Finite Group Databasebeta

← #15872850

Subgroups — #15872850

Order 3328.

1366 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719182C211yesmaximal, characteristic
2C2×#11718585C211yesmaximal, characteristic, complemented
3#13115788C212yesmaximal, complemented
4#13115763C212yesmaximal, complemented
5#13115272C214yesmaximal, complemented
6#13114843C214yesmaximal, complemented
7C13×#2283C211yesmaximal, characteristic, complemented
8C22×#6127—21no—
9C22×C42×C13C2211yescharacteristic
10C2×#88184—21no—
11C2×#88184C2212yes—
12C22×#6099C2212yes—
13#11719182—24no—
14#11719182C418yescomplemented
15C2×#88129—21no—
16C2×#88128C2211yescharacteristic
17C2×C4×D52—21no—
18C2×C4×D52C2212yes—
19C2×#88123C2211yescharacteristic
20C13×#383—21no—
21C13×#382C2212yes—
22C13×#381—21no—
23C13×#380C2212yes—
24C4×C13×C4⋊C4C2212yes—
25C4×C13×C22⋊C4—21no—
26C13×#376—21no—
27C2×#88049C2218yes—
28#11718585—24no—
29C2×C4⋊#1055C2212yes—
30C2×C4⋊#1055C2212yes—
31#11718121C2218yes—
32C23×C4×C13D411yescharacteristic
33C23×C4×C13D411yescharacteristic
34C23×C4×C13C2311yescharacteristic
35C22×C22⋊D13—22no—
36C2×#6127—28no—
37C2×#6127—48no—
38C22×D52—21no—
39C22×D52—21no—
40C22×D52C2312yes—
41C22×C4×D13—22no—
42C22×Dic26—21no—
43C2×C13×C4⋊C4C2314yes—
44C2×C13×C22⋊C4—22no—
45C2×C42×C13—22no—
46C2×C42×C13C2×C414yes—
47C2×C42×C13D414yes—
48C2×C42×C13D411yescharacteristic
49C2×C42×C13D411yescharacteristic
50C2×C42×C13C2311yescharacteristic
51C2×C42×C13C2311yescharacteristic
52#88184—28no—
53#88184—24no—
54#88184C2×C418yes—
55C2×#6103—22no—
56C2×#6103—22no—
57C2×C4⋊Dic13—21no—
58C2×C4⋊Dic13C2312yes—
59C2×#6099—24no—
60C2×#6099—44no—
61C22×#1055C2314yes—
62#88129—28no—
63#88128—22no—
64#88128C2×C414yes—
65C4×D52—28no—
66C4×D52—24no—
67C4×D52C2×C418yes—
68#88123—22no—
69#88123C2×C414yes—
70C13×#95—22no—
71C13×#95—22no—
72C13×#95—22no—
73C13×#95—22no—
74#88049—48no—
75C4⋊#1055—24no—
76C4⋊#1055—24no—
77#21891—131nomaximal, Sylow
78C24×C13C2×D411yescharacteristic
79C23×D13—22no—
80C22×C4×C13—24no—
81C22×C4×C13—22no—
82C22×C4×C13—22no—
83C22×C4×C13—24no—
84C22×C4×C13—42no—
85C22×C4×C13C22⋊C414yes—
86C22×C4×C13C22⋊C414yes—
87C22×C4×C13C22×C412yes—
88C22×C4×C13C2×D412yes—
89C22×C4×C13C2×D411yescharacteristic
90C22×C4×C13C2×D411yescharacteristic
91C22×C4×C13C2×D411yescharacteristic
92C22×C4×C13C2×D411yescharacteristic
93C22×C4×C13C2×D411yescharacteristic
94C22×C4×C13C2411yescharacteristic
95C2×C22⋊D13—28no—
96C2×C22⋊D13—48no—
97C22×Dic13—22no—
98#6127—432no—
99C2×D52—24no—
100C2×D52—24no—
101C2×D52—24no—
102C2×D52—24no—
103C2×D52—22no—
104C2×D52—44no—
105C2×D52—44no—
106C2×D52C22×C414yes—
107C2×C4×D13—28no—
108C2×C4×D13—48no—
109C2×Dic26—24no—
110C2×Dic26—44no—
111C13×C4⋊C4—24no—
112C13×C4⋊C4—24no—
113C13×C22⋊C4—44no—
114C42×C13—44no—
115C42×C13C22⋊C414yes—
116C42×C13C22⋊C414yes—
117C42×C13C22×C412yes—
118C42×C13C22×C412yes—
119C42×C13C2×D412yes—
120C42×C13C2×D412yes—
121#6103—28no—
122#6103—28no—
123C4⋊Dic13—24no—
124C4⋊Dic13—22no—
125C4⋊Dic13C22×C414yes—
126#6099—48no—
127C2×#1055—24no—
128C2×#1055—28no—
129C2×#532—131no—
130C2×#423—131no—
131#2283D1311yescharacteristic, complemented
132#1988—132no—
133#1982—132no—
134#1895—134no—
135#1804—134no—
136C23×C13—22no—
137C23×C13—22no—
138C23×C13—22no—
139C23×C13C2×C22⋊C412yes—
140C23×C13C22×D411yescharacteristic
141C22×D13—24no—
142C22×D13—44no—
143C22×D13—44no—
144C22×D13—44no—
145C2×C4×C13—22no—
146C2×C4×C13—22no—
147C2×C4×C13—22no—
148C2×C4×C13—22no—
149C2×C4×C13—24no—
150C2×C4×C13—22no—
151C2×C4×C13—22no—
152C2×C4×C13—28no—
153C2×C4×C13—44no—
154C2×C4×C13—44no—
155C2×C4×C13—44no—
156C2×C4×C13—44no—
157C2×C4×C13C2×C22⋊C412yes—
158C2×C4×C13C2×C22⋊C412yes—
159C2×C4×C13C2×C22⋊C412yes—
160C2×C4×C13C2×C22⋊C412yes—
161C2×C4×C13C2×C22⋊C412yes—
162C2×C4×C13AGL(1,ℤ/8)14yes—
163C2×C4×C13C23×C411yescharacteristic
164C2×C4×C13C22×D411yescharacteristic
165C2×C4×C13H(5,𝔽2)12yes—
166C2×C4×C132−1+412yes—
167C22⋊D13—416no—
168C2×Dic13—24no—
169C2×Dic13—44no—
170D52—28no—
171D52—48no—
172D52—42no—
173D52—42no—
174D52—48no—
175C4×D13—416no—
176Dic26—48no—
177#1055—44no—
178C22×#41—261no—
179C22×M16—132no—
180#532—138no—
181#532—264no—
182C2×C4⋊Q8—131no—
183C2×#127—131no—
184C2×#124—261no—
185C2×#121—132no—
186C2×#121—261no—
187C2×C4×D4—132no—
188C2×C4×D4—261no—
189C22×C42C2×D1311yescharacteristic
190#423—264no—
191C2×C4⋊C8—132no—
192C2×C4⋊C8—132no—
193C2×#102—138no—
194#383—21no—
195#382C2×D1312yes—
196#381—21no—
197#380C2×D1312yes—
198C4×C4⋊C4C2×D1312yes—
199C4×C22⋊C4—21no—
200#376—21no—
201#340—138no—
202C22×C13—22no—
203C22×C13—22no—
204C22×C13—22no—
205C22×C13—44no—
206C22×C13C22×C22⋊C411yescharacteristic
207C22×C13#53712yes—
208C22×C13C2×AGL(1,ℤ/8)12yes—
209C22×C13#57512yes—
210C2×D13—44no—
211C2×D13—44no—
212C2×D13—44no—
213C2×D13—88no—
214C4×C13—22no—
215C4×C13—22no—
216C4×C13—22no—
217C4×C13—22no—
218C4×C13—84no—
219C4×C13#41814yes—
220C4×C13#52012yes—
221C4×C13#53511yescharacteristic
222C4×C13#53611yescharacteristic
223Dic13—44no—
224C2×#41—268no—
225C2×#41—528no—
226C22×Q8—261no—
227C22×D4—132no—
228C22×D4—262no—
229C22×D4—261no—
230C22×D4—261no—
231C23×C4—262no—
232C23×C4C22⋊D1311yescharacteristic
233C23×C4C22⋊D1311yescharacteristic
234C23×C4C22×D1311yescharacteristic
235C2×M16—264no—
236C2×M16—524no—
237C22×C8—134no—
238C4⋊Q8—134no—
239C4⋊Q8—262no—
240#127—134no—
241#127—262no—
242#124—268no—
243#121—138no—
244#121—268no—
245#121—264no—
246C4×D4—138no—
247C4×D4—268no—
248C4×D4—264no—
249C2×C4⋊C4—132no—
250C2×C4⋊C4—261no—
251C2×C4⋊C4C22×D1314yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—262no—
254C2×C22⋊C4—262no—
255C2×C42—22no—
256C2×C42C4×D1314yes—
257C2×C42D5214yes—
258C2×C42C22⋊D1311yescharacteristic
259C2×C42C22⋊D1311yescharacteristic
260C2×C42C22×D1311yescharacteristic
261C2×C42C22×D1311yescharacteristic
262C4⋊C8—264no—
263C4⋊C8—264no—
264#102—528no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269C2×C13—42no—
270C2×C13#231111yescharacteristic
271C2×C13C2×#41811yescharacteristic
272C2×C13#289611yescharacteristic
273C2×C13#325114yes—
274D13—84no—
275C24—262no—
276C24C2×C22⋊D1311yescharacteristic
277#41—5232no—
278C2×Q8—264no—
279C2×Q8—524no—
280C2×D4—134no—
281C2×D4—268no—
282C2×D4—264no—
283C2×D4—264no—
284C2×D4—262no—
285C2×D4—264no—
286C2×D4—264no—
287C2×D4—524no—
288C2×D4—528no—
289C2×D4—524no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—42no—
295C22×C4—262no—
296C22×C4—268no—
297C22×C4—528no—
298C22×C4#610314yes—
299C22×C4#610314yes—
300C22×C4C2×C4×D1312yes—
301C22×C4C2×D5212yes—
302C22×C4C2×C22⋊D1311yescharacteristic
303C22×C4C2×C22⋊D1311yescharacteristic
304C22×C4C2×C22⋊D1311yescharacteristic
305C22×C4C2×C22⋊D1311yescharacteristic
306C22×C4C2×C22⋊D1311yescharacteristic
307C22×C4C23×D1311yescharacteristic
308M16—528no—
309C2×C8—264no—
310C2×C8—268no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—134no—
314C4⋊C4—264no—
315C4⋊C4—262no—
316C22⋊C4—44no—
317C22⋊C4—268no—
318C22⋊C4—268no—
319C42—44no—
320C42#610314yes—
321C42#610314yes—
322C42C2×C4×D1312yes—
323C42C2×C4×D1312yes—
324C42C2×D5212yes—
325C42C2×D5212yes—
326C13#2189111yescharacteristic, complemented, Sylow
327C23—22no—
328C23—22no—
329C23—22no—
330C23—264no—
331C23—524no—
332C23—524no—
333C23—524no—
334C23C2×#610312yes—
335C23C22×C22⋊D1311yescharacteristic
336Q8—528no—
337D4—268no—
338D4—528no—
339D4—5216no—
340D4—528no—
341D4—522no—
342D4—522no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—28no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—264no—
356C2×C4—5216no—
357C2×C4—524no—
358C2×C4C2×#610312yes—
359C2×C4C2×#610312yes—
360C2×C4C2×#610312yes—
361C2×C4C2×#610312yes—
362C2×C4C2×#610312yes—
363C2×C4#8820314yes—
364C2×C4C22×C4×D1311yescharacteristic
365C2×C4C22×D5211yescharacteristic
366C2×C4#8825612yes—
367C2×C4#8825712yes—
368C8—524no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—44no—
373C22—524no—
374C22—524no—
375C22—524no—
376C22—1048no—
377C22C22×#610311yescharacteristic
378C22C2×#8820312yes—
379C22#1171952712yes—
380C22#1171953812yes—
381C4—22no—
382C4—22no—
383C4—22no—
384C4—22no—
385C4—84no—
386C4—524no—
387C4#1171866214yes—
388C4#1171922812yes—
389C4#1171923011yescharacteristic
390C4#1171923111yescharacteristic
391C2—42no—
392C2—1044no—
393C2#1311742411yescharacteristic
394C2C2×#1171866211yescharacteristic
395C2#1312335311yescharacteristic
396C2#1312619814yes—
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2283C2
socleC23×C13C22×D4
prime radicalC2×#11719182C2
cosocleC22×C4×C13C24
hypercenter#2283D13
hypocenterC13#21891
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
2. C2×#11718585C2semidirect
3. #13115788C2semidirect
4. #13115763C2semidirect
5. #13115272C2semidirect
6. #13114843C2semidirect
7. C13×#2283C2semidirect
14. #11719182C4semidirect
131. #2283D13semidirect
326. C13#21891semidirect

Maximal subgroups

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

subgroupindex
C2×#117191822
C2×#117185852
#131157882
#131157632
#131152722
#131148432
C13×#22832
#2189113