Finite Group Databasebeta

← #15882715

Subgroups — #15882715

Order 3328.

922 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718533C211yesmaximal, characteristic
2#13117781C212yesmaximal, complemented
3#13115712C211yesmaximal, characteristic
4#13115710C211yesmaximal, characteristic
5#13115463C212yesmaximal, complemented
6#13115461C212yesmaximal, complemented
7#13114991C211yesmaximal, characteristic, complemented
8#13114835C212yesmaximal, complemented
9#13114735C211yesmaximal, characteristic, complemented
10#13114704C211yesmaximal, characteristic, complemented
11C13×#1776C211yesmaximal, characteristic, complemented
12C22×D104C2211yescharacteristic
13C2×#88147C2212yes—
14C2×#88126—22no—
15C2×C13×#107C2211yescharacteristic
16C2×C13×#106C2211yescharacteristic
17C2×C13×#97C2211yescharacteristic
18C4×C13×C4⋊C4C2212yes—
19C2×#88061C2212yes—
20C2×#88057C2211yescharacteristic
21C2×#88055C2211yescharacteristic
22C2×#88054C2211yescharacteristic
23C2×C8×Dic13C2211yescharacteristic
24#11718649C2212yes—
25#11718637C2212yes—
26C4⋊C4×Dic13C2212yes—
27C2×#88049C2212yes—
28C2×#88049C2212yes—
29C2×#88048C2212yes—
30#11718601—22no—
31#11718600C2212yes—
32C4×#6101C2211yescharacteristic
33#11718533C418yescomplemented
34#11718330—22no—
35#11718328—22no—
36C13×#340C2212yes—
37#11718142C2211yescharacteristic
38#11718121C2212yes—
39#11718116C2211yescharacteristic
40C22×D52C2312yes—
41C22×C4×D13—22no—
42C22×C8×C13C2311yescharacteristic
43C22×C8×C13C2311yescharacteristic
44C2×C13×C4⋊C4C2312yes—
45C2×C13×C4⋊C4C2312yes—
46C2×C42×C13—22no—
47C2×C42×C13C2311yescharacteristic
48C2×#6103—22no—
49C2×#6103—22no—
50C2×#6103—22no—
51C2×#6103—22no—
52C2×#6103—22no—
53C2×C4⋊Dic13C2312yes—
54C2×#6101—22no—
55C2×#6101—22no—
56C2×#6101D412yes—
57C2×C4×Dic13—22no—
58C2×C4×Dic13C2311yescharacteristic
59C2×C4×Dic13C2311yescharacteristic
60C22×#1055C2311yescharacteristic
61C22×#1055C2311yescharacteristic
62C2×D104—22no—
63C2×D104—22no—
64C2×D104C2×C414yes—
65#88147—24no—
66#88147C2×C418yes—
67#88126—48no—
68C13×#107C2×C414yes—
69C13×#106—22no—
70C13×#97—22no—
71C13×#95—22no—
72#88071—22no—
73#88071D412yes—
74#88061—42no—
75#88057—22no—
76#88055—22no—
77#88054—22no—
78C8×Dic13C2×C414yes—
79#88049—42no—
80#88049C2×C418yes—
81#88048—42no—
82#22813—131nomaximal, Sylow
83C23×D13—22no—
84C22×C4×C13—22no—
85C22×C4×C13—22no—
86C22×C4×C13C2×D412yes—
87C22×C4×C13C2411yescharacteristic
88C22×Dic13—22no—
89C22×Dic13C2×D412yes—
90C22×Dic13C2×D411yescharacteristic
91C22×Dic13C2×D411yescharacteristic
92C2×D52—22no—
93C2×D52—22no—
94C2×D52—22no—
95C2×D52—22no—
96C2×D52—22no—
97C2×D52C22×C414yes—
98C2×C4×D13—28no—
99C2×C4×D13—44no—
100C2×C4×D13—44no—
101C2×C8×C13—21no—
102C2×C8×C13—21no—
103C2×C8×C13—21no—
104C2×C8×C13C22×C412yes—
105C2×C8×C13#4112yes—
106C2×C8×C13#4112yes—
107C13×C4⋊C4—22no—
108C13×C4⋊C4—22no—
109C13×C4⋊C4—22no—
110C13×C4⋊C4C22×C414yes—
111C42×C13—21no—
112C42×C13—21no—
113C42×C13—44no—
114#6103—28no—
115#6103—44no—
116#6103—44no—
117#6103—82no—
118#6103—82no—
119C4⋊Dic13—22no—
120C4⋊Dic13—22no—
121#6101—24no—
122#6101—44no—
123#6101—82no—
124C4×Dic13—21no—
125C4×Dic13—21no—
126C4×Dic13—44no—
127C4×Dic13C22×C412yes—
128C4×Dic13#4112yes—
129C2×#1055—21no—
130C2×#1055—21no—
131C2×#1055—21no—
132C2×#1055—21no—
133C2×#1055—21no—
134C2×#1055C22×C412yes—
135D104—28no—
136C2×C4×D8—131no—
137#2354—132no—
138#1972—131no—
139#1972—131no—
140#1926—132no—
141#1926—132no—
142#1844—131no—
143#1802—132no—
144#1776—131no—
145#1776D1311yescharacteristic, complemented
146#1770—131no—
147C23×C13C22×D411yescharacteristic
148C22×D13—24no—
149C22×D13—42no—
150C22×D13—42no—
151C22×D13—42no—
152C22×D13—42no—
153C22×D13—42no—
154C22×D13—42no—
155C2×C4×C13—24no—
156C2×C4×C13—22no—
157C2×C4×C13—22no—
158C2×C4×C13—22no—
159C2×C4×C13—42no—
160C2×C4×C13—42no—
161C2×C4×C13—42no—
162C2×C4×C13—42no—
163C2×C4×C13—42no—
164C2×C4×C13C23×C411yescharacteristic
165C2×C4×C13C2×#4111yescharacteristic
166C2×C4×C13C2×#4111yescharacteristic
167C2×C4×C13C2×#4111yescharacteristic
168C2×C4×C13H(5,𝔽2)11yescharacteristic
169C2×C4×C132−1+411yescharacteristic
170C2×Dic13—22no—
171C2×Dic13—22no—
172C2×Dic13—22no—
173C2×Dic13—22no—
174C2×Dic13—42no—
175C2×Dic13—42no—
176C2×Dic13—42no—
177C2×Dic13C4×D412yes—
178C2×Dic13C4×D412yes—
179C2×Dic13AGL(1,ℤ/8)14yes—
180D52—24no—
181D52—24no—
182D52—42no—
183D52—42no—
184C4×D13—416no—
185C8×C13—41no—
186C8×C13#11714yescomplemented
187#1055—22no—
188#1055—41no—
189C22×D8—131no—
190C2×C4×D4—132no—
191C2×#117—262no—
192C4×D8—138no—
193C2×#107C2×D1311yescharacteristic
194C2×#106—131no—
195C2×#106—132no—
196C2×#106C2×D1311yescharacteristic
197C2×C4⋊C8—131no—
198C2×#102—132no—
199C2×#102—132no—
200C2×#102—132no—
201C2×#97—131no—
202C2×#97C2×D1311yescharacteristic
203C2×C4×C8—131no—
204#390—132no—
205#390—132no—
206#388—262no—
207#388—262no—
208#384—132no—
209#380—262no—
210C4×C4⋊C4—131no—
211C4×C4⋊C4—132no—
212C4×C4⋊C4C2×D1312yes—
213#340—132no—
214#340C2×D1312yes—
215#336—131no—
216#336—131no—
217C22×C13C2×C4×D411yescharacteristic
218C22×C13#54711yescharacteristic
219C22×C13#55011yescharacteristic
220C22×C13C2×AGL(1,ℤ/8)12yes—
221C22×C13#57712yes—
222C2×D13—44no—
223C2×D13—44no—
224C2×D13—44no—
225C2×D13—82no—
226C2×D13—82no—
227C2×D13—82no—
228C2×D13—82no—
229C4×C13—42no—
230C4×C13—44no—
231C4×C13—82no—
232C4×C13C2×#11711yescharacteristic
233C4×C13#52011yescharacteristic
234C4×C13#55511yescharacteristic
235C4×C13#55611yescharacteristic
236Dic13—22no—
237Dic13—42no—
238Dic13—82no—
239Dic13#44214yes—
240C22×D4—132no—
241C23×C4—262no—
242C2×D8—134no—
243C2×D8—262no—
244C2×D8—262no—
245C22×C8—131no—
246C22×C8—131no—
247C22×C8C22×D1311yescharacteristic
248C22×C8C22×D1311yescharacteristic
249C4×D4—138no—
250C4×D4—264no—
251#117—528no—
252C2×C4⋊C4—132no—
253C2×C4⋊C4—132no—
254C2×C4⋊C4—262no—
255C2×C4⋊C4—262no—
256C2×C4⋊C4C22×D1312yes—
257C2×C4⋊C4C22×D1312yes—
258C2×C22⋊C4—262no—
259C2×C22⋊C4—262no—
260C2×C22⋊C4—262no—
261C2×C22⋊C4—262no—
262C2×C22⋊C4—262no—
263C2×C42—22no—
264C2×C42—131no—
265C2×C42—131no—
266C2×C42—262no—
267C2×C42C22×D1311yescharacteristic
268#107C4×D1314yes—
269#106—22no—
270#106—262no—
271#106—522no—
272C4⋊C8—262no—
273#102—138no—
274#102—522no—
275#102—522no—
276#97—22no—
277#97—262no—
278C4×C8—134no—
279#95—22no—
280#95—132no—
281#95—262no—
282C2×C13#235411yescharacteristic
283C2×C13C2×#44211yescharacteristic
284C2×C13#295211yescharacteristic
285C2×C13#335412yes—
286C2×C13#339712yes—
287D13—84no—
288C24—262no—
289C2×D4—134no—
290C2×D4—262no—
291C2×D4—262no—
292C2×D4—262no—
293C2×D4—262no—
294C2×D4—262no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—131no—
298C22×C4—131no—
299C22×C4—132no—
300C22×C4—268no—
301C22×C4—262no—
302C22×C4—524no—
303C22×C4—524no—
304C22×C4D4×D1312yes—
305C22×C4C23×D1311yescharacteristic
306D8—268no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—132no—
311C2×C8—261no—
312C2×C8—261no—
313C2×C8—261no—
314C2×C8—261no—
315C2×C8—261no—
316C2×C8C2×C4×D1312yes—
317C2×C8#612712yes—
318C2×C8#612712yes—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—264no—
323C4⋊C4—262no—
324C4⋊C4—262no—
325C4⋊C4—524no—
326C4⋊C4—1042no—
327C4⋊C4C2×C4×D1314yes—
328C22⋊C4—268no—
329C22⋊C4—524no—
330C22⋊C4—524no—
331C22⋊C4—1042no—
332C22⋊C4—1042no—
333C42—21no—
334C42—21no—
335C42—44no—
336C42—132no—
337C42—132no—
338C42—261no—
339C42—261no—
340C42—524no—
341C13#2281311yescharacteristic, complemented, Sylow
342C23—264no—
343C23—522no—
344C23—522no—
345C23—522no—
346C23—522no—
347C23—522no—
348C23—522no—
349C23C2×D4×D1311yescharacteristic
350D4—264no—
351D4—264no—
352D4—522no—
353D4—522no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—134no—
364C2×C4—132no—
365C2×C4—132no—
366C2×C4—262no—
367C2×C4—262no—
368C2×C4—262no—
369C2×C4—262no—
370C2×C4—5216no—
371C2×C4—522no—
372C2×C4—522no—
373C2×C4—522no—
374C2×C4C22×C4×D1311yescharacteristic
375C2×C4C2×#612711yescharacteristic
376C2×C4C2×#612711yescharacteristic
377C2×C4#8825411yescharacteristic
378C2×C4#41×D1311yescharacteristic
379C2×C4#8825611yescharacteristic
380C8—41no—
381C8—262no—
382C8—521no—
383C8#8812614yescomplemented
384C22—524no—
385C22—524no—
386C22—524no—
387C22—1042no—
388C22—1042no—
389C22—1042no—
390C22—1042no—
391C22#1171921211yescharacteristic
392C22C4×D4×D1311yescharacteristic
393C22#1171925811yescharacteristic
394C22D13×AGL(1,ℤ/8)12yes—
395C22#1171948312yes—
396C4—42no—
397C4—44no—
398C4—82no—
399C4—134no—
400C4—262no—
401C4—522no—
402C4—1042no—
403C4C2×#8812611yescharacteristic
404C4#1171923711yescharacteristic
405C4#1171927311yescharacteristic
406C4#1171927911yescharacteristic
407C2—1044no—
408C2#1311778111yescharacteristic
409C2D13×#44211yescharacteristic
410C2#1312371611yescharacteristic
411C2#1312742712yes—
412C2#1312785712yes—
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1776C2
socleC23×C13C22×D4
prime radicalC22×D104C22
cosocleC22×C4×C13C24
hypercenter#1776D13
hypocenterC13#22813
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
2. #13117781C2semidirect
5. #13115463C2semidirect
6. #13115461C2semidirect
7. #13114991C2semidirect
8. #13114835C2semidirect
9. #13114735C2semidirect
10. #13114704C2semidirect
11. C13×#1776C2semidirect
33. #11718533C4semidirect
145. #1776D13semidirect
186. C8×C13#117semidirect
341. C13#22813semidirect
383. C8#88126semidirect

Maximal subgroups

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

subgroupindex
C2×#117185332
#131177812
#131157122
#131157102
#131154632
#131154612
#131149912
#131148352
#131147352
#131147042
C13×#17762
#2281313