711yescharacteristic
327C22×C13#57811yescharacteristic
328C4×C13—22no—
329C4×C13—22no—
330C4×C13—41no—
331C4×C13—41no—
332C4×C13—41no—
333C4×C13—42no—
334C4×C13—42no—
335C4×C13—81no—
336C4×C13—81no—
337C4×C13C2×#11711yescharacteristic
338C4×C13#51911yescharacteristic
339C4×C13#55211yescharacteristic
340C4×C13#55211yescharacteristic
341Dic13—42no—
342Dic13—42no—
343Dic13—81no—
344Dic13—81no—
345C23×C4—21no—
346C23×C4C22×D1311yescharacteristic
347C2×M16—134no—
348C2×M16—262no—
349C2×M16—262no—
350C22×C8—131no—
351C22×C8—131no—
352C22×C8—131no—
353C22×C8—131no—
354#117—44no—
355#117—134no—
356#117—262no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—131no—
362C2×C4⋊C4—131no—
363C2×C4⋊C4—131no—
364C2×C4⋊C4—131no—
365C2×C4⋊C4—261no—
366C2×C4⋊C4—261no—
367C2×C4⋊C4C4×D1314yes—
368C2×C4⋊C4C22×D1311yescharacteristic
369C2×C4⋊C4C22×D1311yescharacteristic
370C2×C4⋊C4C22×D1311yescharacteristic
371C2×C4⋊C4C22×D1311yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—261no—
374C2×C22⋊C4—261no—
375C2×C22⋊C4C22⋊D1311yescharacteristic
376C2×C22⋊C4C22⋊D1311yescharacteristic
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—261no—
380C2×C42—261no—
381C2×C42—261no—
382C2×C42C22×D1311yescharacteristic
383C2×C42C22×D1311yescharacteristic
384#107—134no—
385#107—134no—
386#107—262no—
387#106—134no—
388#106—134no—
389#106—262no—
390C22⋊C8—262no—
391#97—262no—
392C4×C8—262no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—261no—
399#95—261no—
400#95—261no—
401#95—261no—
402#95—261no—
403#95C22⋊D1311yescharacteristic
404#95C22⋊D1311yescharacteristic
405C2×C13—42no—
406C2×C13#234411yescharacteristic
407C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
408C2×C13#294711yescharacteristic
409C2×C13#334311yescharacteristic
410C2×C13#334811yescharacteristic
411C2×C13#335211yescharacteristic
412C2×C13#335411yescharacteristic
413C24C2×C22⋊D1311yescharacteristic
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—261no—
426C22×C4—261no—
427C22×C4—261no—
428C22×C4—261no—
429C22×C4C2×C4×D1312yes—
430C22×C4#612712yes—
431C22×C4C2×C22⋊D1311yescharacteristic
432C22×C4C2×C22⋊D1311yescharacteristic
433C22×C4C2×C22⋊D1311yescharacteristic
434C22×C4C2×C22⋊D1311yescharacteristic
435C22×C4C2×C22⋊D1311yescharacteristic
436C22×C4C23×D1311yescharacteristic
437M16—268no—
438C2×C8—132no—
439C2×C8—132no—
440C2×C8—132no—
441C2×C8—132no—
442C2×C8—261no—
443C2×C8—261no—
444C2×C8—261no—
445C2×C8—261no—
446C2×C8—261no—
447C2×C8—261no—
448C2×C8—261no—
449C2×C8—261no—
450C4⋊C4—24no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—132no—
460C4⋊C4—132no—
461C4⋊C4—261no—
462C4⋊C4—261no—
463C4⋊C4—261no—
464C4⋊C4—261no—
465C4⋊C4—261no—
466C4⋊C4—261no—
467C4⋊C4—1041no—
468C4⋊C4—1041no—
469C4⋊C4C2×C4×D1312yes—
470C4⋊C4C2×C4×D1312yes—
471C22⋊C4—22no—
472C22⋊C4—41no—
473C22⋊C4—42no—
474C22⋊C4—264no—
475C22⋊C4—1041no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—42no—
481C42—42no—
482C42—264no—
483C42—522no—
484C42—522no—
485C13#2262411yescharacteristic, complemented, Sylow
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23C4×C22⋊D1312yes—
493C23C22×C22⋊D1311yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—48no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—262no—
525C2×C4—262no—
526C2×C4—521no—
527C2×C4—521no—
528C2×C4—521no—
529C2×C4—521no—
530C2×C4—521no—
531C2×C4—521no—
532C2×C4—521no—
533C2×C4—521no—
534C2×C4—521no—
535C2×C4—521no—
536C2×C4C4×C22⋊D1312yes—
537C2×C4#8818612yes—
538C2×C4C22×C4×D1311yescharacteristic
539C2×C4C2×#612711yescharacteristic
540C2×C4#41×D1311yescharacteristic
541C2×C4#41×D1311yescharacteristic
542C2×C4#8825711yescharacteristic
543C2×C4#8825711yescharacteristic
544C8—134no—
545C8—262no—
546C8—521no—
547C8—521no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×C4×C22⋊D1311yescharacteristic
556C22C2×#8818611yescharacteristic
557C22#1171952711yescharacteristic
558C22#1171952911yescharacteristic
559C22#1171952911yescharacteristic
560C22#1171954211yescharacteristic
561C22#1171954311yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—42no—
566C4—42no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—522no—
572C4—522no—
573C4—1041no—
574C4—1041no—
575C4#1171922111yescharacteristic
576C4D13×#11711yescharacteristic
577C4#1171923611yescharacteristic
578C4#1171923611yescharacteristic
579C2—42no—
580C2#1311770611yescharacteristic
581C2C4×#8818611yescharacteristic
582C2#1312367911yescharacteristic
583C2#1312727011yescharacteristic
584C2#1312734511yescharacteristic
585C2#1312740511yescharacteristic
586C2#1312743111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2344C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#2344D13
hypocenterC13#22624
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2344

Complements

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

NH ≅ G/Ntype
3. #13115576C2semidirect
4. #13115561C2semidirect
5. #13115425C2semidirect
6. #13115421C2semidirect
7. #13115414C2semidirect
8. #13115404C2semidirect
12. #13114734C2semidirect
13. C4×#88047C2semidirect
31. #11718625C4semidirect
269. #1055#117semidirect
485. C13#22624semidirect

Maximal subgroups

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

subgroupindex
C2×#117186252
#131177192
#131155762
#131155612
#131154252
#131154212
#131154142
#131154042
#131149282
#131149262
#131148672
#131147342
C4×#880472
#131146202
C13×#23442
#2262413