Finite Group Databasebeta

← C22×#16839524 #29677408

Subgroups — C22×#16839524 #29677408

Order 15616.

3006 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×#92418C211yesmaximal, characteristic
2C23×C4⋊Dic61C211yesmaximal, characteristic
3C2×#16839524C2112yesmaximal, direct product factor
4C23×C4⋊C4×C61C211yesmaximal, characteristic
5C24×Dic61—21no—
6C23×#92418—24no—
7C23×#92418—24no—
8C23×#92418C418yescomplemented
9C23×#92418C2216yes—
10C22×C4⋊Dic61—24no—
11C22×C4⋊Dic61—23no—
12C22×C4⋊Dic61C418yescomplemented
13C22×C4⋊Dic61C2216yes—
14#16839524C22116yesdirect product factor
15C24×C4×C61—21no—
16C24×C4×C61C2211yescharacteristic
17C22×C4⋊C4×C61—23no—
18C22×C4⋊C4×C61—24no—
19C22×C4⋊C4×C61C418yescomplemented
20C22×C4⋊C4×C61C2216yes—
21C23×Dic61—28no—
22C23×Dic61—26no—
23C23×Dic61—44no—
24C23×Dic61—44no—
25C22×#92418—232no—
26C22×#92418—212no—
27C22×#92418—212no—
28C22×#92418C2×C4124yescomplemented
29C22×#92418C2314yes—
30C2×C4⋊Dic61—224no—
31C2×C4⋊Dic61—212no—
32C2×C4⋊Dic61—26no—
33C2×C4⋊Dic61C2×C4124yescomplemented
34C2×C4⋊Dic61C2314yes—
35C25×C61D411yescharacteristic
36C23×C4×C61—28no—
37C23×C4×C61—26no—
38C23×C4×C61—44no—
39C23×C4×C61—44no—
40C23×C4×C61C2×C414yes—
41C23×C4×C61C2×C414yes—
42C23×C4×C61C2×C414yes—
43C23×C4×C61D414yes—
44C23×C4×C61D413yes—
45C23×C4×C61D414yes—
46C23×C4×C61Q814yes—
47C23×C4×C61C2313yes—
48C2×C4⋊C4×C61—224no—
49C2×C4⋊C4×C61—26no—
50C2×C4⋊C4×C61—212no—
51C2×C4⋊C4×C61C2×C4124yescomplemented
52C2×C4⋊C4×C61C2314yes—
53C24×C61C22⋊C414yes—
54C24×C61C22⋊C414yes—
55C24×C61C4⋊C414yes—
56C24×C61D814yes—
57C24×C61SD1614yes—
58C24×C61SD1614yes—
59C24×C61Q1614yes—
60C24×C61C2×D413yes—
61C22×C4×C61—224no—
62C22×C4×C61—24no—
63C22×C4×C61—432no—
64C22×C4×C61—412no—
65C22×C4×C61—412no—
66C22×C4×C61C42116yes—
67C22×C4×C61C22⋊C4112yes—
68C22×C4×C61C22⋊C4112yes—
69C22×C4×C61C22⋊C4116yes—
70C22×C4×C61C4⋊C4112yes—
71C22×C4×C61C4⋊C4116yes—
72C22×C4×C61C4⋊C4116yes—
73C22×C4×C61C22×C416yes—
74C22×C4×C61C22×C416yes—
75C22×C4×C61C22×C416yes—
76C22×C4×C61C2×D413yes—
77C22×C4×C61C2×D416yes—
78C22×C4×C61C2×D416yes—
79C22×C4×C61C2×Q816yes—
80C22×C4×C61C2411yescharacteristic
81C22×Dic61—224no—
82C22×Dic61—24no—
83C22×Dic61—432no—
84C22×Dic61—412no—
85C22×Dic61—412no—
86C4⋊C4×C61—224no—
87C4⋊C4×C61—24no—
88C4⋊C4×C61C22×C418yescomplemented
89C4⋊Dic61—224no—
90C4⋊Dic61—24no—
91C4⋊Dic61C22×C418yescomplemented
92C2×#92418—248no—
93C2×#92418—24no—
94C2×#92418—24no—
95C2×#92418C22×C418yescomplemented
96C23×C61#95116yes—
97C23×C61#102116yes—
98C23×C61#102116yes—
99C23×C61#103116yes—
100C23×C61#103116yes—
101C23×C61#106116yes—
102C23×C61#107116yes—
103C23×C61C2×C22⋊C416yes—
104C23×C61C2×C22⋊C416yes—
105C23×C61C2×C4⋊C416yes—
106C23×C61C2×D816yes—
107C23×C61C2×SD1616yes—
108C23×C61C2×SD1616yes—
109C23×C61C2×Q1616yes—
110C23×C61C22×D411yescharacteristic
111C2×C4×C61—28no—
112C2×C4×C61—44no—
113C2×C4×C61—44no—
114C2×C4×C61—448no—
115C2×C4×C61#95148yes—
116C2×C4×C61C2×C42112yes—
117C2×C4×C61C2×C22⋊C4112yes—
118C2×C4×C61C2×C22⋊C416yes—
119C2×C4×C61C2×C22⋊C416yes—
120C2×C4×C61C2×C4⋊C4112yes—
121C2×C4×C61C2×C4⋊C4112yes—
122C2×C4×C61C2×C4⋊C416yes—
123C2×C4×C61C23×C411yescharacteristic
124C2×C4×C61C23×C411yescharacteristic
125C2×C4×C61C23×C411yescharacteristic
126C2×C4×C61C22×D411yescharacteristic
127C2×C4×C61C22×D411yescharacteristic
128C2×C4×C61C22×Q811yescharacteristic
129C2×Dic61—28no—
130C2×Dic61—44no—
131C2×Dic61—44no—
132C2×Dic61—448no—
133#92418—28no—
134C22×#340—611nomaximal, Sylow
135C22×C61#340164yescomplemented
136C22×C61C2×#95112yes—
137C22×C61C2×#102112yes—
138C22×C61C2×#102112yes—
139C22×C61C2×#103112yes—
140C22×C61C2×#103112yes—
141C22×C61C2×#106112yes—
142C22×C61C2×#107112yes—
143C22×C61C22×C22⋊C411yescharacteristic
144C22×C61C22×C22⋊C411yescharacteristic
145C22×C61C22×C4⋊C411yescharacteristic
146C22×C61C22×D811yescharacteristic
147C22×C61C22×SD1611yescharacteristic
148C22×C61C22×SD1611yescharacteristic
149C22×C61C22×Q1611yescharacteristic
150C4×C61—48no—
151C4×C61C2×#95112yes—
152C4×C61C22×C4211yescharacteristic
153C4×C61C22×C22⋊C411yescharacteristic
154C4×C61C22×C4⋊C411yescharacteristic
155C4×C61C22×C4⋊C411yescharacteristic
156Dic61—48no—
157C24×C8—611no—
158C23×C4⋊C4—611no—
159C23×C4⋊C4D6111yescharacteristic
160C2×#340—6112no—
161C2×C61C2×#340124yescomplemented
162C2×C61C22×#9511yescharacteristic
163C2×C61C22×#10211yescharacteristic
164C2×C61C22×#10211yescharacteristic
165C2×C61C22×#10311yescharacteristic
166C2×C61C22×#10311yescharacteristic
167C2×C61C22×#10611yescharacteristic
168C2×C61C22×#10711yescharacteristic
169C24×C4—21no—
170C24×C4—1221no—
171C24×C4C2×D6111yescharacteristic
172C23×C8—618no—
173C23×C8—616no—
174C23×C8—1224no—
175C23×C8—1224no—
176C22×C4⋊C4—23no—
177C22×C4⋊C4—24no—
178C22×C4⋊C4—618no—
179C22×C4⋊C4—616no—
180C22×C4⋊C4—1224no—
181C22×C4⋊C4—1223no—
182C22×C4⋊C4Dic6118yescomplemented
183C22×C4⋊C4C2×D6116yes—
184#340—6116no—
185C61C22×#34011yescharacteristic, complemented, Sylow
186C25C22⋊D6111yescharacteristic
187C23×C4—28no—
188C23×C4—26no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—1228no—
192C23×C4—1226no—
193C23×C4—2444no—
194C23×C4—2444no—
195C23×C4Dic12214yes—
196C23×C4C4×D6114yes—
197C23×C4C4×D6114yes—
198C23×C4D24414yes—
199C23×C4C2×Dic6114yes—
200C23×C4C22⋊D6113yes—
201C23×C4C22⋊D6114yes—
202C23×C4C22×D6113yes—
203C22×C8—6124no—
204C22×C8—614no—
205C22×C8—12232no—
206C22×C8—12212no—
207C22×C8—12212no—
208C2×C4⋊C4—224no—
209C2×C4⋊C4—212no—
210C2×C4⋊C4—26no—
211C2×C4⋊C4—6124no—
212C2×C4⋊C4—614no—
213C2×C4⋊C4—12224no—
214C2×C4⋊C4—12212no—
215C2×C4⋊C4—1226no—
216C2×C4⋊C4C2×Dic61124yescomplemented
217C2×C4⋊C4C22×D6114yes—
218C24#1175819614yes—
219C24#1175819814yes—
220C24#1175819914yes—
221C24#1175820014yes—
222C24#1175820114yes—
223C24#1175820214yes—
224C24C22⋊Dic6114yes—
225C24C2×C22⋊D6113yes—
226C22×C4—24no—
227C22×C4—224no—
228C22×C4—432no—
229C22×C4—412no—
230C22×C4—412no—
231C22×C4—1224no—
232C22×C4—12224no—
233C22×C4—24432no—
234C22×C4—24412no—
235C22×C4—24412no—
236C22×C4C4×Dic61116yes—
237C22×C4#11758196112yes—
238C22×C4#11758196116yes—
239C22×C4C4⋊Dic61116yes—
240C22×C4#11758198112yes—
241C22×C4#11758198116yes—
242C22×C4C22⋊Dic61112yes—
243C22×C4C2×Dic12216yes—
244C22×C4C2×C4×D6116yes—
245C22×C4C2×C4×D6116yes—
246C22×C4C2×D24416yes—
247C22×C4C22×Dic6116yes—
248C22×C4C2×C22⋊D6116yes—
249C22×C4C2×C22⋊D6113yes—
250C22×C4C23×D6111yescharacteristic
251C2×C8—618no—
252C2×C8—1224no—
253C2×C8—1224no—
254C2×C8—12248no—
255C4⋊C4—24no—
256C4⋊C4—224no—
257C4⋊C4—618no—
258C4⋊C4—1224no—
259C4⋊C4—12224no—
260C4⋊C4C22×Dic6118yescomplemented
261C23#14519684116yes—
262C23#14519697116yes—
263C23#14519698116yes—
264C23#14519700116yes—
265C23#14519701116yes—
266C23#14519708116yes—
267C23#14519710116yes—
268C23C2×#1175819816yes—
269C23C2×C22⋊Dic6116yes—
270C23C2×#1175819616yes—
271C23C2×#1175819916yes—
272C23C2×#1175820116yes—
273C23C2×#1175820016yes—
274C23C2×#1175820216yes—
275C23C22×C22⋊D6111yescharacteristic
276C2×C4—28no—
277C2×C4—448no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—1228no—
281C2×C4—24448no—
282C2×C4—2444no—
283C2×C4—2444no—
284C2×C4#14519684148yes—
285C2×C4C2×C4×Dic61112yes—
286C2×C4C2×#11758198112yes—
287C2×C4C2×#1175819816yes—
288C2×C4C2×C22⋊Dic6116yes—
289C2×C4C2×#11758196112yes—
290C2×C4C2×#1175819616yes—
291C2×C4C2×C4⋊Dic61112yes—
292C2×C4C22×C4×D6111yescharacteristic
293C2×C4C22×C4×D6111yescharacteristic
294C2×C4C23×Dic6111yescharacteristic
295C2×C4C22×C22⋊D6111yescharacteristic
296C2×C4C22×D24411yescharacteristic
297C2×C4C22×Dic12211yescharacteristic
298C8—1228no—
299C22#16839524164yesdirect product factor
300C22C2×#14519684112yes—
301C22C2×#14519697112yes—
302C22C2×#14519698112yes—
303C22C2×#14519700112yes—
304C22C2×#14519701112yes—
305C22C2×#14519708112yes—
306C22C2×#14519710112yes—
307C22C22×#1175819811yescharacteristic
308C22C22×C22⋊Dic6111yescharacteristic
309C22C22×#1175819611yescharacteristic
310C22C22×#1175819911yescharacteristic
311C22C22×#1175820111yescharacteristic
312C22C22×#1175820011yescharacteristic
313C22C22×#1175820211yescharacteristic
314C4—48no—
315C4—2448no—
316C4C2×#14519684112yes—
317C4C22×C4×Dic6111yescharacteristic
318C4C22×#1175819811yescharacteristic
319C4C22×#1175819611yescharacteristic
320C4C22×C4⋊Dic6111yescharacteristic
321C2C2×#16839524124yesdirect product factor
322C2C22×#1451968411yescharacteristic
323C2C22×#1451969711yescharacteristic
324C2C22×#1451969811yescharacteristic
325C2C22×#1451970011yescharacteristic
326C2C22×#1451970111yescharacteristic
327C2C22×#1451970811yescharacteristic
328C2C22×#1451971011yescharacteristic
subgroupG/H
centerC25C22⋊D61
commutatorC4×C61C22×C42
FrattiniC22×C4C23×D61
FittingC23×C4⋊C4×C61C2
socle, prime radicalC25×C61D4
cosocleC22×C4×C61C24
hypercenterC23×C4⋊C4D61
hypocenterC61C22×#340
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C25 ◁ C24×C4 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#16839524C2direct (N × H)
8. C23×#92418C4semidirect
12. C22×C4⋊Dic61C4semidirect
14. #16839524C22direct (N × H)
19. C22×C4⋊C4×C61C4semidirect
28. C22×#92418C2×C4semidirect
33. C2×C4⋊Dic61C2×C4semidirect
51. C2×C4⋊C4×C61C2×C4semidirect
88. C4⋊C4×C61C22×C4semidirect
91. C4⋊Dic61C22×C4semidirect
95. C2×#92418C22×C4semidirect
135. C22×C61#340semidirect
161. C2×C61C2×#340semidirect
182. C22×C4⋊C4Dic61semidirect
185. C61C22×#340semidirect
216. C2×C4⋊C4C2×Dic61semidirect
260. C4⋊C4C22×Dic61semidirect
299. C22#16839524direct (N × H)
321. C2C2×#16839524direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#924182
C23×C4⋊Dic612
C2×#168395242
C23×C4⋊C4×C612
C22×#34061