Finite Group Databasebeta

← #15758680

Subgroups — #15758680

Order 3328.

630 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718597C211yesmaximal, characteristic
2#13114904C211yesmaximal, characteristic
3C2×#11718130C212yesmaximal
4#13112714C212yesmaximal
5C13×#2855C211yesmaximal, characteristic
6C2×C13×C22⋊Q8C2211yescharacteristic
7C22×C42×C13C2211yescharacteristic
8C22×C4⋊Dic13—21no—
9C13×#391C412yescomplemented
10C13×#389C412yescomplemented
11C13×#387—22no—
12C13×#385—21no—
13C13×#381—21no—
14C13×#380C2211yescharacteristic
15C2×#88071—21no—
16C2×#88071C2212yes—
17C2×C22⋊#1055C2212yes—
18#11718597—24no—
19#11718130—24no—
20#11718116—22no—
21C22×Q8×C13C2×C411yescharacteristic
22C23×C4×C13D411yescharacteristic
23C23×C4×C13D411yescharacteristic
24C23×C4×C13C2311yescharacteristic
25C23×Dic13—22no—
26C13×C22⋊Q8—22no—
27C13×C22⋊Q8—22no—
28C2×C13×C4⋊C4—22no—
29C2×C13×C4⋊C4—21no—
30C2×C13×C4⋊C4C2×C411yescharacteristic
31C2×C13×C4⋊C4C2×C412yes—
32C2×C13×C22⋊C4—21no—
33C2×C13×C22⋊C4—21no—
34C2×C42×C13—22no—
35C2×C42×C13—22no—
36C2×C42×C13C2×C412yes—
37C2×C42×C13D412yes—
38C2×C4⋊Dic13—24no—
39C2×C4⋊Dic13—42no—
40C2×C4⋊Dic13—42no—
41C22×#1055—22no—
42C13×#95—22no—
43C13×#95—22no—
44C13×#95—22no—
45C13×#95—21no—
46#88071—24no—
47#88071—22no—
48#88071D414yes—
49C22⋊#1055—22no—
50C22⋊#1055—22no—
51#9152—131nomaximal, Sylow
52C24×C13C2×D411yescharacteristic
53C2×Q8×C13—22no—
54C2×Q8×C13—42no—
55C22×C4×C13—21no—
56C22×C4×C13—21no—
57C22×C4×C13—21no—
58C22×C4×C13—21no—
59C22×C4×C13—21no—
60C22×C4×C13—21no—
61C22×C4×C13—22no—
62C22×C4×C13—21no—
63C22×C4×C13—21no—
64C22×C4×C13—42no—
65C22×C4×C13—42no—
66C22×C4×C13C22⋊C412yes—
67C22×C4×C13C22⋊C412yes—
68C22×C4×C13SD1612yes—
69C22×C4×C13Q1612yes—
70C22×C4×C13C22×C411yescharacteristic
71C22×C4×C13C2×D412yes—
72C22×C4×C13C2×D411yescharacteristic
73C22×C4×C13#4112yes—
74C22×Dic13—24no—
75C22×Dic13—44no—
76C22×Dic13—44no—
77C22×Dic13—42no—
78C22×Dic13—42no—
79C13×C4⋊C4—22no—
80C13×C4⋊C4—42no—
81C13×C4⋊C4—44no—
82C13×C4⋊C4—42no—
83C13×C22⋊C4—41no—
84C13×C22⋊C4—41no—
85C13×C22⋊C4—41no—
86C13×C22⋊C4—41no—
87C42×C13—24no—
88C42×C13—24no—
89C4⋊Dic13—48no—
90C2×#1055—44no—
91C2×#1055—42no—
92#2855D1311yescharacteristic
93C2×#382—131no—
94#1817—131no—
95C2×#328—132no—
96#1294—132no—
97C23×C13—22no—
98C23×C13—22no—
99C23×C13C2×C22⋊C411yescharacteristic
100C23×C13#12011yescharacteristic
101C23×C13#12311yescharacteristic
102C23×C13C2×SD1611yescharacteristic
103C23×C13C2×Q1611yescharacteristic
104C23×C13AGL(1,ℤ/8)11yescharacteristic
105C23×C13#13711yescharacteristic
106Q8×C13—82no—
107C2×C4×C13—22no—
108C2×C4×C13—22no—
109C2×C4×C13—22no—
110C2×C4×C13—22no—
111C2×C4×C13—22no—
112C2×C4×C13—22no—
113C2×C4×C13—22no—
114C2×C4×C13—22no—
115C2×C4×C13—22no—
116C2×C4×C13—22no—
117C2×C4×C13—22no—
118C2×C4×C13—42no—
119C2×C4×C13—42no—
120C2×C4×C13—42no—
121C2×C4×C13—42no—
122C2×C4×C13—41no—
123C2×C4×C13—41no—
124C2×C4×C13—42no—
125C2×C4×C13—42no—
126C2×C4×C13—42no—
127C2×C4×C13—42no—
128C2×C4×C13—42no—
129C2×C4×C13#10314yes—
130C2×C4×C13C4×D412yes—
131C2×C4×C13#12112yes—
132C2×Dic13—44no—
133C2×Dic13—44no—
134C2×Dic13—44no—
135C2×Dic13—84no—
136C2×Dic13—84no—
137#1055—82no—
138C2×C22⋊Q8C2×D1311yescharacteristic
139C22×C4⋊C4—261no—
140C22×C42C2×D1311yescharacteristic
141C2×C22⋊C8—132no—
142#391Dic1312yescomplemented
143#389Dic1312yescomplemented
144#387—22no—
145#385—21no—
146#382—264no—
147#381—21no—
148#380C2×D1311yescharacteristic
149C2×#95—132no—
150C2×#95—261no—
151#336—262no—
152#328—264no—
153C22×C13—21no—
154C22×C13—21no—
155C22×C13—21no—
156C22×C13—21no—
157C22×C13—22no—
158C22×C13—22no—
159C22×C13—22no—
160C22×C13—22no—
161C22×C13#38611yescharacteristic
162C22×C13#41012yes—
163C22×C13C2×#10311yescharacteristic
164C22×C13#41711yescharacteristic
165C22×C13#42112yes—
166C22×C13#45011yescharacteristic
167C22×C13#45111yescharacteristic
168C22×C13#48311yescharacteristic
169C22×C13#48411yescharacteristic
170C4×C13—24no—
171C4×C13—22no—
172C4×C13—22no—
173C4×C13—42no—
174C4×C13—81no—
175C4×C13—81no—
176C4×C13—82no—
177Dic13—84no—
178C22×Q8C2×Dic1311yescharacteristic
179C23×C4—262no—
180C23×C4C22⋊D1311yescharacteristic
181C23×C4C22⋊D1311yescharacteristic
182C23×C4C22×D1311yescharacteristic
183C22×C8—262no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—264no—
189C2×C4⋊C4—522no—
190C2×C4⋊C4—522no—
191C2×C4⋊C4C2×Dic1311yescharacteristic
192C2×C4⋊C4C2×Dic1312yes—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42C2×Dic1312yes—
198C2×C42C22⋊D1312yes—
199C22⋊C8—262no—
200C22⋊C8—262no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—21no—
205#95—134no—
206#95—264no—
207#95—262no—
208C2×C13—21no—
209C2×C13—21no—
210C2×C13—21no—
211C2×C13—21no—
212C2×C13#151212yes—
213C2×C13#151812yes—
214C2×C13#187211yescharacteristic
215C2×C13#189711yescharacteristic
216C2×C13#189911yescharacteristic
217C24C2×C22⋊D1311yescharacteristic
218C2×Q8—22no—
219C2×Q8—42no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—264no—
232C22×C4—524no—
233C22×C4—524no—
234C22×C4—522no—
235C22×C4—522no—
236C22×C4#610612yes—
237C22×C4#610712yes—
238C22×C4C22⋊Dic1312yes—
239C22×C4C22⋊Dic1312yes—
240C22×C4D4×D1312yes—
241C22×C4#612912yes—
242C22×C4C22×Dic1311yescharacteristic
243C22×C4C2×C22⋊D1311yescharacteristic
244C2×C8—524no—
245C2×C8—522no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—528no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C42—24no—
256C42—24no—
257C13#915211yescharacteristic, complemented, Sylow
258C23—22no—
259C23—22no—
260C23#8818911yescharacteristic
261C23#8819111yescharacteristic
262C23C2×#610611yescharacteristic
263C23C2×#610711yescharacteristic
264C23#8820311yescharacteristic
265C23#8820511yescharacteristic
266C23C2×C22⋊Dic1311yescharacteristic
267Q8—82no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—524no—
291C2×C4—524no—
292C2×C4—524no—
293C2×C4—1044no—
294C2×C4—1044no—
295C2×C4#8807514yes—
296C2×C4D4×Dic1312yes—
297C2×C4#8819312yes—
298C8—1042no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22#1171870411yescharacteristic
308C22#1171870511yescharacteristic
309C22#1171870711yescharacteristic
310C22#1171871111yescharacteristic
311C22#1171888111yescharacteristic
312C22C2×#8807511yescharacteristic
313C22#1171889311yescharacteristic
314C22#1171889712yes—
315C22#1171890112yes—
316C4—24no—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—82no—
321C4—81no—
322C4—81no—
323C4—1044no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2#1311337412yes—
329C2#1311340012yes—
330C2#1311514411yescharacteristic
331C2#1311528511yescharacteristic
332C2#1311529511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic13
commutatorC22×C4×C13C22×C4
FrattiniC23×C4C22×D13
FittingC13×#2855C2
socleC23×C13C2×C22⋊C4
prime radicalC24×C13C2×D4
cosocleC23×C4×C13C23
hypercenter#2855D13
hypocenterC13#9152
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
9. C13×#391C4semidirect
10. C13×#389C4semidirect
142. #391Dic13semidirect
143. #389Dic13semidirect
257. C13#9152semidirect

Maximal subgroups

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

subgroupindex
C2×#117185972
#131149042
C2×#117181302
#131127142
C13×#28552
#915213