Finite Group Databasebeta

← #15846430

Subgroups — #15846430

Order 3328.

640 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13126148C211yesmaximal, characteristic
2#13125964C211yesmaximal, characteristic
3#13123582C211yesmaximal, characteristic
4#13114356C212yesmaximal, complemented
5#13114332C212yesmaximal, complemented
6#13114232C212yesmaximal, complemented
7#13114208C212yesmaximal, complemented
8#13113588C211yesmaximal, characteristic, complemented
9#13113493C211yesmaximal, characteristic, complemented
10C2×#11718138C211yesmaximal, characteristic, complemented
11C13×#3165C211yesmaximal, characteristic, complemented
12C13×#533C2211yescharacteristic
13C2×C13×C4⋊Q8C2211yescharacteristic
14C2×#88161—21no—
15#11719182C2211yescharacteristic
16C13×#501—22no—
17C13×#500C2212yes—
18C13×#498C2212yes—
19C13×#428—21no—
20C13×#428—21no—
21C4×C13×M16C2211yescharacteristic
22#11718790—21no—
23#11718749—21no—
24#11718745C2212yes—
25#11718745C2212yes—
26#11718745C2212yes—
27#11718744C2212yes—
28#11718740—21no—
29#11718709—22no—
30#11718688—21no—
31#11718681—21no—
32#11718629—22no—
33#11718586C2211yescharacteristic
34C2×C4⋊#1055C2211yescharacteristic
35C2×C4⋊#1055C2211yescharacteristic
36#11718367—22no—
37#11718349C2212yes—
38C8×D52C2212yes—
39#11718139C2211yescharacteristic
40#11718139C2211yescharacteristic
41#11718138C2212yes—
42#11718138C2211yescharacteristic
43#11718138C2212yes—
44#11718138C2211yescharacteristic
45#11718050C2212yes—
46#11718049C2212yes—
47C22×Q8×C13—21no—
48C2×#6127—21no—
49C2×C13×M16D411yescharacteristic
50C2×C13×M16D411yescharacteristic
51C13×C4⋊Q8—21no—
52C13×C4⋊Q8—21no—
53C13×C4⋊Q8—21no—
54C13×C4⋊Q8C2311yescharacteristic
55C13×C4⋊Q8C2312yes—
56C13×C4⋊Q8C2311yescharacteristic
57C13×#125—21no—
58C13×C22⋊Q8—21no—
59C13×C22⋊Q8—21no—
60C4×Q8×C13—21no—
61C4×Q8×C13—21no—
62C13×#117—21no—
63C2×C13×C4⋊C4—21no—
64C2×C13×C4⋊C4—21no—
65C2×C42×C13C2311yescharacteristic
66C2×#6106—22no—
67#88184—21no—
68#88184D412yes—
69C22×#1055—21no—
70C22×#1055—21no—
71#88161—44no—
72C2×#6094—22no—
73C2×C8×D13—22no—
74#88129—21no—
75#88128D411yescharacteristic
76C4×D52—21no—
77C4×D52C2312yes—
78#88123D411yescharacteristic
79C13×#107—22no—
80C13×#107—22no—
81C13×#106—22no—
82C13×#106—22no—
83C13×#97D412yes—
84C4×C8×C13C2312yes—
85#88075—22no—
86C22⋊#1055—21no—
87#88060—22no—
88#88060—22no—
89#88049—22no—
90#88049—22no—
91#88049—22no—
92#88048—22no—
93#88048—22no—
94#88048—22no—
95C4⋊#1055—21no—
96C4⋊#1055—21no—
97C4⋊#1055C2312yes—
98C4⋊#1055C2311yescharacteristic
99C4⋊#1055C2311yescharacteristic
100C4⋊#1055C2311yescharacteristic
101C4⋊#1055C2311yescharacteristic
102C4×#1055—21no—
103#19233—131nomaximal, Sylow
104C2×Q8×C13—21no—
105C2×Q8×C13—21no—
106C2×Q8×C13—22no—
107C2×Q8×C13—41no—
108C2×Q8×C13—82no—
109C22×C4×C13—41no—
110C22×C4×C13C2×D411yescharacteristic
111C22×C4×C13C2×D411yescharacteristic
112C22×C4×C13#4111yescharacteristic
113C2×C22⋊D13—22no—
114#6127—42no—
115#6127—42no—
116C2×D52—21no—
117C2×D52—21no—
118C2×D52C2×D412yes—
119C2×C4×D13—22no—
120C2×Dic26—21no—
121C13×M16—22no—
122C13×M16SD1614yescomplemented
123C2×C8×C13C2×D412yes—
124C2×C8×C13C2×D412yes—
125C13×C4⋊C4—21no—
126C13×C4⋊C4—22no—
127C13×C4⋊C4—21no—
128C13×C4⋊C4—22no—
129C13×C4⋊C4—21no—
130C13×C4⋊C4—21no—
131C13×C4⋊C4—41no—
132C13×C4⋊C4—41no—
133C13×C4⋊C4—41no—
134C13×C4⋊C4—41no—
135C13×C22⋊C4—41no—
136C42×C13—41no—
137C42×C13C2×D411yescharacteristic
138C42×C13C2×D411yescharacteristic
139C42×C13#4111yescharacteristic
140C42×C13C2411yescharacteristic
141#6106—82no—
142#6103—22no—
143#6103—22no—
144C4⋊Dic13—21no—
145C4⋊Dic13C2×D412yes—
146C2×#1055—21no—
147C2×#1055—21no—
148C2×#1055—22no—
149C2×#1055—21no—
150C2×#1055—21no—
151C2×#1055—42no—
152C2×#1055—81no—
153#6094—44no—
154C8×D13—44no—
155#3246—131no—
156#3227—131no—
157#3165D1311yescharacteristic, complemented
158#2935—131no—
159#1695—132no—
160#1691—132no—
161#1675—132no—
162#1671—132no—
163#1565—131no—
164#1542—131no—
165C2×#333—131no—
166C23×C13#12111yescharacteristic
167C22×D13—22no—
168Q8×C13—41no—
169Q8×C13—41no—
170Q8×C13—82no—
171Q8×C13—81no—
172Q8×C13—82no—
173C2×C4×C13—21no—
174C2×C4×C13—21no—
175C2×C4×C13—41no—
176C2×C4×C13—41no—
177C2×C4×C13—41no—
178C2×C4×C13—42no—
179C2×C4×C13—82no—
180C2×C4×C13#12111yescharacteristic
181C2×C4×C13#12111yescharacteristic
182C2×C4×C13#12111yescharacteristic
183C2×C4×C13C2×SD1612yes—
184C2×C4×C13#13712yes—
185C2×C4×C13C22×D411yescharacteristic
186C2×C4×C13C22×D411yescharacteristic
187C2×C4×C13C2×#4111yescharacteristic
188C22⋊D13—44no—
189C2×Dic13—22no—
190D52—22no—
191D52—22no—
192D52—42no—
193D52—81no—
194C4×D13—42no—
195C4×D13—42no—
196Dic26—42no—
197C8×C13—22no—
198C8×C13C2×SD1614yes—
199#1055—41no—
200#1055—41no—
201#1055—41no—
202#1055—42no—
203#1055—41no—
204C2×#131—261no—
205#533C2×D1311yescharacteristic
206#532—131no—
207C2×C4⋊Q8C2×D1311yescharacteristic
208#501—22no—
209#500C2×D1312yes—
210#498—261no—
211#498C2×D1312yes—
212#486—261no—
213#481—262no—
214#476—132no—
215#476—132no—
216#476—132no—
217#475—261no—
218#465—262no—
219#461—132no—
220#460—261no—
221#436—132no—
222#435—262no—
223C8×D4—132no—
224#431—131no—
225#428—21no—
226#428—21no—
227C2×C4⋊C8—131no—
228C2×C4⋊C8—131no—
229#408—261no—
230C4×M16C2×D1311yescharacteristic
231#335—132no—
232#334—132no—
233#333—132no—
234#333—132no—
235#333—131no—
236#333—131no—
237#332—131no—
238#332—131no—
239C22×C13—21no—
240C22×C13—21no—
241C22×C13#46112yes—
242C22×C13C2×#12111yescharacteristic
243C2×D13—42no—
244C2×D13—42no—
245C2×D13—42no—
246C4×C13—21no—
247C4×C13—21no—
248C4×C13—21no—
249C4×C13—81no—
250C4×C13—82no—
251C4×C13—81no—
252C4×C13#46112yes—
253C4×C13C22×SD1611yescharacteristic
254C4×C13C2×#13711yescharacteristic
255C4×C13#57711yescharacteristic
256C4×C13#57811yescharacteristic
257Dic13—42no—
258C2×#41—261no—
259C22×Q8—21no—
260C2×SD16—262no—
261#131—524no—
262C2×M16—262no—
263C2×M16C22⋊D1311yescharacteristic
264C2×M16C22⋊D1311yescharacteristic
265C22×C8—261no—
266C22×C8—261no—
267C22×C8—262no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8—131no—
272C4⋊Q8C22×D1312yes—
273C4⋊Q8C22×D1311yescharacteristic
274C4⋊Q8C22×D1311yescharacteristic
275#127—131no—
276#125—21no—
277#124—261no—
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280#121—132no—
281#121—261no—
282C4×Q8—21no—
283C4×Q8—21no—
284C4×D4—132no—
285C4×D4—261no—
286#117—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C42C22×D1311yescharacteristic
290#107—22no—
291#107—22no—
292#106—22no—
293#106—22no—
294#106—262no—
295#106—262no—
296#106—262no—
297C4⋊C8—131no—
298C4⋊C8—131no—
299C4⋊C8—131no—
300C4⋊C8—131no—
301C4⋊C8—132no—
302C4⋊C8—261no—
303C4⋊C8—261no—
304#103—262no—
305#102—262no—
306#102—262no—
307#102—262no—
308C22⋊C8—261no—
309C22⋊C8—262no—
310C22⋊C8—262no—
311#97C22⋊D1312yes—
312C4×C8—261no—
313C4×C8C22×D1312yes—
314C2×C13—22no—
315C2×C13C2×#46111yescharacteristic
316C2×C13#306311yescharacteristic
317C2×C13#306811yescharacteristic
318D13—82no—
319#41—522no—
320#41—522no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—22no—
324C2×Q8—41no—
325C2×Q8—82no—
326C2×Q8—261no—
327C2×D4—132no—
328C2×D4—262no—
329C2×D4—261no—
330C2×D4—261no—
331C22×C4—41no—
332C22×C4—262no—
333C22×C4D4×D1311yescharacteristic
334C22×C4#612911yescharacteristic
335C22×C4C2×C22⋊D1311yescharacteristic
336SD16—1042no—
337M16—22no—
338M16—524no—
339M16#610614yescomplemented
340C2×C8—261no—
341C2×C8—261no—
342C2×C8—262no—
343C2×C8—261no—
344C2×C8—261no—
345C2×C8—522no—
346C2×C8—524no—
347C2×C8—1041no—
348C2×C8C2×C22⋊D1312yes—
349C2×C8C2×C22⋊D1312yes—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—132no—
361C4⋊C4—261no—
362C22⋊C4—41no—
363C22⋊C4—262no—
364C22⋊C4—262no—
365C42—41no—
366C42D4×D1311yescharacteristic
367C42#612911yescharacteristic
368C42C2×C22⋊D1311yescharacteristic
369C42C23×D1311yescharacteristic
370C13#1923311yescharacteristic, complemented, Sylow
371C23—262no—
372C23#8819211yescharacteristic
373Q8—41no—
374Q8—41no—
375Q8—82no—
376Q8—81no—
377Q8—82no—
378Q8—522no—
379D4—262no—
380D4—262no—
381D4—524no—
382D4—522no—
383D4—1041no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—262no—
392C2×C4—522no—
393C2×C4—522no—
394C2×C4#8819211yescharacteristic
395C2×C4#8819211yescharacteristic
396C2×C4#8819211yescharacteristic
397C2×C4C2×#610612yes—
398C2×C4#8819612yes—
399C2×C4C2×D4×D1311yescharacteristic
400C2×C4C2×#612911yescharacteristic
401C2×C4C22×C22⋊D1311yescharacteristic
402C8—22no—
403C8—521no—
404C8—522no—
405C8—521no—
406C8—521no—
407C8—521no—
408C8C2×#610614yes—
409C22—21no—
410C22—21no—
411C22—522no—
412C22—522no—
413C22—522no—
414C22#1171874412yes—
415C22C2×#8819211yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—81no—
420C4—82no—
421C4—81no—
422C4—522no—
423C4#1171874412yes—
424C4#1171948311yescharacteristic
425C4#1171948411yescharacteristic
426C4C22×#610611yescharacteristic
427C4C2×#8819611yescharacteristic
428C2—22no—
429C2—1042no—
430C2C2×#1171874411yescharacteristic
431C2#1312452111yescharacteristic
432C2#1312456011yescharacteristic
subgroupG/H
centerC22C2×#88192
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC13×#3165C2
socleC22×C13C2×#121
prime radical#11719182C22
hypercenter#3165D13
hypocenterC13#19233
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
4. #13114356C2semidirect
5. #13114332C2semidirect
6. #13114232C2semidirect
7. #13114208C2semidirect
8. #13113588C2semidirect
9. #13113493C2semidirect
10. C2×#11718138C2semidirect
11. C13×#3165C2semidirect
122. C13×M16SD16semidirect
157. #3165D13semidirect
339. M16#6106semidirect
370. C13#19233semidirect

Maximal subgroups

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

subgroupindex
#131261482
#131259642
#131235822
#131143562
#131143322
#131142322
#131142082
#131135882
#131134932
C2×#117181382
C13×#31652
#1923313