Finite Group Databasebeta

← C22×#11718679 #16726950

Subgroups — C22×#11718679 #16726950

Order 3328.

3222 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#6105C211yesmaximal, characteristic
2C22×C4×Dic26C211yesmaximal, characteristic, complemented
3C2×#11718679C2124yesmaximal, direct product factor
4C22×#88047C211yesmaximal, characteristic, complemented
5C22×#88050C211yesmaximal, characteristic, complemented
6C22×#88072C211yesmaximal, characteristic
7C22×#88043C211yesmaximal, characteristic, complemented
8C22×C4×D4×C13C211yesmaximal, characteristic
9C23×D4×C13C2211yescharacteristic
10C24×C4×C13—21no—
11C23×Dic26C2211yescharacteristic
12C2×C4×D4×C13—26no—
13C2×C4×D4×C13C22112yes—
14C22×C13×C4⋊C4C2211yescharacteristic
15C22×C13×C22⋊C4—21no—
16C22×C42×C13C2211yescharacteristic
17C22×#6105C4116yescomplemented
18C22×#6105C22112yes—
19C22×C4⋊Dic13C2211yescharacteristic
20C22×#6101—21no—
21C22×C4×Dic13—21no—
22C23×#1055C2211yescharacteristic
23C23×#1055C2211yescharacteristic
24C2×C4×Dic26—26no—
25C2×C4×Dic26C22112yescomplemented
26C2×#88072C22112yes—
27#11718679C22164yesdirect product factor
28C2×#88050C22112yescomplemented
29C2×#88047C22112yescomplemented
30C2×#88043C22112yescomplemented
31C25×C13—21no—
32C22×D4×C13—23no—
33C22×D4×C13—24no—
34C22×D4×C13C2×C418yes—
35C22×D4×C13C2316yes—
36C23×C4×C13—21no—
37C23×C4×C13—212no—
38C23×C4×C13—48no—
39C23×C4×C13D411yescharacteristic
40C23×C4×C13D411yescharacteristic
41C23×C4×C13C2311yescharacteristic
42C23×Dic13—21no—
43C23×Dic13—21no—
44C22×Dic26—23no—
45C22×Dic26—24no—
46C22×Dic26C2×C418yescomplemented
47C22×Dic26C2316yes—
48C4×D4×C13—224no—
49C4×D4×C13C23116yes—
50C2×C13×C4⋊C4—23no—
51C2×C13×C4⋊C4C2316yes—
52C2×C13×C22⋊C4—212no—
53C2×C42×C13D416yes—
54C2×C42×C13C2316yes—
55C2×#6105C2×C4196yescomplemented
56C2×#6105C23116yes—
57C2×C4⋊Dic13—23no—
58C2×C4⋊Dic13C2316yes—
59C2×#6101—212no—
60C2×C4×Dic13—212no—
61C22×#1055—24no—
62C22×#1055C2×C418yescomplemented
63C22×#1055C2316yes—
64C22×#1055C2316yes—
65C4×Dic26—224no—
66C4×Dic26C23116yescomplemented
67#88072C23116yes—
68#88050C23116yescomplemented
69#88047C23116yescomplemented
70#88043C23116yescomplemented
71C22×#440—131nomaximal, Sylow
72C24×C13—28no—
73C24×C13—26no—
74C24×C13—44no—
75C24×C13—44no—
76C24×C13C2×D411yescharacteristic
77C2×D4×C13—224no—
78C2×D4×C13—26no—
79C2×D4×C13—212no—
80C2×D4×C13C22×C4124yes—
81C2×D4×C13C2414yes—
82C22×C4×C13—24no—
83C22×C4×C13—24no—
84C22×C4×C13—26no—
85C22×C4×C13—216no—
86C22×C4×C13—448no—
87C22×C4×C13—44no—
88C22×C4×C13C22×C414yes—
89C22×C4×C13C2×D416yes—
90C22×C4×C13C2×D416yes—
91C22×C4×C13C2×D413yes—
92C22×C4×C13#4114yes—
93C22×C4×C13C2413yes—
94C22×Dic13—28no—
95C22×Dic13—26no—
96C22×Dic13—26no—
97C22×Dic13—44no—
98C2×Dic26—224no—
99C2×Dic26—26no—
100C2×Dic26—212no—
101C2×Dic26C22×C4124yescomplemented
102C2×Dic26C2414yes—
103C13×C4⋊C4—26no—
104C13×C4⋊C4C2414yes—
105C13×C22⋊C4—216no—
106C42×C13C2×D4112yes—
107C42×C13C2414yes—
108#6105C22×C4164yescomplemented
109C4⋊Dic13—26no—
110C4⋊Dic13C2414yes—
111#6101—216no—
112C4×Dic13—216no—
113C2×#1055—212no—
114C2×#1055C22×C4124yescomplemented
115C2×#1055C2414yes—
116C2×#1055C2414yes—
117C23×SD16—131no—
118C22×C4×Q8—131no—
119C22×C4×D4D1311yescharacteristic
120C2×#440—1324no—
121C22×#107—131no—
122C22×#103—131no—
123C22×#102—131no—
124C22×#97—131no—
125C23×C13—224no—
126C23×C13—24no—
127C23×C13—412no—
128C23×C13—412no—
129C23×C13—432no—
130C23×C13C4×D414yes—
131C23×C13AGL(1,ℤ/8)14yes—
132C23×C13#13714yes—
133C23×C13C22×D413yes—
134D4×C13—224no—
135D4×C13—24no—
136D4×C13C23×C418yes—
137C2×C4×C13—212no—
138C2×C4×C13—212no—
139C2×C4×C13—24no—
140C2×C4×C13—432no—
141C2×C4×C13—412no—
142C2×C4×C13C4×D4112yes—
143C2×C4×C13C23×C416yes—
144C2×C4×C13C22×D414yes—
145C2×C4×C13C22×D414yes—
146C2×C4×C13C22×D413yes—
147C2×C4×C13C2×#4116yes—
148C2×C4×C13C2511yescharacteristic
149C2×Dic13—224no—
150C2×Dic13—24no—
151C2×Dic13—24no—
152C2×Dic13—412no—
153Dic26—224no—
154Dic26—24no—
155Dic26C23×C418yescomplemented
156#1055—24no—
157#1055C23×C418yescomplemented
158C23×Q8—131no—
159C23×D4C2×D1311yescharacteristic
160C24×C4—21no—
161C22×SD16—1316no—
162C22×SD16—1312no—
163C23×C8—131no—
164C23×C8—131no—
165C2×C4×Q8—1312no—
166C2×C4×Q8—266no—
167C2×C4×D4—26no—
168C2×C4×D4C2×D13112yes—
169C22×C4⋊C4—131no—
170C22×C4⋊C4—261no—
171C22×C4⋊C4C2×D1311yescharacteristic
172C22×C22⋊C4—21no—
173C22×C42—261no—
174C22×C42C2×D1311yescharacteristic
175#440—1364no—
176C2×#107—1312no—
177C2×#103—1312no—
178C2×#102—1312no—
179C2×#97—1312no—
180C22×C13—28no—
181C22×C13—448no—
182C22×C13—44no—
183C22×C13—44no—
184C22×C13#440116yescomplemented
185C22×C13C2×C4×D416yes—
186C22×C13C2×AGL(1,ℤ/8)16yes—
187C22×C13C2×#13716yes—
188C22×C13C23×D411yescharacteristic
189C4×C13—24no—
190C4×C13—24no—
191C4×C13—44no—
192C4×C13C2×C4×D416yes—
193C4×C13C24×C411yescharacteristic
194C4×C13C22×#4111yescharacteristic
195Dic13—28no—
196Dic13—44no—
197C25—21no—
198C22×Q8—138no—
199C22×Q8—136no—
200C22×Q8—264no—
201C22×Q8—263no—
202C22×D4—24no—
203C22×D4—23no—
204C22×D4C4×D1318yes—
205C22×D4C22×D1316yes—
206C23×C4—21no—
207C23×C4—212no—
208C23×C4—48no—
209C23×C4—261no—
210C23×C4—261no—
211C23×C4C22⋊D1311yescharacteristic
212C23×C4C22⋊D1311yescharacteristic
213C23×C4C22×D1311yescharacteristic
214C2×SD16—1396no—
215C2×SD16—1316no—
216C22×C8—138no—
217C22×C8—136no—
218C22×C8—136no—
219C22×C8—264no—
220C4×Q8—1316no—
221C4×Q8—2624no—
222C4×D4—224no—
223C4×D4C22×D13116yes—
224C2×C4⋊C4—23no—
225C2×C4⋊C4—136no—
226C2×C4⋊C4—263no—
227C2×C4⋊C4—2612no—
228C2×C4⋊C4C22×D1316yes—
229C2×C22⋊C4—212no—
230C2×C42—2612no—
231C2×C42C22⋊D1316yes—
232C2×C42C22×D1316yes—
233#107—1316no—
234#103—1316no—
235#102—1316no—
236#97—1316no—
237C2×C13—48no—
238C2×C13C2×#440112yescomplemented
239C2×C13C22×C4×D411yescharacteristic
240C2×C13C22×AGL(1,ℤ/8)11yescharacteristic
241C2×C13C22×#13711yescharacteristic
242C24—28no—
243C24—26no—
244C24—44no—
245C24—44no—
246C24C2×C22⋊D1311yescharacteristic
247C2×Q8—1324no—
248C2×Q8—134no—
249C2×Q8—2624no—
250C2×Q8—2612no—
251C2×Q8—266no—
252C2×D4—224no—
253C2×D4—212no—
254C2×D4—26no—
255C2×D4C2×C4×D13124yes—
256C2×D4C23×D1314yes—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—26no—
260C22×C4—216no—
261C22×C4—448no—
262C22×C4—44no—
263C22×C4—268no—
264C22×C4—266no—
265C22×C4—266no—
266C22×C4—524no—
267C22×C4C2×C4×D1314yes—
268C22×C4#612714yes—
269C22×C4C2×C22⋊D1316yes—
270C22×C4C2×C22⋊D1316yes—
271C22×C4C2×C22⋊D1313yes—
272C22×C4C23×D1313yes—
273SD16—1364no—
274C2×C8—1324no—
275C2×C8—134no—
276C2×C8—134no—
277C2×C8—2612no—
278C4⋊C4—26no—
279C4⋊C4—134no—
280C4⋊C4—266no—
281C4⋊C4—2616no—
282C4⋊C4C23×D1314yes—
283C22⋊C4—216no—
284C42—2616no—
285C42C2×C22⋊D13112yes—
286C42C23×D1314yes—
287C13C22×#44011yescharacteristic, complemented, Sylow
288C23—224no—
289C23—24no—
290C23—432no—
291C23—412no—
292C23—412no—
293C23C4×C22⋊D1314yes—
294C23#8818614yes—
295C23#8820514yes—
296C23C22×C22⋊D1313yes—
297Q8—138no—
298Q8—2624no—
299Q8—264no—
300D4—224no—
301D4—24no—
302D4C22×C4×D1318yes—
303C2×C4—212no—
304C2×C4—212no—
305C2×C4—24no—
306C2×C4—432no—
307C2×C4—412no—
308C2×C4—2624no—
309C2×C4—264no—
310C2×C4—264no—
311C2×C4—5212no—
312C2×C4C4×C22⋊D13112yes—
313C2×C4C22×C4×D1316yes—
314C2×C4C2×#612716yes—
315C2×C4C22×C22⋊D1314yes—
316C2×C4C22×C22⋊D1314yes—
317C2×C4C22×C22⋊D1313yes—
318C2×C4C24×D1311yescharacteristic
319C8—138no—
320C8—264no—
321C22—28no—
322C22—448no—
323C22—44no—
324C22—44no—
325C22#11718679116yesdirect product factor
326C22C2×C4×C22⋊D1316yes—
327C22C2×#8818616yes—
328C22C2×#8820516yes—
329C22C23×C22⋊D1311yescharacteristic
330C4—24no—
331C4—24no—
332C4—44no—
333C4—268no—
334C4—524no—
335C4C2×C4×C22⋊D1316yes—
336C4C23×C4×D1311yescharacteristic
337C4C22×#612711yescharacteristic
338C2—48no—
339C2C2×#11718679112yesdirect product factor
340C2C22×C4×C22⋊D1311yescharacteristic
341C2C22×#8818611yescharacteristic
342C2C22×#8820511yescharacteristic
subgroupG/H
centerC24C2×C22⋊D13
commutatorC4×C13C24×C4
FrattiniC2×C4C24×D13
FittingC22×C4×D4×C13C2
socleC24×C13C2×D4
prime radicalC23×D4×C13C22
cosocleC2×C4×C13C25
hypercenterC22×C4×D4D13
hypocenterC13C22×#440
LCSG ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C24 ◁ C22×C42 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic26C2semidirect
3. C2×#11718679C2direct (N × H)
4. C22×#88047C2semidirect
5. C22×#88050C2semidirect
7. C22×#88043C2semidirect
17. C22×#6105C4semidirect
25. C2×C4×Dic26C22semidirect
27. #11718679C22direct (N × H)
28. C2×#88050C22semidirect
29. C2×#88047C22semidirect
30. C2×#88043C22semidirect
46. C22×Dic26C2×C4semidirect
55. C2×#6105C2×C4semidirect
62. C22×#1055C2×C4semidirect
66. C4×Dic26C23semidirect
68. #88050C23semidirect
69. #88047C23semidirect
70. #88043C23semidirect
101. C2×Dic26C22×C4semidirect
108. #6105C22×C4semidirect
114. C2×#1055C22×C4semidirect
155. Dic26C23×C4semidirect
157. #1055C23×C4semidirect
184. C22×C13#440semidirect
238. C2×C13C2×#440semidirect
287. C13C22×#440semidirect
325. C22#11718679direct (N × H)
339. C2C2×#11718679direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#61052
C22×C4×Dic262
C2×#117186792
C22×#880472
C22×#880502
C22×#880722
C22×#880432
C22×C4×D4×C132
C22×#44013