Finite Group Databasebeta

← #16029239

Subgroups — #16029239

Order 3328.

1962 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13118143C216yesmaximal, complemented
2#13118130C213yesmaximal, complemented
3#13118127C213yesmaximal, complemented
4#13118112C216yesmaximal
5#13118108C213yesmaximal
6#13118106C213yesmaximal
7C42×Dic26C213yesmaximal, complemented
8C4×#88126C213yesmaximal
9C42×Q8×C13C211yesmaximal, characteristic, complemented
10C2×C4×Q8×C13C2213yes—
11C2×#88126C2213yes—
12C2×#88126C2213yes—
13C2×C42×D13C2211yescharacteristic
14C2×C4×Dic26C2213yes—
15C2×C4×Dic26C2216yes—
16C4×C13×C4⋊C4C2213yes—
17C4×C13×C4⋊C4C2216yes—
18C43×C13C2213yes—
19#11718892C2211yescharacteristic
20#11718890C2213yes—
21#11718655C2216yes—
22#11718653C2213yes—
23#11718645C2216yes—
24#11718644C2216yes—
25#11718643C2213yes—
26#11718641C2216yes—
27#11718640C2216yes—
28C4×#6103C2216yes—
29C4×#6103C2216yes—
30C4×C4⋊Dic13C2213yes—
31C4×C4⋊Dic13C2216yes—
32#11718593C2213yes—
33#11718593C2213yes—
34C4×#6101C2216yes—
35C4×#6101C2216yes—
36C4×#6101C2216yes—
37C4×#6101C2216yes—
38C4×#6101C2216yes—
39C42×Dic13C2213yes—
40#11718338C2211yescharacteristic
41#11718337C2212yes—
42#11718334C2213yes—
43#11718320C2213yes—
44#11718319C2216yes—
45#11718318C2216yes—
46#11718317C2213yes—
47C22×Q8×C13C2311yescharacteristic
48C22×C4×D13C2313yes—
49C22×Dic26C2313yes—
50C4×Q8×C13—212no—
51C2×C13×C4⋊C4C2316yes—
52C2×C13×C4⋊C4C2313yes—
53C2×C42×C13C2316yes—
54C2×C42×C13C2313yes—
55C2×C42×C13C2316yes—
56C2×C42×C13C2313yes—
57C2×C42×C13C2311yescharacteristic
58C2×#6103C2316yes—
59C2×#6103C2313yes—
60C2×#6103C2313yes—
61C2×C4⋊Dic13C2316yes—
62C2×C4⋊Dic13C2313yes—
63C2×#6101C2316yes—
64C2×#6101C2316yes—
65C2×#6101C2316yes—
66C2×#6101C2313yes—
67C2×#6101C2316yes—
68C2×#6101C2313yes—
69C2×C4×Dic13C2316yes—
70C2×C4×Dic13C2316yes—
71C2×C4×Dic13C2313yes—
72C2×C4×Dic13C2313yes—
73C2×C4×Dic13C2313yes—
74#88126—212no—
75#88126—212no—
76C42×D13Q818yescomplemented
77C4×Dic26—212no—
78C4×Dic26—224no—
79C13×#95C2311yescharacteristic
80C13×#95C2313yes—
81C13×#95C2312yes—
82#88071C2316yes—
83#88071C2316yes—
84#88071C2316yes—
85#88071C2316yes—
86#88071C2316yes—
87#88071C2313yes—
88#88071C2313yes—
89#88071C2316yes—
90#34552—131nomaximal, Sylow
91C23×D13C2411yescharacteristic
92C2×Q8×C13—26no—
93C22×C4×C13C2416yes—
94C22×C4×C13C2413yes—
95C22×C4×C13C2413yes—
96C22×C4×C13C2413yes—
97C22×Dic13C2413yes—
98C22×Dic13C2413yes—
99C22×Dic13C2413yes—
100C22×Dic13C2416yes—
101C2×C4×D13—26no—
102C2×C4×D13—26no—
103C2×C4×D13C2×Q8112yes—
104C2×Dic26—26no—
105C2×Dic26—212no—
106C13×C4⋊C4—212no—
107C13×C4⋊C4—26no—
108C42×C13—212no—
109C42×C13—26no—
110C42×C13—212no—
111C42×C13C2×Q814yes—
112C42×C13#41112yes—
113#6103—26no—
114#6103—26no—
115#6103—212no—
116C4⋊Dic13—26no—
117C4⋊Dic13—212no—
118#6101—212no—
119#6101—26no—
120#6101—26no—
121#6101—212no—
122#6101—212no—
123#6101—212no—
124C4×Dic13—212no—
125C4×Dic13—212no—
126C4×Dic13—26no—
127C4×Dic13C2×Q8112yes—
128C4×Dic13#41112yes—
129#2402—136no—
130#2399—133no—
131#2399—133no—
132#2396—133no—
133#2396—136no—
134#2396—133no—
135C42×Q8—133no—
136C42×Q8D1311yescharacteristic, complemented
137C4×#117—133no—
138C23×C13C2511yescharacteristic
139C22×D13—23no—
140C22×D13—23no—
141C22×D13C22×Q812yes—
142Q8×C13—28no—
143C2×C4×C13—26no—
144C2×C4×C13—26no—
145C2×C4×C13—26no—
146C2×C4×C13—26no—
147C2×C4×C13—26no—
148C2×C4×C13C22×Q816yes—
149C2×C4×C13C2×#4116yes—
150C2×C4×C13C2×#4116yes—
151C2×C4×C13C2×#4116yes—
152C2×C4×C13C2×#4116yes—
153C2×Dic13—26no—
154C2×Dic13—26no—
155C2×Dic13—23no—
156C2×Dic13—23no—
157C2×Dic13—26no—
158C2×Dic13—26no—
159C2×Dic13—23no—
160C2×Dic13—23no—
161C2×Dic13C22×Q816yes—
162C2×Dic13C2×#41112yes—
163C4×D13—224no—
164Dic26—224no—
165C2×C4×Q8—136no—
166C2×C4×Q8—133no—
167C2×C4×Q8C2×D1313yes—
168C2×#117—133no—
169C2×#117—133no—
170C22×C42—131no—
171#400—133no—
172#400—136no—
173#400—133no—
174#399—133no—
175#399—131no—
176#398—136no—
177#398—136no—
178#398—136no—
179#398—136no—
180#396—136no—
181#396—132no—
182#395—133no—
183#395—136no—
184#395—133no—
185#393—131no—
186#393—133no—
187C4×C4⋊C4—136no—
188C4×C4⋊C4—133no—
189C4×C4⋊C4—136no—
190C4×C4⋊C4—136no—
191C4×C4⋊C4—136no—
192C4×C4⋊C4—136no—
193C4×C4⋊C4—136no—
194C4×C4⋊C4C2×D1313yes—
195C4×C4⋊C4C2×D1316yes—
196C4×C22⋊C4—136no—
197C4×C22⋊C4—136no—
198#376—133no—
199#376—133no—
200C43—133no—
201C43C2×D1313yes—
202C22×C13C23×Q811yescharacteristic
203C22×C13C22×#4113yes—
204C22×C13C22×#4113yes—
205C2×D13—26no—
206C2×D13—43no—
207C2×D13—41no—
208C4×C13—26no—
209C4×C13—26no—
210C4×C13—46no—
211C4×C13#533112yes—
212Dic13—26no—
213Dic13—212no—
214Dic13—43no—
215Dic13—43no—
216C22×Q8—133no—
217C22×Q8C22×D1311yescharacteristic
218C23×C4—133no—
219C4×Q8—212no—
220C4×Q8—2624no—
221C4×Q8—2612no—
222#117—2612no—
223#117—2612no—
224C2×C4⋊C4—136no—
225C2×C4⋊C4—133no—
226C2×C4⋊C4—133no—
227C2×C4⋊C4—136no—
228C2×C4⋊C4—136no—
229C2×C4⋊C4—136no—
230C2×C4⋊C4—136no—
231C2×C4⋊C4—133no—
232C2×C4⋊C4C22×D1316yes—
233C2×C4⋊C4C22×D1313yes—
234C2×C22⋊C4—136no—
235C2×C22⋊C4—133no—
236C2×C22⋊C4—133no—
237C2×C42—133no—
238C2×C42—133no—
239C2×C42—138no—
240C2×C42—133no—
241C2×C42—136no—
242C2×C42—136no—
243C2×C42C22×D1313yes—
244C2×C42C22×D1316yes—
245C2×C42C22×D1316yes—
246C2×C42C22×D1313yes—
247C2×C42C22×D1311yescharacteristic
248#95—136no—
249#95—136no—
250#95—136no—
251#95—136no—
252#95—136no—
253#95—133no—
254#95—133no—
255#95—136no—
256#95C22×D1313yes—
257#95C22×D1311yescharacteristic
258#95C22×D1312yes—
259C2×C13C2×#53313yes—
260C2×C13#344413yes—
261C2×C13#344411yescharacteristic
262D13—42no—
263C24—131no—
264C2×Q8—26no—
265C2×Q8—2612no—
266C2×Q8—266no—
267C22×C4—133no—
268C22×C4—136no—
269C22×C4—1312no—
270C22×C4—133no—
271C22×C4—133no—
272C22×C4—266no—
273C22×C4—266no—
274C22×C4C23×D1316yes—
275C22×C4C23×D1313yes—
276C22×C4C23×D1313yes—
277C22×C4C23×D1313yes—
278C4⋊C4—212no—
279C4⋊C4—26no—
280C4⋊C4—2612no—
281C4⋊C4—2612no—
282C4⋊C4—2612no—
283C4⋊C4—2612no—
284C4⋊C4—266no—
285C4⋊C4—266no—
286C4⋊C4—2612no—
287C4⋊C4—266no—
288C22⋊C4—2612no—
289C22⋊C4—266no—
290C22⋊C4—266no—
291C42—212no—
292C42—26no—
293C42—212no—
294C42—1312no—
295C42—1312no—
296C42—266no—
297C42—2612no—
298C42—2612no—
299C42#6127112yes—
300C42Q8×D1314yes—
301C13#3455211yescharacteristic, complemented, Sylow
302C23—132no—
303C23—263no—
304C23—263no—
305C23C24×D1311yescharacteristic
306Q8—28no—
307Q8—2624no—
308C2×C4—26no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—26no—
312C2×C4—26no—
313C2×C4—136no—
314C2×C4—1312no—
315C2×C4—266no—
316C2×C4—263no—
317C2×C4—266no—
318C2×C4—263no—
319C2×C4—2624no—
320C2×C4—263no—
321C2×C4—263no—
322C2×C4—266no—
323C2×C4—266no—
324C2×C4C2×#612716yes—
325C2×C4C2×#612716yes—
326C2×C4C2×#612716yes—
327C2×C4C2×Q8×D1316yes—
328C2×C4#41×D1316yes—
329C22—266no—
330C22—523no—
331C22—521no—
332C22C22×#612713yes—
333C22C22×Q8×D1311yescharacteristic
334C22C2×#41×D1313yes—
335C4—26no—
336C4—26no—
337C4—46no—
338C4—266no—
339C4—2612no—
340C4—523no—
341C4—523no—
342C4#11719283112yes—
343C2—522no—
344C2C2×#1171928313yes—
345C2#1312812913yes—
346C2#1312813211yescharacteristic
subgroupG/H
centerC2×C42C22×D13
commutator, socle, cosocleC23×C13C25
FrattiniC23C24×D13
FittingC42×Q8×C13C2
prime radicalC23×D13C24
hypercenterC42×Q8D13
hypocenterC13#34552
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C2×C42 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
1. #13118143C2semidirect
2. #13118130C2semidirect
3. #13118127C2semidirect
7. C42×Dic26C2semidirect
9. C42×Q8×C13C2semidirect
76. C42×D13Q8semidirect
136. C42×Q8D13semidirect
301. C13#34552semidirect

Maximal subgroups

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

subgroupindex
#131181432
#131181302
#131181272
#131181122
#131181082
#131181062
C42×Dic262
C4×#881262
C42×Q8×C132
#3455213