Finite Group Databasebeta

← C2×#14521238 #16855098

Subgroups — C2×#14521238 #16855098

Order 3968.

794 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758561C211yesmaximal, characteristic
2C2×#11758506C212yesmaximal, complemented
3#14521238C218yesmaximal, direct product factor
4C2×#11758422C211yesmaximal, characteristic, complemented
5C2×#11758394C212yesmaximal, complemented
6C2×C31×#123C211yesmaximal, characteristic, complemented
7C22×D4×C31C2211yescharacteristic
8C23×C4×C31C2211yescharacteristic, complemented
9C22×C22⋊D31—21no—
10C22×C22⋊D31C2212yes—
11#11758561—22no—
12#11758561—24no—
13#11758561C2214yes—
14C2×#92497—22no—
15C2×D4×D31—22no—
16C2×#92495—21no—
17C31×#123—22no—
18C31×#123C2214yescomplemented
19C2×C4⋊C4×C31—21no—
20C2×C22⋊C4×C31—21no—
21C2×C22⋊C4×C31C2211yescharacteristic, complemented
22C2×C22⋊Dic31C2212yescomplemented
23#11758506—24no—
24#11758506C2218yescomplemented
25C2×(C2×D31)⋊C4—22no—
26C2×Dic31⋊C4—22no—
27#11758422C2214yescomplemented
28#11758394C2218yescomplemented
29C24×C31D411yescharacteristic
30C24×C31C2311yescharacteristic
31C23×D31—21no—
32C23×D31D412yes—
33C2×D4×C31—22no—
34C2×D4×C31—22no—
35C2×D4×C31D412yes—
36C2×D4×C31C2312yes—
37C22×C4×C31—22no—
38C22×C4×C31—22no—
39C22×C4×C31—21no—
40C22×C4×C31—21no—
41C22×C4×C31—21no—
42C22×C4×C31D411yescharacteristic
43C22×C4×C31C2312yescomplemented
44C2×C22⋊D31—22no—
45C2×C22⋊D31—24no—
46C2×C22⋊D31—22no—
47C2×C22⋊D31—22no—
48C2×C22⋊D31—22no—
49C2×C22⋊D31—22no—
50C2×C22⋊D31—24no—
51C2×C22⋊D31D414yes—
52C2×C22⋊D31C2314yes—
53C22×Dic31—21no—
54C22×Dic31—22no—
55C22×Dic31D412yes—
56#92497—28no—
57#92497—28no—
58D4×D31—28no—
59D4×D31—28no—
60#92495—24no—
61#92495—24no—
62C2×D124—21no—
63C2×C4×D31—22no—
64C2×Dic62—21no—
65C4⋊C4×C31—24no—
66C22⋊C4×C31—24no—
67C22⋊C4×C31—21no—
68C22⋊C4×C31C2312yescomplemented
69C22⋊Dic31—22no—
70C22⋊Dic31C2314yescomplemented
71(C2×D31)⋊C4—28no—
72Dic31⋊C4—28no—
73C23×C31—21no—
74C23×C31—21no—
75C23×C31—42no—
76C23×C31—42no—
77C23×C31C2×D412yes—
78C23×C31C2×D412yes—
79C23×C31C2×D411yescharacteristic
80C23×C31C2×D412yes—
81C23×C31C2×D411yescharacteristic
82C23×C31C2411yescharacteristic
83C22×D31—22no—
84C22×D31—24no—
85C22×D31—22no—
86C22×D31—44no—
87C22×D31—82no—
88C22×D31C2×D414yes—
89D4×C31—24no—
90D4×C31—24no—
91C2×C4×C31—21no—
92C2×C4×C31—21no—
93C2×C4×C31—24no—
94C2×C4×C31—22no—
95C2×C4×C31—22no—
96C2×C4×C31—42no—
97C2×C4×C31—42no—
98C2×C4×C31—41no—
99C2×C4×C31—41no—
100C2×C4×C31C2×D412yes—
101C22⋊D31—28no—
102C22⋊D31—24no—
103C22⋊D31—24no—
104C22⋊D31—24no—
105C22⋊D31—24no—
106C2×Dic31—24no—
107C2×Dic31—22no—
108C2×Dic31—22no—
109C2×Dic31—24no—
110C2×Dic31—44no—
111C2×Dic31C2×D414yes—
112D124—24no—
113C4×D31—28no—
114Dic62—24no—
115C2×#458—311nomaximal, Sylow
116C22×C31—24no—
117C22×C31—22no—
118C22×C31—22no—
119C22×C31—42no—
120C22×C31—41no—
121C22×C31—41no—
122C22×C31—42no—
123C22×C31—42no—
124C22×C31—81no—
125C22×C31#12012yes—
126C22×C31#12011yescharacteristic
127C22×C31#12011yescharacteristic
128C22×C31C22×D412yes—
129C22×C31C22×D411yescharacteristic
130C2×D31—24no—
131C2×D31—44no—
132C2×D31—44no—
133C2×D31—82no—
134C2×D31—84no—
135C4×C31—22no—
136C4×C31—42no—
137C4×C31—42no—
138Dic31—24no—
139Dic31—44no—
140C2×H(5,𝔽2)—311no—
141C2×#123—312no—
142C2×#123D3111yescharacteristic, complemented
143#458—318no—
144C2×#99—311no—
145C2×#99—312no—
146C2×C31—22no—
147C2×C31—21no—
148C2×C31—22no—
149C2×C31—21no—
150C2×C31—42no—
151C2×C31#45812yescomplemented
152C2×C31C2×#12011yescharacteristic
153D31—44no—
154H(5,𝔽2)—314no—
155H(5,𝔽2)—624no—
156H(5,𝔽2)—622no—
157C2×#41—622no—
158C2×#41—621no—
159C22×D4—312no—
160C22×D4—622no—
161C22×D4—621no—
162C22×D4C2×D3111yescharacteristic
163C23×C4C2×D3111yescharacteristic, complemented
164#123—22no—
165#123—318no—
166#123—624no—
167#123C2×D3114yescomplemented
168C2×C4⋊C4—21no—
169C2×C4⋊C4—622no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—312no—
172C2×C22⋊C4—622no—
173C2×C22⋊C4C2×D3111yescharacteristic, complemented
174#99—314no—
175#99—318no—
176C31C2×#45811yescharacteristic, complemented, Sylow
177C24—312no—
178C24—621no—
179C24C22⋊D3111yescharacteristic
180C24C22×D3111yescharacteristic
181#41—624no—
182#41—628no—
183#41—628no—
184#41—624no—
185C2×Q8—621no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—314no—
189C2×D4—314no—
190C2×D4—621no—
191C2×D4—628no—
192C2×D4—628no—
193C2×D4—622no—
194C2×D4—624no—
195C2×D4—622no—
196C2×D4—622no—
197C2×D4—622no—
198C2×D4—622no—
199C2×D4—624no—
200C2×D4C22⋊D3112yes—
201C2×D4C22×D3112yes—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—312no—
208C22×C4—622no—
209C22×C4—621no—
210C22×C4—622no—
211C22×C4C22⋊D3111yescharacteristic
212C22×C4C22×D3112yescomplemented
213C4⋊C4—24no—
214C4⋊C4—628no—
215C22⋊C4—24no—
216C22⋊C4—21no—
217C22⋊C4—314no—
218C22⋊C4—628no—
219C22⋊C4—622no—
220C22⋊C4C22×D3112yescomplemented
221C23—21no—
222C23—21no—
223C23—42no—
224C23—42no—
225C23—314no—
226C23—624no—
227C23—622no—
228C23—622no—
229C23—1244no—
230C23—2482no—
231C23D4×D3112yes—
232C23D4×D3112yes—
233C23C2×C22⋊D3111yescharacteristic
234C23C2×C22⋊D3112yes—
235C23C2×C22⋊D3111yescharacteristic
236C23C23×D3111yescharacteristic
237Q8—624no—
238D4—24no—
239D4—24no—
240D4—624no—
241D4—628no—
242D4—624no—
243D4—624no—
244D4—624no—
245D4—624no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—314no—
256C2×C4—628no—
257C2×C4—624no—
258C2×C4—622no—
259C2×C4—622no—
260C2×C4—624no—
261C2×C4—1244no—
262C2×C4C2×C22⋊D3112yes—
263C22—22no—
264C22—24no—
265C22—22no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—42no—
270C22—42no—
271C22—81no—
272C22—624no—
273C22—1244no—
274C22—1244no—
275C22—2482no—
276C22—2484no—
277C22#1175851411yescharacteristic
278C22#1175851412yes—
279C22#1175851411yescharacteristic
280C22C2×D4×D3112yes—
281C22C22×C22⋊D3111yescharacteristic
282C4—22no—
283C4—42no—
284C4—42no—
285C4—624no—
286C4—1244no—
287C2—21no—
288C2—22no—
289C2—21no—
290C2—22no—
291C2—42no—
292C2—1244no—
293C2#1452123812yesdirect product factor
294C2C2×#1175851411yescharacteristic
subgroupG/H
centerC22#11758514
commutator, cosocleC23×C31C24
FrattiniC23C23×D31
FittingC2×C31×#123C2
socleC22×C31#120
prime radicalC2×#11758561C2
hypercenterC2×#123D31
hypocenterC31C2×#458
LCSG ▷ C23×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#11758506C2semidirect
3. #14521238C2direct (N × H)
4. C2×#11758422C2semidirect
5. C2×#11758394C2semidirect
6. C2×C31×#123C2semidirect
8. C23×C4×C31C22semidirect
18. C31×#123C22semidirect
21. C2×C22⋊C4×C31C22semidirect
22. C2×C22⋊Dic31C22semidirect
24. #11758506C22semidirect
27. #11758422C22semidirect
28. #11758394C22semidirect
43. C22×C4×C31C23semidirect
68. C22⋊C4×C31C23semidirect
70. C22⋊Dic31C23semidirect
142. C2×#123D31semidirect
151. C2×C31#458semidirect
163. C23×C4C2×D31semidirect
167. #123C2×D31semidirect
173. C2×C22⋊C4C2×D31semidirect
176. C31C2×#458semidirect
212. C22×C4C22×D31semidirect
220. C22⋊C4C22×D31semidirect
293. C2#14521238direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117585612
C2×#117585062
#145212382
C2×#117584222
C2×#117583942
C2×C31×#1232
C2×#45831