Finite Group Databasebeta

← C22×#11718146 #15860360

Subgroups — C22×#11718146 #15860360

Order 3328.

2494 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#6099C211yesmaximal, characteristic
2C23×C4⋊Dic13C211yesmaximal, characteristic
3C2×#11718146C2112yesmaximal, direct product factor
4C23×C13×M16C211yesmaximal, characteristic
5C24×C4×C13C2211yescharacteristic
6C24×Dic13C412yes—
7C22×C13×M16—24no—
8C22×C13×M16—23no—
9C22×C13×M16C418yescomplemented
10C22×C13×M16C2216yes—
11C23×C8×C13—21no—
12C22×C4⋊Dic13—21no—
13C22×C4⋊Dic13—23no—
14C22×C4⋊Dic13—21no—
15C22×C4⋊Dic13—23no—
16C22×C4⋊Dic13C416yes—
17C22×C4⋊Dic13C2216yes—
18C22×#6099—23no—
19C22×#6099—24no—
20C22×#6099C418yescomplemented
21C22×#6099C2216yes—
22C23×#1055—21no—
23#11718146C22116yesdirect product factor
24C25×C13C2×C411yescharacteristic
25C23×C4×C13C2×C413yes—
26C23×C4×C13C2×C414yes—
27C23×C4×C13C2×C414yes—
28C23×C4×C13D411yescharacteristic
29C23×C4×C13D413yes—
30C23×C4×C13D414yes—
31C23×C4×C13D411yescharacteristic
32C23×C4×C13D413yes—
33C23×C4×C13Q814yes—
34C23×C4×C13C2313yes—
35C23×Dic13—26no—
36C23×Dic13—22no—
37C23×Dic13—48no—
38C23×Dic13C2×C4112yes—
39C2×C13×M16—224no—
40C2×C13×M16—212no—
41C2×C13×M16—26no—
42C2×C13×M16C2×C4124yescomplemented
43C2×C13×M16C2314yes—
44C22×C8×C13—28no—
45C22×C8×C13—26no—
46C2×C4⋊Dic13—212no—
47C2×C4⋊Dic13—26no—
48C2×C4⋊Dic13—26no—
49C2×C4⋊Dic13—26no—
50C2×C4⋊Dic13—26no—
51C2×C4⋊Dic13—26no—
52C2×C4⋊Dic13—26no—
53C2×C4⋊Dic13C2×C4112yes—
54C2×C4⋊Dic13C2314yes—
55C2×#6099—224no—
56C2×#6099—26no—
57C2×#6099—212no—
58C2×#6099C2×C4124yescomplemented
59C2×#6099C2314yes—
60C22×#1055—28no—
61C22×#1055—26no—
62C22×#343—131nomaximal, Sylow
63C24×C13—24no—
64C24×C13—24no—
65C24×C13C4214yes—
66C24×C13C22⋊C413yes—
67C24×C13C22⋊C411yescharacteristic
68C24×C13C4⋊C414yes—
69C24×C13C22×C413yes—
70C22×C4×C13C42112yes—
71C22×C4×C13C22⋊C416yes—
72C22×C4×C13C22⋊C413yes—
73C22×C4×C13C22⋊C413yes—
74C22×C4×C13C22⋊C414yes—
75C22×C4×C13C22⋊C4112yes—
76C22×C4×C13C22⋊C414yes—
77C22×C4×C13C22⋊C4112yes—
78C22×C4×C13C4⋊C4112yes—
79C22×C4×C13C4⋊C4112yes—
80C22×C4×C13C4⋊C414yes—
81C22×C4×C13C4⋊C4112yes—
82C22×C4×C13C4⋊C414yes—
83C22×C4×C13C22×C413yes—
84C22×C4×C13C22×C416yes—
85C22×C4×C13C22×C416yes—
86C22×C4×C13C2×D413yes—
87C22×C4×C13C2×D413yes—
88C22×C4×C13C2×D416yes—
89C22×C4×C13C2×D413yes—
90C22×C4×C13C2×D413yes—
91C22×C4×C13C2×Q816yes—
92C22×C4×C13C2411yescharacteristic
93C22×Dic13—212no—
94C22×Dic13—212no—
95C22×Dic13—424no—
96C22×Dic13—48no—
97C22×Dic13—424no—
98C22×Dic13C22×C418yes—
99C13×M16—224no—
100C13×M16—24no—
101C13×M16C22×C418yescomplemented
102C2×C8×C13—224no—
103C2×C8×C13—24no—
104C4⋊Dic13—212no—
105C4⋊Dic13—212no—
106C4⋊Dic13—24no—
107C4⋊Dic13—24no—
108#6099—224no—
109#6099—24no—
110#6099C22×C418yescomplemented
111C2×#1055—224no—
112C2×#1055—24no—
113C23×M16—131no—
114C23×M16D1311yescharacteristic
115C23×C4⋊C4—131no—
116C2×#343—1312no—
117C23×C13—216no—
118C23×C13—216no—
119C23×C13—26no—
120C23×C13—26no—
121C23×C13—26no—
122C23×C13—26no—
123C23×C13#95112yes—
124C23×C13#9514yes—
125C23×C13#10014yes—
126C23×C13#10114yes—
127C23×C13C2×C4216yes—
128C23×C13C2×C22⋊C413yes—
129C23×C13C2×C22⋊C413yes—
130C23×C13C2×C4⋊C416yes—
131C23×C13C23×C411yescharacteristic
132C2×C4×C13#95124yes—
133C2×C4×C13#95112yes—
134C2×C4×C13#95112yes—
135C2×C4×C13C2×C4216yes—
136C2×C4×C13C2×C22⋊C416yes—
137C2×C4×C13C2×C22⋊C416yes—
138C2×C4×C13C2×C22⋊C416yes—
139C2×C4×C13C2×C22⋊C416yes—
140C2×C4×C13C2×C22⋊C413yes—
141C2×C4×C13C2×C22⋊C413yes—
142C2×C4×C13C2×C4⋊C416yes—
143C2×C4×C13C2×C4⋊C416yes—
144C2×C4×C13C2×C4⋊C416yes—
145C2×C4×C13C2×C4⋊C416yes—
146C2×C4×C13C2×C4⋊C416yes—
147C2×C4×C13C23×C411yescharacteristic
148C2×C4×C13C23×C411yescharacteristic
149C2×C4×C13C22×D411yescharacteristic
150C2×C4×C13C22×D411yescharacteristic
151C2×C4×C13C22×D411yescharacteristic
152C2×C4×C13C22×Q811yescharacteristic
153C2×Dic13—28no—
154C2×Dic13—424no—
155C2×Dic13—424no—
156C2×Dic13—48no—
157C8×C13—28no—
158#1055—28no—
159C24×C4—132no—
160C24×C4C2×D1311yescharacteristic
161C22×M16—24no—
162C22×M16—23no—
163C22×M16—138no—
164C22×M16—136no—
165C22×M16—263no—
166C22×M16—264no—
167C22×M16Dic1318yescomplemented
168C22×M16C2×D1316yes—
169C23×C8—21no—
170C23×C8—261no—
171C22×C4⋊C4—136no—
172C22×C4⋊C4—136no—
173C22×C4⋊C4—263no—
174C22×C4⋊C4—261no—
175C22×C4⋊C4—263no—
176C22×C4⋊C4—261no—
177#343—1316no—
178C22×C13—23no—
179C22×C13—21no—
180C22×C13—23no—
181C22×C13—21no—
182C22×C13—224no—
183C22×C13—212no—
184C22×C13—212no—
185C22×C13#343116yescomplemented
186C22×C13C2×#9516yes—
187C22×C13C2×#9516yes—
188C22×C13C2×#10016yes—
189C22×C13C2×#10116yes—
190C22×C13C22×C4211yescharacteristic
191C22×C13C22×C22⋊C411yescharacteristic
192C22×C13C22×C4⋊C411yescharacteristic
193C4×C13C2×#9516yes—
194C4×C13C2×#9516yes—
195C4×C13C22×C22⋊C411yescharacteristic
196C4×C13C22×C22⋊C411yescharacteristic
197C4×C13C22×C4⋊C411yescharacteristic
198C4×C13C22×C4⋊C411yescharacteristic
199Dic13—48no—
200C25C4×D1311yescharacteristic
201C23×C4—1312no—
202C23×C4—266no—
203C23×C4—262no—
204C23×C4—528no—
205C23×C4Dic2614yes—
206C23×C4C4×D1313yes—
207C23×C4C4×D1314yes—
208C23×C4D5213yes—
209C23×C4D5211yescharacteristic
210C23×C4C2×Dic1314yes—
211C23×C4C22⋊D1313yes—
212C23×C4C22⋊D1311yescharacteristic
213C23×C4C22⋊D1314yes—
214C23×C4C22×D1313yes—
215C2×M16—224no—
216C2×M16—212no—
217C2×M16—26no—
218C2×M16—1324no—
219C2×M16—134no—
220C2×M16—2624no—
221C2×M16—266no—
222C2×M16—2612no—
223C2×M16C2×Dic13124yescomplemented
224C2×M16C22×D1314yes—
225C22×C8—28no—
226C22×C8—26no—
227C22×C8—268no—
228C22×C8—266no—
229C2×C4⋊C4—1312no—
230C2×C4⋊C4—134no—
231C2×C4⋊C4—2612no—
232C2×C4⋊C4—266no—
233C2×C4⋊C4—266no—
234C2×C4⋊C4—266no—
235C2×C4⋊C4—266no—
236C2×C4⋊C4—266no—
237C2×C4⋊C4—266no—
238C2×C13—21no—
239C2×C13—26no—
240C2×C13—21no—
241C2×C13C2×#343112yescomplemented
242C2×C13C22×#9511yescharacteristic
243C2×C13C22×#10011yescharacteristic
244C2×C13C22×#10111yescharacteristic
245C24—24no—
246C24—24no—
247C24C4×Dic1314yes—
248C24#610114yes—
249C24#610313yes—
250C24#610311yescharacteristic
251C24C2×C4×D1313yes—
252C22×C4—138no—
253C22×C4—2612no—
254C22×C4—2612no—
255C22×C4—5224no—
256C22×C4—528no—
257C22×C4—5224no—
258C22×C4C4×Dic13112yes—
259C22×C4#6101112yes—
260C22×C4#6101112yes—
261C22×C4#610114yes—
262C22×C4C4⋊Dic13112yes—
263C22×C4C4⋊Dic1314yes—
264C22×C4#610316yes—
265C22×C4#610313yes—
266C22×C4#610313yes—
267C22×C4#610314yes—
268C22×C4#6103112yes—
269C22×C4C22⋊Dic1314yes—
270C22×C4C22⋊Dic13112yes—
271C22×C4C2×Dic2616yes—
272C22×C4C2×C4×D1313yes—
273C22×C4C2×C4×D1316yes—
274C22×C4C2×D5213yes—
275C22×C4C2×D5213yes—
276C22×C4C22×Dic1316yes—
277C22×C4C2×C22⋊D1313yes—
278C22×C4C2×C22⋊D1313yes—
279C22×C4C2×C22⋊D1316yes—
280C22×C4C23×D1311yescharacteristic
281M16—224no—
282M16—24no—
283M16—138no—
284M16—2624no—
285M16—264no—
286M16C22×Dic1318yescomplemented
287C2×C8—224no—
288C2×C8—24no—
289C2×C8—2624no—
290C2×C8—264no—
291C4⋊C4—2612no—
292C4⋊C4—2612no—
293C4⋊C4—264no—
294C4⋊C4—264no—
295C13C22×#34311yescharacteristic, complemented, Sylow
296C23—216no—
297C23—216no—
298C23—26no—
299C23—26no—
300C23—26no—
301C23—26no—
302C23#8806314yes—
303C23#8806414yes—
304C23#88071112yes—
305C23#8807114yes—
306C23C2×C4×Dic1316yes—
307C23C2×#610116yes—
308C23C2×#610313yes—
309C23C2×#610313yes—
310C23C22×C4×D1311yescharacteristic
311C2×C4—268no—
312C2×C4—5224no—
313C2×C4—5224no—
314C2×C4—528no—
315C2×C4#88071124yes—
316C2×C4#88071112yes—
317C2×C4#88071112yes—
318C2×C4C2×C4×Dic1316yes—
319C2×C4C2×#610116yes—
320C2×C4C2×#610116yes—
321C2×C4C2×#610116yes—
322C2×C4C2×C4⋊Dic1316yes—
323C2×C4C2×C4⋊Dic1316yes—
324C2×C4C2×#610316yes—
325C2×C4C2×#610316yes—
326C2×C4C2×#610313yes—
327C2×C4C2×#610313yes—
328C2×C4C2×C22⋊Dic1316yes—
329C2×C4C2×C22⋊Dic1316yes—
330C2×C4C22×Dic2611yescharacteristic
331C2×C4C22×C4×D1311yescharacteristic
332C2×C4C22×D5211yescharacteristic
333C2×C4C23×Dic1311yescharacteristic
334C2×C4C22×C22⋊D1311yescharacteristic
335C2×C4C22×C22⋊D1311yescharacteristic
336C8—28no—
337C8—268no—
338C22—224no—
339C22—212no—
340C22—212no—
341C22—23no—
342C22—21no—
343C22—23no—
344C22—21no—
345C22#11718146116yesdirect product factor
346C22C2×#8806316yes—
347C22C2×#8806416yes—
348C22C2×#8807116yes—
349C22C2×#8807116yes—
350C22C22×C4×Dic1311yescharacteristic
351C22C22×#610111yescharacteristic
352C22C22×#610311yescharacteristic
353C4—528no—
354C4C2×#8807116yes—
355C4C2×#8807116yes—
356C4C22×#610111yescharacteristic
357C4C22×C4⋊Dic1311yescharacteristic
358C4C22×#610311yescharacteristic
359C4C22×C22⋊Dic1311yescharacteristic
360C2—21no—
361C2—21no—
362C2—26no—
363C2C2×#11718146112yesdirect product factor
364C2C22×#8807111yescharacteristic
365C2C22×#8806311yescharacteristic
366C2C22×#8806411yescharacteristic
subgroupG/H
centerC24#6103
commutatorC22×C13C22×C42
FrattiniC22×C4C23×D13
FittingC23×C13×M16C2
socleC24×C13C22⋊C4
prime radicalC25×C13C2×C4
cosocleC22×C4×C13C24
hypercenterC23×M16D13
hypocenterC13C22×#343
LCSG ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C24 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
3. C2×#11718146C2direct (N × H)
9. C22×C13×M16C4semidirect
20. C22×#6099C4semidirect
23. #11718146C22direct (N × H)
42. C2×C13×M16C2×C4semidirect
58. C2×#6099C2×C4semidirect
101. C13×M16C22×C4semidirect
110. #6099C22×C4semidirect
167. C22×M16Dic13semidirect
185. C22×C13#343semidirect
223. C2×M16C2×Dic13semidirect
241. C2×C13C2×#343semidirect
286. M16C22×Dic13semidirect
295. C13C22×#343semidirect
345. C22#11718146direct (N × H)
363. C2C2×#11718146direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#60992
C23×C4⋊Dic132
C2×#117181462
C23×C13×M162
C22×#34313