Finite Group Databasebeta

← #15839181

Subgroups — #15839181

Order 3328.

1004 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 195 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
1#13125293C211yesmaximal, characteristic
2#13124958C212yesmaximal
3#13113976C214yesmaximal, complemented
4#13113914C214yesmaximal, complemented
5#13113562C211yesmaximal, characteristic, complemented
6#13113283C212yesmaximal, complemented
7C2×Q82×C13C211yesmaximal, characteristic, complemented
8Q82×C13—22no—
9Q82×C13—24no—
10Q82×C13C2214yes—
11C2×C13×C4⋊Q8—22no—
12C2×C13×C4⋊Q8—22no—
13C2×C13×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8×C13—22no—
15C2×C4×Q8×C13C2212yes—
16C2×#88196—22no—
17#11719182C2211yescharacteristic
18#11718888—22no—
19#11718753C2212yes—
20#11718747—22no—
21#11718746C2212yes—
22#11718688C2214yes—
23#11718687—24no—
24#11718682C2214yes—
25#11718628—22no—
26#11718587C2212yes—
27C4×#6099C2211yescharacteristic
28#11718138C2214yes—
29Q8⋊#1055C2218yes—
30C22×Q8×C13—22no—
31C22×Q8×C13D412yes—
32C22×Q8×C13D412yes—
33C2×#6127—21no—
34C13×C4⋊Q8—28no—
35C13×C4⋊Q8—28no—
36C13×C4⋊Q8—28no—
37C13×C4⋊Q8—28no—
38C13×C4⋊Q8—22no—
39C13×C4⋊Q8D412yes—
40C13×C4⋊Q8C2312yes—
41C4×Q8×C13—28no—
42C4×Q8×C13—28no—
43C4×Q8×C13—22no—
44C4×Q8×C13—22no—
45C4×Q8×C13D414yes—
46C4×Q8×C13C2314yes—
47C2×C13×C4⋊C4—24no—
48C2×C13×C4⋊C4—22no—
49C2×C13×C4⋊C4—22no—
50C2×C13×C4⋊C4D412yes—
51C2×C42×C13—22no—
52C2×C42×C13—22no—
53C2×C42×C13C2311yescharacteristic
54C2×#6107—24no—
55#88196—48no—
56C2×#6106—24no—
57C2×#6106—24no—
58#88184—22no—
59C2×#6099—21no—
60#88129—21no—
61#88128C2311yescharacteristic
62C4×D52—22no—
63#88123C2311yescharacteristic
64#88075—24no—
65C22⋊#1055—22no—
66#88049—24no—
67#88047—24no—
68C4⋊#1055C2314yes—
69#88043—22no—
70#88043—21no—
71C4×#1055C2312yes—
72#18546—131nomaximal, Sylow
73C2×Q8×C13—28no—
74C2×Q8×C13—28no—
75C2×Q8×C13—22no—
76C2×Q8×C13—22no—
77C2×Q8×C13—44no—
78C2×Q8×C13—84no—
79C2×Q8×C13C2×D414yes—
80C2×Q8×C13C2×D414yes—
81C22×C4×C13—22no—
82C22×C4×C13—22no—
83C22×C4×C13—22no—
84C22×C4×C13C2×D412yes—
85C22×C4×C13C2×D411yescharacteristic
86C2×C22⋊D13—41no—
87#6127—44no—
88C2×D52—22no—
89C2×D52—21no—
90C2×C4×D13—41no—
91C2×Dic26—21no—
92C13×C4⋊C4—216no—
93C13×C4⋊C4—24no—
94C13×C4⋊C4—24no—
95C13×C4⋊C4—22no—
96C13×C4⋊C4—24no—
97C13×C4⋊C4—24no—
98C13×C4⋊C4C2×D414yes—
99C42×C13—28no—
100C42×C13—24no—
101C42×C13—24no—
102C42×C13C2×D412yes—
103C42×C13C2×D411yescharacteristic
104C42×C13C2411yescharacteristic
105#6107—48no—
106#6106—48no—
107#6106—48no—
108#6103—42no—
109C4⋊Dic13—22no—
110#6099—24no—
111C2×#1055—22no—
112C2×#1055—24no—
113C2×Q82D1311yescharacteristic, complemented
114#3152—131no—
115#3112—132no—
116#1633—134no—
117#1623—134no—
118#1558—131no—
119#1491—132no—
120C23×C13#12011yescharacteristic
121C22×D13—41no—
122Q8×C13—28no—
123Q8×C13—24no—
124Q8×C13—44no—
125Q8×C13—82no—
126Q8×C13—88no—
127Q8×C13#13718yescomplemented
128C2×C4×C13—28no—
129C2×C4×C13—24no—
130C2×C4×C13—22no—
131C2×C4×C13—24no—
132C2×C4×C13—24no—
133C2×C4×C13—24no—
134C2×C4×C13—44no—
135C2×C4×C13—44no—
136C2×C4×C13#12012yes—
137C2×C4×C13#12011yescharacteristic
138C2×C4×C13C22×D411yescharacteristic
139C2×C4×C13C22×D412yes—
140C22⋊D13—82no—
141C2×Dic13—41no—
142D52—42no—
143D52—44no—
144C4×D13—82no—
145Dic26—42no—
146#1055—24no—
147#1055—44no—
148C2×#137—262no—
149Q82—22no—
150Q82—24no—
151Q82C2×D1314yes—
152#532—131no—
153C2×C4⋊Q8—22no—
154C2×C4⋊Q8—22no—
155C2×C4⋊Q8C2×D1311yescharacteristic
156C2×C4×Q8—22no—
157C2×C4×Q8C2×D1312yes—
158#497—262no—
159#494—132no—
160#492—132no—
161#480—262no—
162#477—134no—
163#460—134no—
164#448—262no—
165#440—264no—
166#432—132no—
167C4×M16—131no—
168#333—134no—
169Q8⋊C8—138no—
170C22×C13—22no—
171C22×C13#45312yes—
172C22×C13C2×#12011yescharacteristic
173C2×D13—82no—
174C2×D13—81no—
175C4×C13—28no—
176C4×C13—22no—
177C4×C13—21no—
178C4×C13—21no—
179C4×C13—44no—
180C4×C13—44no—
181C4×C13C2×#13714yes—
182Dic13—81no—
183C2×#41—261no—
184C22×Q8—22no—
185C22×Q8C22⋊D1312yes—
186C22×Q8C22⋊D1312yes—
187#137—528no—
188C2×Q16—264no—
189C2×SD16—264no—
190C2×SD16—264no—
191C2×M16—261no—
192C4⋊Q8—28no—
193C4⋊Q8—28no—
194C4⋊Q8—28no—
195C4⋊Q8—28no—
196C4⋊Q8—22no—
197C4⋊Q8—131no—
198C4⋊Q8C22⋊D1312yes—
199C4⋊Q8C22×D1312yes—
200#127—131no—
201#124—261no—
202#121—262no—
203C4×Q8—28no—
204C4×Q8—28no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8C22⋊D1314yes—
208C4×Q8C22×D1314yes—
209C4×D4—262no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4C22⋊D1312yes—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42C22×D1311yescharacteristic
217#107—264no—
218C4⋊C8—134no—
219#103—264no—
220#102—264no—
221C22⋊C8—262no—
222#97—261no—
223#97—262no—
224C4×C8—132no—
225C2×C13—22no—
226C2×C13#300412yes—
227C2×C13C2×#45311yescharacteristic
228D13—161no—
229#41—524no—
230C2×Q8—28no—
231C2×Q8—28no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—44no—
235C2×Q8—84no—
236C2×Q8—261no—
237C2×Q8C2×C22⋊D1314yes—
238C2×Q8C2×C22⋊D1314yes—
239C2×D4—262no—
240C2×D4—261no—
241C2×D4—521no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—521no—
246C22×C4C2×C22⋊D1312yes—
247C22×C4C2×C22⋊D1311yescharacteristic
248Q16—528no—
249SD16—528no—
250SD16—528no—
251M16—264no—
252C2×C8—262no—
253C2×C8—264no—
254C4⋊C4—216no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—262no—
261C4⋊C4C2×C22⋊D1314yes—
262C22⋊C4—522no—
263C42—28no—
264C42—24no—
265C42—24no—
266C42C2×C22⋊D1312yes—
267C42C2×C22⋊D1311yescharacteristic
268C42C23×D1311yescharacteristic
269C13#1854611yescharacteristic, complemented, Sylow
270C23—521no—
271C23#8820711yescharacteristic
272Q8—28no—
273Q8—24no—
274Q8—44no—
275Q8—82no—
276Q8—88no—
277Q8—522no—
278Q8#8819618yescomplemented
279D4—522no—
280D4—524no—
281D4—1042no—
282C2×C4—28no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—521no—
291C2×C4—1042no—
292C2×C4#8820712yes—
293C2×C4#8820711yescharacteristic
294C2×C4C22×C22⋊D1311yescharacteristic
295C2×C4C22×C22⋊D1312yes—
296C8—264no—
297C8—524no—
298C22—22no—
299C22—1041no—
300C22—1042no—
301C22#1171890212yes—
302C22C2×#8820711yescharacteristic
303C4—21no—
304C4—21no—
305C4—28no—
306C4—22no—
307C4—44no—
308C4—44no—
309C4—1041no—
310C4C2×#8819614yes—
311C2—22no—
312C2—2081no—
313C2#1312407012yes—
314C2C2×#1171890211yescharacteristic
subgroupG/H
centerC22C2×#88207
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC2×Q82×C13C2
socleC22×C13C2×#120
prime radical#11719182C22
hypercenterC2×Q82D13
hypocenterC13#18546
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
3. #13113976C2semidirect
4. #13113914C2semidirect
5. #13113562C2semidirect
6. #13113283C2semidirect
7. C2×Q82×C13C2semidirect
113. C2×Q82D13semidirect
127. Q8×C13#137semidirect
269. C13#18546semidirect
278. Q8#88196semidirect

Maximal subgroups

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

subgroupindex
#131252932
#131249582
#131139762
#131139142
#131135622
#131132832
C2×Q82×C132
#1854613