Finite Group Databasebeta

← #15780434

Subgroups — #15780434

Order 3328.

820 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718901C211yesmaximal, characteristic, complemented
2C2×#11718128C212yesmaximal, complemented
3#13114764C211yesmaximal, characteristic, complemented
4#13112954C212yesmaximal, complemented
5C13×#2404C211yesmaximal, characteristic
6C22×C13×#41C2211yescharacteristic
7C2×C13×#120C411yescharacteristic, complemented
8C2×C13×#120C2211yescharacteristic
9C22×C13×C22⋊C4C411yescharacteristic, complemented
10C2×#88180—21no—
11C13×#393—21no—
12C13×#392—21no—
13C13×#386—21no—
14C13×#386—21no—
15C13×#386C411yescharacteristic, complemented
16C13×#386C2211yescharacteristic
17C13×#381C411yescharacteristic, complemented
18#11718901—24no—
19C2×#88076—21no—
20C2×#88074—21no—
21C2×#88074C2212yes—
22C2×#88073C2212yes—
23#11718868—21no—
24#11718128—24no—
25C25×C13C2×C411yescharacteristic
26C2×C13×#41—22no—
27C2×C13×#41—22no—
28C2×C13×#41—41no—
29C2×C13×#41—41no—
30C2×C13×#41—42no—
31C22×Q8×C13D411yescharacteristic
32C22×D4×C13—21no—
33C22×D4×C13—21no—
34C22×D4×C13C2×C411yescharacteristic
35C22×D4×C13D411yescharacteristic
36C22×D4×C13C2311yescharacteristic
37C23×C4×C13C2×C411yescharacteristic
38C23×C4×C13C2×C411yescharacteristic
39C23×C4×C13D411yescharacteristic
40C23×C4×C13D411yescharacteristic
41C13×#120—22no—
42C13×#120—22no—
43C13×#120—42no—
44C2×C13×C22⋊C4—21no—
45C2×C13×C22⋊C4—21no—
46C2×C13×C22⋊C4—21no—
47C2×C13×C22⋊C4—21no—
48C2×C13×C22⋊C4—21no—
49C2×C13×C22⋊C4—21no—
50C2×C13×C22⋊C4—41no—
51C2×C13×C22⋊C4—41no—
52C2×C13×C22⋊C4C2×C411yescharacteristic
53C2×C13×C22⋊C4C2×C411yescharacteristic
54C2×C22⋊Dic13—22no—
55#88180—24no—
56#88180—42no—
57C2×C4⋊Dic13—41no—
58C2×C4×Dic13—41no—
59C2×#6099—22no—
60C13×#95—21no—
61C13×#95—21no—
62C13×#95—21no—
63C13×#95—21no—
64#88076—42no—
65#88074—24no—
66#88074—22no—
67#88074D414yescomplemented
68#88073—22no—
69#88073—22no—
70C22⋊#1055—41no—
71C22⋊#1055—41no—
72#13087—131nomaximal, Sylow
73C24×C13—21no—
74C24×C13—22no—
75C24×C13—41no—
76C24×C13—42no—
77C24×C13—41no—
78C24×C13—81no—
79C24×C13C22⋊C411yescharacteristic
80C24×C13C22×C411yescharacteristic
81C24×C13C2×D411yescharacteristic
82C13×#41—44no—
83C13×#41—44no—
84C13×#41—44no—
85C13×#41—44no—
86C2×Q8×C13—22no—
87C2×Q8×C13—22no—
88C2×Q8×C13—41no—
89C2×D4×C13—22no—
90C2×D4×C13—21no—
91C2×D4×C13—21no—
92C2×D4×C13—22no—
93C2×D4×C13—41no—
94C2×D4×C13—41no—
95C2×D4×C13—41no—
96C2×D4×C13—41no—
97C2×D4×C13—41no—
98C2×D4×C13—42no—
99C2×D4×C13—42no—
100C2×D4×C13—42no—
101C2×D4×C13—41no—
102C2×D4×C13—41no—
103C2×D4×C13—81no—
104C2×D4×C13—81no—
105C2×D4×C13C2×D412yes—
106C2×D4×C13#4112yes—
107C22×C4×C13—21no—
108C22×C4×C13—21no—
109C22×C4×C13—21no—
110C22×C4×C13—21no—
111C22×C4×C13—22no—
112C22×C4×C13—22no—
113C22×C4×C13—21no—
114C22×C4×C13—21no—
115C22×C4×C13—21no—
116C22×C4×C13—21no—
117C22×C4×C13—22no—
118C22×C4×C13—21no—
119C22×C4×C13—21no—
120C22×C4×C13—21no—
121C22×C4×C13—41no—
122C22×C4×C13—41no—
123C22×C4×C13—41no—
124C22×C4×C13—81no—
125C22×C4×C13C22⋊C411yescharacteristic
126C22×C4×C13C22⋊C411yescharacteristic
127C22×C4×C13C22⋊C411yescharacteristic
128C22×C4×C13C2×D411yescharacteristic
129C22×Dic13—42no—
130C13×C22⋊C4—22no—
131C13×C22⋊C4—42no—
132C13×C22⋊C4—41no—
133C13×C22⋊C4—41no—
134C13×C22⋊C4—42no—
135C13×C22⋊C4—42no—
136C13×C22⋊C4—41no—
137C13×C22⋊C4—81no—
138C13×C22⋊C4—81no—
139C22⋊Dic13—24no—
140C22⋊Dic13—42no—
141C4⋊Dic13—44no—
142C4×Dic13—44no—
143#6099—42no—
144#6099—42no—
145C2×#1055—42no—
146C2×#410—131no—
147#2404D1311yescharacteristic
148C2×AΣL(1,𝔽16)—132no—
149#1785—131no—
150#1390—132no—
151C23×C13—21no—
152C23×C13—21no—
153C23×C13—21no—
154C23×C13—21no—
155C23×C13—22no—
156C23×C13—21no—
157C23×C13—21no—
158C23×C13—21no—
159C23×C13—41no—
160C23×C13—41no—
161C23×C13—41no—
162C23×C13—41no—
163C23×C13—41no—
164C23×C13—41no—
165C23×C13—41no—
166C23×C13—41no—
167C23×C13—41no—
168C23×C13—42no—
169C23×C13—41no—
170C23×C13—41no—
171C23×C13—41no—
172C23×C13—42no—
173C23×C13—82no—
174C23×C13—84no—
175C23×C13—84no—
176C23×C13—82no—
177C23×C13C2×C22⋊C411yescharacteristic
178C23×C13C4×D412yes—
179C23×C13#12011yescharacteristic
180C23×C13#12311yescharacteristic
181Q8×C13—44no—
182D4×C13—42no—
183D4×C13—42no—
184D4×C13—44no—
185D4×C13—42no—
186D4×C13—42no—
187D4×C13—82no—
188D4×C13—82no—
189D4×C13—84no—
190C2×C4×C13—22no—
191C2×C4×C13—22no—
192C2×C4×C13—22no—
193C2×C4×C13—22no—
194C2×C4×C13—21no—
195C2×C4×C13—21no—
196C2×C4×C13—21no—
197C2×C4×C13—21no—
198C2×C4×C13—21no—
199C2×C4×C13—21no—
200C2×C4×C13—21no—
201C2×C4×C13—21no—
202C2×C4×C13—21no—
203C2×C4×C13—42no—
204C2×C4×C13—42no—
205C2×C4×C13—42no—
206C2×C4×C13—42no—
207C2×C4×C13—41no—
208C2×C4×C13—41no—
209C2×C4×C13—41no—
210C2×C4×C13—41no—
211C2×C4×C13—41no—
212C2×C4×C13—41no—
213C2×C4×C13—41no—
214C2×C4×C13—41no—
215C2×C4×C13—41no—
216C2×C4×C13—82no—
217C2×C4×C13#9911yescharacteristic
218C2×C4×C13#9911yescharacteristic
219C2×C4×C13#9912yes—
220C2×C4×C13#12112yes—
221C2×Dic13—44no—
222C2×Dic13—82no—
223C2×Dic13—82no—
224#1055—82no—
225C22×#41C2×D1311yescharacteristic
226C2×#120Dic1311yescharacteristic, complemented
227C2×#120C2×D1311yescharacteristic
228C2×#117—261no—
229C22×C22⋊C4Dic1311yescharacteristic, complemented
230C2×#101—261no—
231C2×#100—132no—
232#410—264no—
233C2×#99—132no—
234C2×#99—261no—
235#407—261no—
236#393—21no—
237#392—21no—
238#386—21no—
239#386—21no—
240#386Dic1311yescharacteristic, complemented
241#386C2×D1311yescharacteristic
242#381Dic1311yescharacteristic, complemented
243AΣL(1,𝔽16)—264no—
244C22×C13—21no—
245C22×C13—21no—
246C22×C13—21no—
247C22×C13—21no—
248C22×C13—22no—
249C22×C13—41no—
250C22×C13—41no—
251C22×C13—42no—
252C22×C13—42no—
253C22×C13—42no—
254C22×C13—42no—
255C22×C13—42no—
256C22×C13—41no—
257C22×C13—41no—
258C22×C13—41no—
259C22×C13—41no—
260C22×C13—41no—
261C22×C13—42no—
262C22×C13—41no—
263C22×C13—41no—
264C22×C13—81no—
265C22×C13—81no—
266C22×C13—81no—
267C22×C13—81no—
268C22×C13—84no—
269C22×C13—84no—
270C22×C13—81no—
271C22×C13—81no—
272C22×C13#38611yescharacteristic
273C22×C13C2×#9911yescharacteristic
274C22×C13C2×#9911yescharacteristic
275C4×C13—22no—
276C4×C13—21no—
277C4×C13—21no—
278C4×C13—22no—
279C4×C13—41no—
280C4×C13—81no—
281C4×C13—81no—
282C4×C13—81no—
283Dic13—84no—
284C25C2×Dic1311yescharacteristic
285C2×#41—22no—
286C2×#41—22no—
287C2×#41—41no—
288C2×#41—41no—
289C2×#41—42no—
290C22×Q8C22⋊D1311yescharacteristic
291C22×D4—21no—
292C22×D4—21no—
293C22×D4C2×Dic1311yescharacteristic
294C22×D4C22⋊D1311yescharacteristic
295C22×D4C22×D1311yescharacteristic
296C23×C4C2×Dic1311yescharacteristic
297C23×C4C2×Dic1311yescharacteristic
298C23×C4C22⋊D1311yescharacteristic
299C23×C4C22⋊D1311yescharacteristic
300C2×M16—262no—
301#120—22no—
302#120—22no—
303#120—42no—
304#117—264no—
305#117—522no—
306C2×C4⋊C4—521no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—41no—
314C2×C22⋊C4—41no—
315C2×C22⋊C4—262no—
316C2×C22⋊C4C2×Dic1311yescharacteristic
317C2×C22⋊C4C2×Dic1311yescharacteristic
318C2×C42—521no—
319#101—522no—
320#100—262no—
321#100—262no—
322#99—134no—
323#99—264no—
324#99—262no—
325C22⋊C8—521no—
326C22⋊C8—521no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331C2×C13—21no—
332C2×C13—21no—
333C2×C13—41no—
334C2×C13—42no—
335C2×C13—41no—
336C2×C13—81no—
337C2×C13—82no—
338C2×C13#189811yescharacteristic
339C2×C13#212011yescharacteristic
340C2×C13#212211yescharacteristic
341C24—21no—
342C24—22no—
343C24—41no—
344C24—42no—
345C24—41no—
346C24—81no—
347C24C22⋊Dic1311yescharacteristic
348C24C22×Dic1311yescharacteristic
349C24C2×C22⋊D1311yescharacteristic
350#41—44no—
351#41—44no—
352#41—44no—
353#41—44no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—41no—
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—22no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—81no—
372C2×D4—81no—
373C2×D4D4×D1312yes—
374C2×D4#612912yes—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—81no—
393C22×C4—522no—
394C22×C4C22⋊Dic1311yescharacteristic
395C22×C4C22⋊Dic1311yescharacteristic
396C22×C4C22⋊Dic1311yescharacteristic
397C22×C4C2×C22⋊D1311yescharacteristic
398M16—522no—
399M16—522no—
400C2×C8—522no—
401C4⋊C4—524no—
402C22⋊C4—22no—
403C22⋊C4—42no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—41no—
409C22⋊C4—81no—
410C22⋊C4—81no—
411C22⋊C4—264no—
412C22⋊C4—522no—
413C42—524no—
414C13#1308711yescharacteristic, complemented, Sylow
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—22no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—82no—
438C23—84no—
439C23—84no—
440C23—82no—
441C23D4×Dic1312yes—
442C23#8818911yescharacteristic
443C23#8819111yescharacteristic
444C23C2×C22⋊Dic1311yescharacteristic
445Q8—44no—
446D4—42no—
447D4—42no—
448D4—44no—
449D4—42no—
450D4—42no—
451D4—82no—
452D4—82no—
453D4—84no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—82no—
481C2×C4—524no—
482C2×C4—1042no—
483C2×C4—1042no—
484C2×C4#8807412yes—
485C2×C4#8807411yescharacteristic
486C2×C4#8807411yescharacteristic
487C2×C4#8819312yes—
488C8—1042no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—22no—
494C22—41no—
495C22—41no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—42no—
507C22—41no—
508C22—41no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—84no—
514C22—84no—
515C22—81no—
516C22—81no—
517C22C2×#8807411yescharacteristic
518C22C2×#8807411yescharacteristic
519C22#1171888111yescharacteristic
520C4—22no—
521C4—21no—
522C4—21no—
523C4—22no—
524C4—41no—
525C4—81no—
526C4—81no—
527C4—81no—
528C4—1044no—
529C2—21no—
530C2—21no—
531C2—41no—
532C2—42no—
533C2—41no—
534C2—81no—
535C2—82no—
536C2#1311529011yescharacteristic
537C2#1311647711yescharacteristic
538C2#1311648611yescharacteristic
subgroupG/H
centerC22C2×#88074
commutatorC24×C13C22×C4
FrattiniC22×D4C22×D13
Fitting, prime radicalC13×#2404C2
socleC22×C13C2×#99
cosocleC22×D4×C13C23
hypercenter#2404D13
hypocenterC13#13087
LCSG ▷ C24×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2404

Complements

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

NH ≅ G/Ntype
1. C2×#11718901C2semidirect
2. C2×#11718128C2semidirect
3. #13114764C2semidirect
4. #13112954C2semidirect
7. C2×C13×#120C4semidirect
9. C22×C13×C22⋊C4C4semidirect
15. C13×#386C4semidirect
17. C13×#381C4semidirect
67. #88074D4semidirect
226. C2×#120Dic13semidirect
229. C22×C22⋊C4Dic13semidirect
240. #386Dic13semidirect
242. #381Dic13semidirect
414. C13#13087semidirect

Maximal subgroups

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

subgroupindex
C2×#117189012
C2×#117181282
#131147642
#131129542
C13×#24042
#1308713