Finite Group Databasebeta

← C22×#11718894 #16726437

Subgroups — C22×#11718894 #16726437

Order 3328.

5326 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×#1055C211yesmaximal, characteristic, complemented
2C22×#88180C211yesmaximal, characteristic, complemented
3C2×#11718894C2124yesmaximal, direct product factor
4C22×#88075C212yesmaximal, complemented
5C22×#88072C212yesmaximal, complemented
6C23×C13×#41C211yesmaximal, characteristic
7C22×C13×#41—28no—
8C22×C13×#41—26no—
9C22×C13×#41C4116yescomplemented
10C22×C13×#41C22112yes—
11C23×Q8×C13C2211yescharacteristic
12C23×D4×C13—21no—
13C23×D4×C13C2211yescharacteristic
14C24×C4×C13—21no—
15C24×C4×C13C2211yescharacteristic
16C22×C22⋊Dic13—21no—
17C2×#88180—26no—
18C2×#88180C22112yescomplemented
19C22×C4⋊Dic13C2212yes—
20C22×C4×Dic13—21no—
21C23×#1055—28no—
22C23×#1055C2212yes—
23C23×#1055C22112yescomplemented
24#11718894C22164yesdirect product factor
25C2×#88075C22124yescomplemented
26C2×#88072C22124yescomplemented
27C25×C13—21no—
28C25×C13D411yescharacteristic
29C2×C13×#41—296no—
30C2×C13×#41—248no—
31C2×C13×#41—224no—
32C2×C13×#41C2×C4196yescomplemented
33C2×C13×#41C23116yes—
34C22×Q8×C13—24no—
35C22×Q8×C13—23no—
36C22×Q8×C13C2×C418yes—
37C22×Q8×C13C2316yes—
38C22×D4×C13—216no—
39C22×D4×C13—212no—
40C22×D4×C13—23no—
41C22×D4×C13—24no—
42C22×D4×C13C2×C418yes—
43C22×D4×C13C2316yes—
44C23×C4×C13—21no—
45C23×C4×C13—216no—
46C23×C4×C13—212no—
47C23×C4×C13C2×C418yes—
48C23×C4×C13D416yes—
49C23×C4×C13D418yes—
50C23×C4×C13D411yescharacteristic
51C23×C4×C13C2311yescharacteristic
52C23×C4×C13C2316yes—
53C23×Dic13—22no—
54C2×C22⋊Dic13—212no—
55#88180—224no—
56#88180C23116yescomplemented
57C2×C4⋊Dic13—26no—
58C2×C4⋊Dic13C23112yes—
59C2×C4×Dic13—212no—
60C22×#1055—28no—
61C22×#1055—248no—
62C22×#1055C23112yes—
63C22×#1055C23116yescomplemented
64#88075C23132yescomplemented
65#88072C23132yescomplemented
66C22×#416—131nomaximal, Sylow
67C24×C13—28no—
68C24×C13—28no—
69C24×C13—26no—
70C24×C13—48no—
71C24×C13C22⋊C418yes—
72C24×C13C2×D416yes—
73C24×C13C2×D411yescharacteristic
74C13×#41—2192no—
75C13×#41—232no—
76C13×#41C22×C4164yescomplemented
77C2×Q8×C13—224no—
78C2×Q8×C13—212no—
79C2×Q8×C13—26no—
80C2×Q8×C13C22×C4124yes—
81C2×Q8×C13C2414yes—
82C2×D4×C13—296no—
83C2×D4×C13—216no—
84C2×D4×C13—224no—
85C2×D4×C13—26no—
86C2×D4×C13—212no—
87C2×D4×C13C22×C4124yes—
88C2×D4×C13C2414yes—
89C22×C4×C13—28no—
90C22×C4×C13—26no—
91C22×C4×C13—296no—
92C22×C4×C13—216no—
93C22×C4×C13C22⋊C418yes—
94C22×C4×C13C22⋊C4148yes—
95C22×C4×C13C22×C414yes—
96C22×C4×C13C22×C4124yes—
97C22×C4×C13C2×D416yes—
98C22×C4×C13C2×D413yes—
99C22×C4×C13C2×D4112yes—
100C22×C4×C13C2×D414yes—
101C22×C4×C13C2×D4124yes—
102C22×C4×C13C2413yes—
103C22×C4×C13C2414yes—
104C22×Dic13—212no—
105C22×Dic13—48no—
106C22⋊Dic13—216no—
107C4⋊Dic13—212no—
108C4⋊Dic13C2418yes—
109C4×Dic13—216no—
110C2×#1055—224no—
111C2×#1055—232no—
112C2×#1055C2418yes—
113C23×#41D1311yescharacteristic
114C24×C8—131no—
115C22×#117—131no—
116C2×#416—1324no—
117C22×#103—132no—
118C22×#102—132no—
119C23×C13—224no—
120C23×C13—232no—
121C23×C13—224no—
122C23×C13—24no—
123C23×C13—424no—
124C23×C13—432no—
125C23×C13C2×C22⋊C4124yes—
126C23×C13C2×C22⋊C414yes—
127C23×C13#13518yes—
128C23×C13C22×D414yes—
129C23×C13C22×D413yes—
130Q8×C13—224no—
131Q8×C13—24no—
132Q8×C13C23×C418yes—
133D4×C13—264no—
134D4×C13—224no—
135D4×C13—24no—
136D4×C13C23×C418yes—
137C2×C4×C13—224no—
138C2×C4×C13—24no—
139C2×C4×C13—264no—
140C2×C4×C13C2×C22⋊C4124yes—
141C2×C4×C13C2×C22⋊C4112yes—
142C2×C4×C13C2×C22⋊C4148yes—
143C2×C4×C13C23×C416yes—
144C2×C4×C13C23×C418yes—
145C2×C4×C13C22×D414yes—
146C2×C4×C13C22×D413yes—
147C2×C4×C13C22×D416yes—
148C2×C4×C13C22×D418yes—
149C2×C4×C13C2511yescharacteristic
150C2×Dic13—28no—
151C2×Dic13—424no—
152#1055—28no—
153C22×#41—28no—
154C22×#41—26no—
155C22×#41Dic13116yescomplemented
156C22×#41C2×D13112yes—
157C23×Q8C2×D1311yescharacteristic
158C23×D4—21no—
159C23×D4C2×D1311yescharacteristic
160C24×C4—21no—
161C24×C4C2×D1311yescharacteristic
162C23×C8—132no—
163C23×C8—1312no—
164C23×C8—268no—
165C2×#117—1312no—
166C2×#117—266no—
167C22×C4⋊C4—132no—
168C22×C22⋊C4—261no—
169C22×C42—261no—
170#416—1364no—
171C2×#103—1324no—
172C2×#102—1324no—
173C22×C13—248no—
174C22×C13—28no—
175C22×C13—28no—
176C22×C13—448no—
177C22×C13—48no—
178C22×C13#416116yescomplemented
179C22×C13C22×C22⋊C418yes—
180C22×C13C22×C22⋊C416yes—
181C22×C13C2×#135112yes—
182C22×C13C23×D411yescharacteristic
183C4×C13—28no—
184C4×C13C22×C22⋊C418yes—
185C4×C13C22×C22⋊C416yes—
186C4×C13C24×C411yescharacteristic
187C4×C13C23×D411yescharacteristic
188Dic13—48no—
189C25—21no—
190C25C22⋊D1311yescharacteristic
191C2×#41—296no—
192C2×#41—248no—
193C2×#41—224no—
194C2×#41C2×Dic13196yescomplemented
195C2×#41C22×D13116yes—
196C22×Q8—24no—
197C22×Q8—23no—
198C22×Q8C2×Dic1318yes—
199C22×Q8C22×D1316yes—
200C22×D4—216no—
201C22×D4—212no—
202C22×D4—23no—
203C22×D4—24no—
204C22×D4C2×Dic1318yes—
205C22×D4C22×D1316yes—
206C23×C4—21no—
207C23×C4—216no—
208C23×C4—212no—
209C23×C4—262no—
210C23×C4C2×Dic1318yes—
211C23×C4C22⋊D1316yes—
212C23×C4C22⋊D1318yes—
213C23×C4C22⋊D1311yescharacteristic
214C23×C4C22×D1311yescharacteristic
215C23×C4C22×D1316yes—
216C22×C8—1312no—
217C22×C8—1316no—
218C22×C8—268no—
219C22×C8—2648no—
220#117—1316no—
221#117—2624no—
222C2×C4⋊C4—1312no—
223C2×C4⋊C4—266no—
224C2×C22⋊C4—2612no—
225C2×C42—2612no—
226#103—1332no—
227#102—1332no—
228C2×C13—28no—
229C2×C13—48no—
230C2×C13C2×#416112yescomplemented
231C2×C13C23×C22⋊C411yescharacteristic
232C2×C13C22×#13512yes—
233C24—28no—
234C24—28no—
235C24—26no—
236C24—48no—
237C24C22⋊Dic1318yes—
238C24C2×C22⋊D1316yes—
239C24C2×C22⋊D1311yescharacteristic
240#41—2192no—
241#41—232no—
242#41C22×Dic13164yescomplemented
243C2×Q8—224no—
244C2×Q8—212no—
245C2×Q8—26no—
246C2×Q8C22×Dic13124yes—
247C2×Q8C23×D1314yes—
248C2×D4—296no—
249C2×D4—216no—
250C2×D4—224no—
251C2×D4—26no—
252C2×D4—212no—
253C2×D4C22×Dic13124yes—
254C2×D4C23×D1314yes—
255C22×C4—28no—
256C22×C4—26no—
257C22×C4—296no—
258C22×C4—216no—
259C22×C4—2612no—
260C22×C4—528no—
261C22×C4C22⋊Dic1318yes—
262C22×C4C22⋊Dic13148yes—
263C22×C4C22×Dic1314yes—
264C22×C4C22×Dic13124yes—
265C22×C4C2×C22⋊D1316yes—
266C22×C4C2×C22⋊D1313yes—
267C22×C4C2×C22⋊D13112yes—
268C22×C4C2×C22⋊D1314yes—
269C22×C4C2×C22⋊D13124yes—
270C22×C4C23×D1313yes—
271C22×C4C23×D1314yes—
272C2×C8—138no—
273C2×C8—2624no—
274C2×C8—2632no—
275C4⋊C4—138no—
276C4⋊C4—2612no—
277C22⋊C4—2616no—
278C42—2616no—
279C13C22×#41611yescharacteristic, complemented, Sylow
280C23—224no—
281C23—232no—
282C23—224no—
283C23—24no—
284C23—424no—
285C23—432no—
286C23#8820418yes—
287C23C2×C22⋊Dic13124yes—
288C23C2×C22⋊Dic1314yes—
289C23C22×C22⋊D1314yes—
290C23C22×C22⋊D1313yes—
291Q8—224no—
292Q8—24no—
293Q8C23×Dic1318yes—
294D4—264no—
295D4—224no—
296D4—24no—
297D4C23×Dic1318yes—
298C2×C4—224no—
299C2×C4—24no—
300C2×C4—264no—
301C2×C4—268no—
302C2×C4—5224no—
303C2×C4C2×C22⋊Dic13124yes—
304C2×C4C2×C22⋊Dic13112yes—
305C2×C4C2×C22⋊Dic13148yes—
306C2×C4C23×Dic1316yes—
307C2×C4C23×Dic1318yes—
308C2×C4C22×C22⋊D1314yes—
309C2×C4C22×C22⋊D1313yes—
310C2×C4C22×C22⋊D1316yes—
311C2×C4C22×C22⋊D1318yes—
312C2×C4C24×D1311yescharacteristic
313C8—268no—
314C22—248no—
315C22—28no—
316C22—28no—
317C22—448no—
318C22—48no—
319C22#11718894116yesdirect product factor
320C22C2×#88204112yes—
321C22C22×C22⋊Dic1318yes—
322C22C22×C22⋊Dic1316yes—
323C22C23×C22⋊D1311yescharacteristic
324C4—28no—
325C4—528no—
326C4C22×C22⋊Dic1318yes—
327C4C22×C22⋊Dic1316yes—
328C4C24×Dic1311yescharacteristic
329C4C23×C22⋊D1311yescharacteristic
330C2—28no—
331C2—48no—
332C2C2×#11718894112yesdirect product factor
333C2C23×C22⋊Dic1311yescharacteristic
334C2C22×#8820412yes—
subgroupG/H
centerC23×C4C22⋊D13
commutatorC4×C13C24×C4
FrattiniC2×C4C24×D13
Fitting, prime radicalC23×C13×#41C2
socleC24×C13C2×D4
cosocleC2×C4×C13C25
hypercenterC23×#41D13
hypocenterC13C22×#416
LCSG ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23×C4 ◁ C24×C4 ◁ C23×#41

Complements

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

NH ≅ G/Ntype
1. C24×#1055C2semidirect
2. C22×#88180C2semidirect
3. C2×#11718894C2direct (N × H)
4. C22×#88075C2semidirect
5. C22×#88072C2semidirect
9. C22×C13×#41C4semidirect
18. C2×#88180C22semidirect
23. C23×#1055C22semidirect
24. #11718894C22direct (N × H)
25. C2×#88075C22semidirect
26. C2×#88072C22semidirect
32. C2×C13×#41C2×C4semidirect
56. #88180C23semidirect
63. C22×#1055C23semidirect
64. #88075C23semidirect
65. #88072C23semidirect
76. C13×#41C22×C4semidirect
155. C22×#41Dic13semidirect
178. C22×C13#416semidirect
194. C2×#41C2×Dic13semidirect
230. C2×C13C2×#416semidirect
242. #41C22×Dic13semidirect
279. C13C22×#416semidirect
319. C22#11718894direct (N × H)
332. C2C2×#11718894direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#10552
C22×#881802
C2×#117188942
C22×#880752
C22×#880722
C23×C13×#412
C22×#41613