Finite Group Databasebeta

← C22×#11718774 #16726438

Subgroups — C22×#11718774 #16726438

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
1C23×#6127C211yesmaximal, characteristic
2C22×#88180C211yesmaximal, characteristic, complemented
3C2×#11718774C2124yesmaximal, direct product factor
4C22×#88056C212yesmaximal, complemented
5C22×#88061C212yesmaximal, complemented
6C24×C8×C13C211yesmaximal, characteristic, complemented
7C24×C4×C13C2211yescharacteristic
8C23×C22⋊D13—21no—
9C22×#6127—26no—
10C22×#6127—28no—
11C22×#6127C4116yescomplemented
12C22×#6127C22112yes—
13C23×D52C2211yescharacteristic
14C23×C4×D13—21no—
15C23×Dic26C2211yescharacteristic
16C23×C8×C13—28no—
17C23×C8×C13C2212yes—
18C23×C8×C13C22112yescomplemented
19C22×C22⋊Dic13—21no—
20C2×#88180—26no—
21C2×#88180C22112yescomplemented
22C22×C4⋊Dic13C2212yes—
23C22×C4×Dic13—21no—
24#11718774C22164yesdirect product factor
25C2×#88061C22124yescomplemented
26C2×#88056C22124yescomplemented
27C25×C13D411yescharacteristic
28C24×D13—21no—
29C23×C4×C13C2×C418yes—
30C23×C4×C13D416yes—
31C23×C4×C13D411yescharacteristic
32C23×C4×C13D418yes—
33C23×C4×C13C2311yescharacteristic
34C23×C4×C13C2316yes—
35C22×C22⋊D13—216no—
36C22×C22⋊D13—212no—
37C23×Dic13—21no—
38C23×Dic13—22no—
39C2×#6127—296no—
40C2×#6127—224no—
41C2×#6127—248no—
42C2×#6127C2×C4196yescomplemented
43C2×#6127C23116yes—
44C22×D52—23no—
45C22×D52—24no—
46C22×D52C2×C418yes—
47C22×D52C2316yes—
48C22×C4×D13—216no—
49C22×C4×D13—212no—
50C22×Dic26—23no—
51C22×Dic26—24no—
52C22×Dic26C2×C418yes—
53C22×Dic26C2316yes—
54C22×C8×C13—28no—
55C22×C8×C13—248no—
56C22×C8×C13C23112yes—
57C22×C8×C13C23116yescomplemented
58C2×C22⋊Dic13—212no—
59#88180—224no—
60#88180C23116yescomplemented
61C2×C4⋊Dic13—26no—
62C2×C4⋊Dic13C23112yes—
63C2×C4×Dic13—212no—
64#88061C23132yescomplemented
65#88056C23132yescomplemented
66C22×#416—131nomaximal, Sylow
67C24×C13—28no—
68C24×C13C22⋊C418yes—
69C24×C13C2×D416yes—
70C24×C13C2×D411yescharacteristic
71C23×D13—28no—
72C23×D13—26no—
73C23×D13—48no—
74C22×C4×C13C22⋊C418yes—
75C22×C4×C13C22⋊C4148yes—
76C22×C4×C13C22×C414yes—
77C22×C4×C13C22×C4124yes—
78C22×C4×C13C2×D416yes—
79C22×C4×C13C2×D413yes—
80C22×C4×C13C2×D4112yes—
81C22×C4×C13C2×D414yes—
82C22×C4×C13C2×D4124yes—
83C22×C4×C13C2413yes—
84C22×C4×C13C2414yes—
85C2×C22⋊D13—296no—
86C2×C22⋊D13—216no—
87C22×Dic13—28no—
88C22×Dic13—26no—
89C22×Dic13—212no—
90C22×Dic13—48no—
91#6127—2192no—
92#6127—232no—
93#6127C22×C4164yescomplemented
94C2×D52—224no—
95C2×D52—26no—
96C2×D52—212no—
97C2×D52C22×C4124yes—
98C2×D52C2414yes—
99C2×C4×D13—296no—
100C2×C4×D13—216no—
101C2×Dic26—224no—
102C2×Dic26—26no—
103C2×Dic26—212no—
104C2×Dic26C22×C4124yes—
105C2×Dic26C2414yes—
106C2×C8×C13—224no—
107C2×C8×C13—232no—
108C2×C8×C13C2418yes—
109C22⋊Dic13—216no—
110C4⋊Dic13—212no—
111C4⋊Dic13C2418yes—
112C4×Dic13—216no—
113C23×#41—131no—
114C24×C8D1311yescharacteristic, complemented
115C22×#117—131no—
116C2×#416—1324no—
117C22×#103—132no—
118C22×#102—132no—
119C23×C13—224no—
120C23×C13—232no—
121C23×C13C2×C22⋊C4124yes—
122C23×C13C2×C22⋊C414yes—
123C23×C13#13518yes—
124C23×C13C22×D414yes—
125C23×C13C22×D413yes—
126C22×D13—224no—
127C22×D13—24no—
128C22×D13—424no—
129C22×D13—432no—
130C2×C4×C13C2×C22⋊C4112yes—
131C2×C4×C13C2×C22⋊C4124yes—
132C2×C4×C13C2×C22⋊C4148yes—
133C2×C4×C13C23×C416yes—
134C2×C4×C13C23×C418yes—
135C2×C4×C13C22×D414yes—
136C2×C4×C13C22×D413yes—
137C2×C4×C13C22×D416yes—
138C2×C4×C13C22×D418yes—
139C2×C4×C13C2511yescharacteristic
140C22⋊D13—264no—
141C2×Dic13—224no—
142C2×Dic13—24no—
143C2×Dic13—28no—
144C2×Dic13—424no—
145D52—224no—
146D52—24no—
147D52C23×C418yes—
148C4×D13—264no—
149Dic26—224no—
150Dic26—24no—
151Dic26C23×C418yes—
152C8×C13—28no—
153C22×#41—1316no—
154C22×#41—1312no—
155C22×#41—266no—
156C22×#41—268no—
157C23×Q8—131no—
158C23×D4—131no—
159C23×D4—261no—
160C24×C4—261no—
161C24×C4C2×D1311yescharacteristic
162C23×C8—28no—
163C23×C8C2×D1312yes—
164C23×C8C2×D13112yescomplemented
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#416116yescomplemented
176C22×C13C22×C22⋊C418yes—
177C22×C13C22×C22⋊C416yes—
178C22×C13C2×#135112yes—
179C22×C13C23×D411yescharacteristic
180C2×D13—28no—
181C2×D13—448no—
182C2×D13—48no—
183C4×C13C22×C22⋊C416yes—
184C4×C13C22×C22⋊C418yes—
185C4×C13C24×C411yescharacteristic
186C4×C13C23×D411yescharacteristic
187Dic13—28no—
188Dic13—48no—
189C25—261no—
190C25D5211yescharacteristic
191C2×#41—1396no—
192C2×#41—1316no—
193C2×#41—2696no—
194C2×#41—2624no—
195C2×#41—2648no—
196C22×Q8—138no—
197C22×Q8—136no—
198C22×Q8—263no—
199C22×Q8—264no—
200C22×D4—138no—
201C22×D4—136no—
202C22×D4—2616no—
203C22×D4—2612no—
204C22×D4—263no—
205C22×D4—264no—
206C23×C4—261no—
207C23×C4—2616no—
208C23×C4—2612no—
209C23×C4—262no—
210C23×C4C4×D1318yes—
211C23×C4D5216yes—
212C23×C4D5211yescharacteristic
213C23×C4C22⋊D1318yes—
214C23×C4C22×D1311yescharacteristic
215C23×C4C22×D1316yes—
216C22×C8—28no—
217C22×C8—248no—
218C22×C8C22×D13112yes—
219C22×C8C22×D13116yescomplemented
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×C13C2×#416112yescomplemented
230C2×C13C23×C22⋊C411yescharacteristic
231C2×C13C22×#13512yes—
232D13—48no—
233C24—28no—
234C24—268no—
235C24—266no—
236C24—528no—
237C24#610318yes—
238C24C2×D5216yes—
239C24C2×D5211yescharacteristic
240#41—1364no—
241#41—26192no—
242#41—2632no—
243C2×Q8—1324no—
244C2×Q8—134no—
245C2×Q8—2624no—
246C2×Q8—266no—
247C2×Q8—2612no—
248C2×D4—1324no—
249C2×D4—134no—
250C2×D4—2696no—
251C2×D4—2616no—
252C2×D4—2624no—
253C2×D4—266no—
254C2×D4—2612no—
255C22×C4—268no—
256C22×C4—266no—
257C22×C4—2696no—
258C22×C4—2616no—
259C22×C4—2612no—
260C22×C4—528no—
261C22×C4#610318yes—
262C22×C4#6103148yes—
263C22×C4C2×C4×D1314yes—
264C22×C4C2×C4×D13124yes—
265C22×C4C2×D5216yes—
266C22×C4C2×D5213yes—
267C22×C4C2×D52112yes—
268C22×C4C2×C22⋊D1314yes—
269C22×C4C2×C22⋊D13124yes—
270C22×C4C23×D1313yes—
271C22×C4C23×D1314yes—
272C2×C8—224no—
273C2×C8—232no—
274C2×C8C23×D1318yes—
275C4⋊C4—138no—
276C4⋊C4—2612no—
277C22⋊C4—2616no—
278C42—2616no—
279C13C22×#41611yescharacteristic, complemented, Sylow
280C23—224no—
281C23—232no—
282C23—2624no—
283C23—264no—
284C23—5224no—
285C23—5232no—
286C23#8815818yes—
287C23C2×#6103124yes—
288C23C2×#610314yes—
289C23C22×D5214yes—
290C23C22×D5213yes—
291Q8—138no—
292Q8—2624no—
293Q8—264no—
294D4—138no—
295D4—2664no—
296D4—2624no—
297D4—264no—
298C2×C4—2624no—
299C2×C4—264no—
300C2×C4—2664no—
301C2×C4—268no—
302C2×C4—5224no—
303C2×C4C2×#6103124yes—
304C2×C4C2×#6103112yes—
305C2×C4C2×#6103148yes—
306C2×C4C22×C4×D1316yes—
307C2×C4C22×C4×D1318yes—
308C2×C4C22×D5214yes—
309C2×C4C22×D5213yes—
310C2×C4C22×C22⋊D1316yes—
311C2×C4C22×C22⋊D1318yes—
312C2×C4C24×D1311yescharacteristic
313C8—28no—
314C22—248no—
315C22—28no—
316C22—268no—
317C22—5248no—
318C22—528no—
319C22#11718774116yesdirect product factor
320C22C2×#88158112yes—
321C22C22×#610318yes—
322C22C22×#610316yes—
323C22C23×D5211yescharacteristic
324C4—268no—
325C4—528no—
326C4C22×#610318yes—
327C4C22×#610316yes—
328C4C23×C4×D1311yescharacteristic
329C4C23×C22⋊D1311yescharacteristic
330C2—28no—
331C2—528no—
332C2C2×#11718774112yesdirect product factor
333C2C23×#610311yescharacteristic
334C2C22×#8815812yes—
subgroupG/H
centerC23×C4D52
commutatorC4×C13C24×C4
FrattiniC2×C4C24×D13
FittingC24×C8×C13C2
socleC24×C13C2×D4
prime radicalC23×#6127C2
cosocleC2×C4×C13C25
hypercenterC24×C8D13
hypocenterC13C22×#416
LCSG ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23×C4 ◁ C24×C4 ◁ C24×C8

Complements

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

NH ≅ G/Ntype
2. C22×#88180C2semidirect
3. C2×#11718774C2direct (N × H)
4. C22×#88056C2semidirect
5. C22×#88061C2semidirect
6. C24×C8×C13C2semidirect
11. C22×#6127C4semidirect
18. C23×C8×C13C22semidirect
21. C2×#88180C22semidirect
24. #11718774C22direct (N × H)
25. C2×#88061C22semidirect
26. C2×#88056C22semidirect
42. C2×#6127C2×C4semidirect
57. C22×C8×C13C23semidirect
60. #88180C23semidirect
64. #88061C23semidirect
65. #88056C23semidirect
93. #6127C22×C4semidirect
114. C24×C8D13semidirect
164. C23×C8C2×D13semidirect
175. C22×C13#416semidirect
219. C22×C8C22×D13semidirect
229. C2×C13C2×#416semidirect
279. C13C22×#416semidirect
319. C22#11718774direct (N × H)
332. C2C2×#11718774direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#61272
C22×#881802
C2×#117187742
C22×#880562
C22×#880612
C24×C8×C132
C22×#41613