Finite Group Databasebeta

← C2×#21444900 #29655893

Subgroups — C2×#21444900 #29655893

Order 15616.

1190 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1190 conjugacy classes of subgroups — too many to draw. Only its 203 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×#16840478C211yesmaximal, characteristic, complemented
2C2×#16839954C212yesmaximal, complemented
3#21444900C218yesmaximal, direct product factor
4C2×#16839511C211yesmaximal, characteristic, complemented
5C2×#16839491C212yesmaximal, complemented
6C2×C61×#492C211yesmaximal, characteristic, complemented
7C22×#11758199—22no—
8#16840478—24no—
9#16840478—22no—
10#16840478C2214yescomplemented
11C2×#14519766—22no—
12C2×#14519752—22no—
13C2×C4×D244C2212yes—
14C2×#14519735—22no—
15#16839954—24no—
16#16839954C2218yescomplemented
17C2×C4⋊#92418C2212yes—
18C2×#14519698—22no—
19#16839511C2214yescomplemented
20#16839491C2218yescomplemented
21C22×SD16×C61—22no—
22C2×C4⋊Q8×C61C2211yescharacteristic
23C2×#127×C61C2211yescharacteristic
24C61×#492—22no—
25C61×#492C2214yescomplemented
26C2×C4×C8×C61C2211yescharacteristic
27C22×D244—22no—
28C22×D244D412yes—
29C22×C4×D61—22no—
30C2×#11758199—28no—
31C2×#11758199—48no—
32C22×#92418—22no—
33#14519766—28no—
34#14519766—28no—
35#14519752—28no—
36#14519752—28no—
37C4×D244—22no—
38C4×D244—22no—
39C4×D244D414yes—
40C4×D244C2314yes—
41#14519735—28no—
42#14519735—28no—
43C2×C4⋊Dic61D412yes—
44C2×#11758198—24no—
45C2×#11758198—22no—
46C2×C4×Dic61—22no—
47C4⋊#92418—22no—
48C4⋊#92418C2314yes—
49#14519698—28no—
50C22×Q8×C61D412yes—
51C22×D4×C61—21no—
52C22×D4×C61—22no—
53C2×SD16×C61—28no—
54C2×SD16×C61—48no—
55C22×C8×C61—21no—
56C4⋊Q8×C61—22no—
57C4⋊Q8×C61D412yes—
58C4⋊Q8×C61C2312yes—
59#127×C61—22no—
60#127×C61—21no—
61#127×C61C2312yes—
62C2×C4⋊C4×C61—22no—
63C2×C42×C61C2311yescharacteristic
64C4×C8×C61—21no—
65C4×C8×C61C2312yes—
66C24×C61—41no—
67C23×D61—22no—
68C2×Q8×C61—22no—
69C2×Q8×C61—44no—
70C2×Q8×C61C2×D414yes—
71C2×D4×C61—22no—
72C2×D4×C61—24no—
73C2×D4×C61—44no—
74C2×D4×C61—41no—
75C2×D4×C61—42no—
76C2×D4×C61—44no—
77C2×D4×C61—44no—
78C22×C4×C61—22no—
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D412yes—
81C22×Dic61—22no—
82C2×D244—24no—
83C2×D244—24no—
84C2×D244—22no—
85C2×D244—28no—
86C2×D244—84no—
87C2×D244C2×D414yes—
88C2×C4×D61—24no—
89C2×C4×D61—24no—
90C2×C4×D61—44no—
91C2×C4×D61—44no—
92SD16×C61—416no—
93C2×C8×C61—24no—
94C2×C8×C61—22no—
95C4⋊C4×C61—24no—
96C4⋊C4×C61—24no—
97C42×C61C2×D412yes—
98C42×C61C2×D411yescharacteristic
99C42×C61C2411yescharacteristic
100#11758199—416no—
101#11758198—216no—
102#11758198—24no—
103#11758198—24no—
104C4⋊Dic61—22no—
105C4⋊Dic61C2×D414yes—
106C4×Dic61—28no—
107C2×#92418—24no—
108C2×#92418—44no—
109C23×C61—42no—
110C23×C61—84no—
111C23×C61—82no—
112C23×C61#12011yescharacteristic
113C22×D61—24no—
114C22×D61—44no—
115C22×D61—44no—
116C22×D61—44no—
117Q8×C61—44no—
118Q8×C61—82no—
119D4×C61—44no—
120D4×C61—44no—
121D4×C61—44no—
122D4×C61—82no—
123D4×C61—84no—
124C2×C4×C61—22no—
125C2×C4×C61—24no—
126C2×C4×C61—44no—
127C2×C4×C61#12011yescharacteristic
128C2×C4×C61#12012yes—
129C2×C4×C61#13514yes—
130C2×C4×C61AGL(1,ℤ/8)14yes—
131C2×C4×C61C22×D411yescharacteristic
132C2×C4×C61C22×D412yes—
133C2×Dic61—24no—
134C2×Dic61—44no—
135D244—24no—
136D244—24no—
137D244—44no—
138D244—88no—
139C4×D61—44no—
140C4×D61—44no—
141C4×D61—48no—
142C8×C61—24no—
143#92418—44no—
144C2×#1624—611nomaximal, Sylow
145C22×C61—84no—
146C22×C61—82no—
147C22×C61—164no—
148C22×C61#44914yes—
149C22×C61#45312yes—
150C22×C61C2×#12011yescharacteristic
151C2×D61—44no—
152C2×D61—44no—
153C2×D61—44no—
154C2×D61—88no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—44no—
158C4×C61#44914yes—
159C4×C61C2×#13512yes—
160C4×C61C2×AGL(1,ℤ/8)12yes—
161Dic61—44no—
162C2×#555—611no—
163C2×#492D6111yescharacteristic, complemented
164C2×#459—612no—
165#1624—618no—
166C2×#333—611no—
167C2×D4⋊C8—612no—
168C2×C61—162no—
169C2×C61#162414yescomplemented
170C2×C61C2×#44912yes—
171C2×C61C2×#45311yescharacteristic
172D61—84no—
173C22×SD16—22no—
174C22×D8—1222no—
175#555—614no—
176#555—1224no—
177#555—1222no—
178C2×C4⋊Q8C2×D6111yescharacteristic
179C2×#127C2×D6111yescharacteristic
180C2×#124—1222no—
181C2×#121—1222no—
182C2×C4×D4—612no—
183C2×#117—1222no—
184#492—22no—
185#492C2×D6114yescomplemented
186#459—618no—
187#459—1224no—
188C2×C4⋊C8—612no—
189C2×#102—1222no—
190C2×C4×C8C2×D6111yescharacteristic
191#333—614no—
192D4⋊C8—618no—
193C61C2×#162411yescharacteristic, complemented, Sylow
194C22×Q8C22⋊D6112yes—
195C22×D4—22no—
196C22×D4—21no—
197C22×D4—612no—
198C22×D4—1222no—
199C23×C4—1222no—
200C2×SD16—28no—
201C2×SD16—48no—
202C2×D8—1228no—
203C2×D8—2448no—
204C22×C8—21no—
205C22×C8—1222no—
206C4⋊Q8—22no—
207C4⋊Q8C22⋊D6112yes—
208C4⋊Q8C22×D6112yes—
209#127—22no—
210#127—21no—
211#127C22×D6112yes—
212#124—1228no—
213#124—1228no—
214#121—1228no—
215#121—1228no—
216C4×D4—614no—
217C4×D4—614no—
218C4×D4—1222no—
219C4×D4—1222no—
220#117—1228no—
221#117—1228no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—612no—
224C2×C22⋊C4—1224no—
225C2×C22⋊C4—1222no—
226C2×C42—1222no—
227C2×C42C22×D6111yescharacteristic
228C4⋊C8—614no—
229C4⋊C8—1222no—
230#102—1228no—
231C4×C8—21no—
232C4×C8C22×D6112yes—
233C24—41no—
234C24—1222no—
235C2×Q8—22no—
236C2×Q8—44no—
237C2×Q8C2×C22⋊D6114yes—
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—44no—
241C2×D4—41no—
242C2×D4—42no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—614no—
246C2×D4—1228no—
247C2×D4—1222no—
248C2×D4—1224no—
249C2×D4—1224no—
250C2×D4—4884no—
251C22×C4—22no—
252C22×C4—1222no—
253C22×C4—1224no—
254C22×C4—1224no—
255C22×C4—2444no—
256C22×C4—2444no—
257C22×C4D4×D6112yes—
258C22×C4C2×C22⋊D6111yescharacteristic
259SD16—416no—
260D8—24416no—
261C2×C8—24no—
262C2×C8—22no—
263C2×C8—1224no—
264C2×C8—2444no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—614no—
268C4⋊C4—1222no—
269C22⋊C4—12216no—
270C22⋊C4—1224no—
271C22⋊C4—1224no—
272C42—1228no—
273C42D4×D6112yes—
274C42C2×C22⋊D6111yescharacteristic
275C42C23×D6111yescharacteristic
276C23—42no—
277C23—84no—
278C23—82no—
279C23—1224no—
280C23—2444no—
281C23—2444no—
282C23—2444no—
283C23#1451974511yescharacteristic
284Q8—44no—
285Q8—82no—
286D4—44no—
287D4—44no—
288D4—44no—
289D4—82no—
290D4—84no—
291D4—1224no—
292D4—1224no—
293D4—4888no—
294D4—2444no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—44no—
298C2×C4—1224no—
299C2×C4—2444no—
300C2×C4—2444no—
301C2×C4—2444no—
302C2×C4—2448no—
303C2×C4#1451974511yescharacteristic
304C2×C4#1451974512yes—
305C2×C4#1451979814yes—
306C2×C4#1451980414yes—
307C2×C4C2×D4×D6112yes—
308C2×C4C22×C22⋊D6111yescharacteristic
309C8—24no—
310C8—2444no—
311C22—82no—
312C22—84no—
313C22—164no—
314C22—4888no—
315C22—2444no—
316C22—2444no—
317C22—2444no—
318C22#1683990614yes—
319C22#1683992912yes—
320C22C2×#1451974511yescharacteristic
321C4—21no—
322C4—21no—
323C4—44no—
324C4—2444no—
325C4#1683990614yes—
326C4C2×#1451979812yes—
327C4C2×#1451980412yes—
328C2—162no—
329C2—4884no—
330C2#2144490014yesdirect product factor
331C2C2×#1683990612yes—
332C2C2×#1683992911yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#492C2
socleC23×C61#120
hypercenterC2×#492D61
hypocenterC61C2×#1624
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#492

Complements

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

NH ≅ G/Ntype
1. C2×#16840478C2semidirect
2. C2×#16839954C2semidirect
3. #21444900C2direct (N × H)
4. C2×#16839511C2semidirect
5. C2×#16839491C2semidirect
6. C2×C61×#492C2semidirect
10. #16840478C22semidirect
16. #16839954C22semidirect
19. #16839511C22semidirect
20. #16839491C22semidirect
25. C61×#492C22semidirect
163. C2×#492D61semidirect
169. C2×C61#1624semidirect
185. #492C2×D61semidirect
193. C61C2×#1624semidirect
330. C2#21444900direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168404782
C2×#168399542
#214449002
C2×#168395112
C2×#168394912
C2×C61×#4922
C2×#162461