Finite Group Databasebeta

← #15873315

Subgroups — #15873315

Order 3328.

830 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718585C211yesmaximal, characteristic
2#13117243C211yesmaximal, characteristic
3#13115602C214yesmaximal, complemented
4#13115599C212yesmaximal, complemented
5#13115594C212yesmaximal, complemented
6#13114867C214yesmaximal
7C13×#2283C211yesmaximal, characteristic
8C22×C42×C13C2211yescharacteristic
9C2×#88180—21no—
10C22×#6099C2212yes—
11C13×#383—21no—
12C13×#382C2212yes—
13C13×#381—21no—
14C13×#380C2212yes—
15C4×C13×C4⋊C4C2211yescharacteristic
16C4×C13×C4⋊C4C2211yescharacteristic
17C4×C13×C22⋊C4—21no—
18C13×#376—21no—
19#11718873—21no—
20#11718873C2212yes—
21C2×#88048C2214yes—
22C2×#88047C2214yes—
23#11718597—21no—
24#11718596C2212yes—
25C4×C4⋊Dic13—21no—
26C4×C4⋊Dic13C2212yes—
27#11718585C418yescomplemented
28C2×C4⋊#1055C2212yes—
29C2×C4⋊#1055C2212yes—
30#11718121C2214yes—
31#11718121C2214yes—
32C23×C4×C13D411yescharacteristic
33C23×C4×C13D411yescharacteristic
34C23×C4×C13C2311yescharacteristic
35C2×C13×C4⋊C4C2312yes—
36C2×C13×C4⋊C4C2312yes—
37C2×C13×C22⋊C4—22no—
38C2×C42×C13—21no—
39C2×C42×C13—21no—
40C2×C42×C13C2×C414yes—
41C2×C42×C13D411yescharacteristic
42C2×C42×C13D411yescharacteristic
43C2×C42×C13Q814yes—
44C2×C42×C13C2311yescharacteristic
45C2×C42×C13C2311yescharacteristic
46C2×C22⋊Dic13—22no—
47#88180—44no—
48C2×C4⋊Dic13—22no—
49C2×C4⋊Dic13—22no—
50C2×C4⋊Dic13C2314yes—
51C2×C4×Dic13—22no—
52C2×#6099—44no—
53C2×#6099C2×C418yes—
54C22×#1055C2314yes—
55C13×#95—22no—
56C13×#95—21no—
57C13×#95—21no—
58C13×#95—22no—
59C13×#95—22no—
60#88071—22no—
61#88071—22no—
62#88048—44no—
63#88047—44no—
64C4⋊#1055C2×C418yes—
65C4⋊#1055C2×C418yes—
66#21963—131nomaximal, Sylow
67C24×C13C2×D411yescharacteristic
68C22×C4×C13—22no—
69C22×C4×C13—22no—
70C22×C4×C13—22no—
71C22×C4×C13—22no—
72C22×C4×C13—22no—
73C22×C4×C13—22no—
74C22×C4×C13—42no—
75C22×C4×C13C4⋊C414yes—
76C22×C4×C13C4⋊C414yes—
77C22×C4×C13C22×C412yes—
78C22×C4×C13C2×D411yescharacteristic
79C22×C4×C13C2×D411yescharacteristic
80C22×C4×C13C2×D411yescharacteristic
81C22×C4×C13C2×D411yescharacteristic
82C22×C4×C13C2×D411yescharacteristic
83C22×C4×C13C2×Q812yes—
84C22×C4×C13C2411yescharacteristic
85C22×Dic13—24no—
86C13×C4⋊C4—22no—
87C13×C4⋊C4—22no—
88C13×C4⋊C4—22no—
89C13×C4⋊C4—22no—
90C13×C22⋊C4—44no—
91C42×C13—42no—
92C42×C13—42no—
93C42×C13C4⋊C414yes—
94C42×C13C4⋊C414yes—
95C42×C13C22×C412yes—
96C42×C13C22×C412yes—
97C42×C13C2×Q812yes—
98C42×C13C2×Q812yes—
99C22⋊Dic13—44no—
100C4⋊Dic13—24no—
101C4⋊Dic13—24no—
102C4⋊Dic13—44no—
103C4⋊Dic13—44no—
104C4×Dic13—44no—
105#6099—48no—
106C2×#1055—24no—
107C2×#1055—24no—
108C2×#1055C22×C418yes—
109C2×#423—131no—
110#2285—131no—
111#2283D1311yescharacteristic
112#1949—134no—
113#1948—132no—
114#1947—132no—
115#1809—134no—
116C23×C13—22no—
117C23×C13—21no—
118C23×C13—21no—
119C23×C13—22no—
120C23×C13C2×C4⋊C412yes—
121C23×C13C22×D411yescharacteristic
122C2×C4×C13—22no—
123C2×C4×C13—21no—
124C2×C4×C13—21no—
125C2×C4×C13—22no—
126C2×C4×C13—22no—
127C2×C4×C13—22no—
128C2×C4×C13—22no—
129C2×C4×C13—24no—
130C2×C4×C13—28no—
131C2×C4×C13—44no—
132C2×C4×C13—42no—
133C2×C4×C13—42no—
134C2×C4×C13—42no—
135C2×C4×C13—42no—
136C2×C4×C13—42no—
137C2×C4×C13—42no—
138C2×C4×C13C2×C4⋊C412yes—
139C2×C4×C13C2×C4⋊C412yes—
140C2×C4×C13C2×C4⋊C412yes—
141C2×C4×C13C2×C4⋊C412yes—
142C2×C4×C13C2×C4⋊C412yes—
143C2×C4×C13AGL(1,ℤ/8)12yes—
144C2×C4×C13#13712yes—
145C2×C4×C13C23×C411yescharacteristic
146C2×C4×C13C22×Q811yescharacteristic
147C2×C4×C13H(5,𝔽2)12yes—
148C2×C4×C132−1+412yes—
149C2×Dic13—44no—
150C2×Dic13—44no—
151C2×Dic13—44no—
152#1055—28no—
153C22×M16—132no—
154C2×#117—261no—
155C22×C42C2×D1311yescharacteristic
156C2×#107—134no—
157C2×#106—134no—
158#423—138no—
159C2×C4⋊C8—132no—
160C2×C4⋊C8—132no—
161#384—132no—
162#383—21no—
163#382—261no—
164#382C2×D1312yes—
165#381—21no—
166#380—132no—
167#380—261no—
168#380C2×D1312yes—
169C4×C4⋊C4—132no—
170C4×C4⋊C4—261no—
171C4×C4⋊C4C2×D1311yescharacteristic
172C4×C4⋊C4C2×D1311yescharacteristic
173C4×C22⋊C4—21no—
174#376—21no—
175#340—134no—
176#340—134no—
177C22×C13—22no—
178C22×C13—22no—
179C22×C13—22no—
180C22×C13—42no—
181C22×C13—42no—
182C22×C13C22×C4⋊C411yescharacteristic
183C22×C13#53712yes—
184C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
185C22×C13C2×#13711yescharacteristic
186C22×C13#57512yes—
187C4×C13—22no—
188C4×C13—22no—
189C4×C13—22no—
190C4×C13—22no—
191C4×C13—82no—
192C4×C13—82no—
193C4×C13#42814yes—
194C4×C13#51811yescharacteristic
195C4×C13#51911yescharacteristic
196C4×C13#54412yes—
197Dic13—84no—
198C23×C4C22⋊D1311yescharacteristic
199C23×C4C22⋊D1311yescharacteristic
200C23×C4C22×D1311yescharacteristic
201C2×M16—138no—
202C2×M16—524no—
203C22×C8—134no—
204#117—524no—
205C2×C4⋊C4—134no—
206C2×C4⋊C4—262no—
207C2×C4⋊C4—262no—
208C2×C4⋊C4C22×D1312yes—
209C2×C4⋊C4C22×D1312yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—262no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—262no—
215C2×C42Dic2614yes—
216C2×C42C4×D1314yes—
217C2×C42C22⋊D1311yescharacteristic
218C2×C42C22⋊D1311yescharacteristic
219C2×C42C22×D1311yescharacteristic
220C2×C42C22×D1311yescharacteristic
221#107—524no—
222#106—524no—
223C4⋊C8—138no—
224C4⋊C8—138no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95—262no—
231#95—262no—
232C2×C13—42no—
233C2×C13#231611yescharacteristic
234C2×C13C2×#42811yescharacteristic
235C2×C13#291111yescharacteristic
236C2×C13#325112yes—
237C2×C13#325212yes—
238C24C2×C22⋊D1311yescharacteristic
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—42no—
246C22×C4—264no—
247C22×C4#610114yes—
248C22×C4#610114yes—
249C22×C4C2×Dic2612yes—
250C22×C4C2×C4×D1312yes—
251C22×C4C2×C22⋊D1311yescharacteristic
252C22×C4C2×C22⋊D1311yescharacteristic
253C22×C4C2×C22⋊D1311yescharacteristic
254C22×C4C2×C22⋊D1311yescharacteristic
255C22×C4C2×C22⋊D1311yescharacteristic
256C22×C4C23×D1311yescharacteristic
257M16—528no—
258C2×C8—138no—
259C2×C8—264no—
260C2×C8—264no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—264no—
266C4⋊C4—264no—
267C4⋊C4—524no—
268C4⋊C4—524no—
269C22⋊C4—44no—
270C22⋊C4—524no—
271C42—42no—
272C42—42no—
273C42—524no—
274C42#610114yes—
275C42#610114yes—
276C42C2×Dic2612yes—
277C42C2×Dic2612yes—
278C42C2×C4×D1312yes—
279C42C2×C4×D1312yes—
280C13#2196311yescharacteristic, complemented, Sylow
281C23—21no—
282C23—22no—
283C23—21no—
284C23—22no—
285C23C2×#610112yes—
286C23C22×C22⋊D1311yescharacteristic
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—28no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—524no—
304C2×C4—524no—
305C2×C4—524no—
306C2×C4C2×#610112yes—
307C2×C4C2×#610112yes—
308C2×C4C2×#610112yes—
309C2×C4C2×#610112yes—
310C2×C4C2×#610112yes—
311C2×C4#8820312yes—
312C2×C4#8820512yes—
313C2×C4C22×Dic2611yescharacteristic
314C2×C4C22×C4×D1311yescharacteristic
315C2×C4#8825612yes—
316C2×C4#8825712yes—
317C8—268no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—42no—
322C22—42no—
323C22C22×#610111yescharacteristic
324C22C2×#8820311yescharacteristic
325C22#1171952712yes—
326C22C2×#8820511yescharacteristic
327C22#1171953812yes—
328C4—22no—
329C4—22no—
330C4—22no—
331C4—22no—
332C4—82no—
333C4—82no—
334C4—1044no—
335C4#1171865614yes—
336C4#1171922111yescharacteristic
337C4#1171922412yes—
338C4#1171922611yescharacteristic
339C2—42no—
340C2#1311745211yescharacteristic
341C2C2×#1171865611yescharacteristic
342C2#1312342511yescharacteristic
343C2#1312619812yes—
344C2#1312620912yes—
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2283C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#2283D13
hypocenterC13#21963
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2283

Complements

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

NH ≅ G/Ntype
3. #13115602C2semidirect
4. #13115599C2semidirect
5. #13115594C2semidirect
27. #11718585C4semidirect
280. C13#21963semidirect

Maximal subgroups

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

subgroupindex
C2×#117185852
#131172432
#131156022
#131155992
#131155942
#131148672
C13×#22832
#2196313