Finite Group Databasebeta

← C2×#21444933 #29655621

Subgroups — C2×#21444933 #29655621

Order 15616.

1722 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1722 conjugacy classes of subgroups — too many to draw. Only its 219 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×#16840397C211yesmaximal, characteristic
2C2×#16840049C212yesmaximal, complemented
3#21444933C218yesmaximal, direct product factor
4C2×#16839506C211yesmaximal, characteristic, complemented
5C2×#16839491C212yesmaximal, complemented
6C2×C61×#492C211yesmaximal, characteristic, complemented
7C22×D4×D61—22no—
8#16840397—24no—
9#16840397—22no—
10#16840397C2214yes—
11C2×#14519748—22no—
12C2×#14519745—22no—
13C2×C4×D244C2212yes—
14#16840049—24no—
15#16840049C2218yescomplemented
16C2×#14519708—22no—
17C2×C4⋊#92418C2212yescomplemented
18C2×#14519697—22no—
19#16839506C2214yescomplemented
20#16839491C2218yescomplemented
21C22×SD16×C61—22no—
22C2×C4⋊Q8×C61C2211yescharacteristic, complemented
23C2×#127×C61C2211yescharacteristic
24C61×#492—22no—
25C61×#492C2214yescomplemented
26C2×C4×C8×C61C2211yescharacteristic, complemented
27C24×D61—22no—
28C22×D244D412yes—
29C22×C22⋊D61—24no—
30C2×D4×D61—28no—
31C2×D4×D61—28no—
32C2×D4×D61—48no—
33C2×D4×D61—48no—
34C22×C4×D61—22no—
35C22×#92418—22no—
36#14519748—28no—
37#14519748—28no—
38#14519745—28no—
39#14519745—28no—
40C4×D244—22no—
41C4×D244—22no—
42C4×D244D414yes—
43C4×D244C2314yes—
44C2×C4⋊Dic61D412yes—
45C2×C22⋊Dic61—22no—
46C2×#11758198—22no—
47#14519708—28no—
48C4⋊#92418—22no—
49C4⋊#92418C2314yescomplemented
50#14519697—28no—
51C22×Q8×C61—21no—
52C22×D4×C61—22no—
53C22×D4×C61D412yes—
54C2×SD16×C61—28no—
55C2×SD16×C61—48no—
56C22×C8×C61—21no—
57C4⋊Q8×C61—22no—
58C4⋊Q8×C61—21no—
59C4⋊Q8×C61C2312yescomplemented
60#127×C61—22no—
61#127×C61D412yes—
62#127×C61C2312yes—
63C2×C4⋊C4×C61—22no—
64C2×C42×C61C2311yescharacteristic
65C4×C8×C61—21no—
66C4×C8×C61C2312yescomplemented
67C24×C61—22no—
68C23×D61—22no—
69C23×D61—28no—
70C23×D61—416no—
71C23×D61—84no—
72C2×Q8×C61—22no—
73C2×Q8×C61—44no—
74C2×Q8×C61—41no—
75C2×D4×C61—24no—
76C2×D4×C61—22no—
77C2×D4×C61—24no—
78C2×D4×C61—44no—
79C2×D4×C61—44no—
80C2×D4×C61—44no—
81C2×D4×C61C2×D414yes—
82C22×C4×C61—41no—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D412yes—
85C2×C22⋊D61—216no—
86C2×C22⋊D61—416no—
87C22×Dic61—22no—
88D4×D61—416no—
89D4×D61—432no—
90D4×D61—416no—
91C2×D244—22no—
92C2×D244—24no—
93C2×D244SD1618yescomplemented
94C2×D244C2×D414yes—
95C2×C4×D61—24no—
96C2×C4×D61—24no—
97C2×C4×D61—44no—
98C2×C4×D61—44no—
99SD16×C61—416no—
100C2×C8×C61—24no—
101C2×C8×C61—22no—
102C4⋊C4×C61—24no—
103C4⋊C4×C61—42no—
104C42×C61C2×D412yes—
105C42×C61C2×D411yescharacteristic
106C42×C61C2411yescharacteristic
107C22⋊Dic61—28no—
108#11758198—24no—
109#11758198—24no—
110C4⋊Dic61—22no—
111C4⋊Dic61C2×D414yes—
112C2×#92418—24no—
113C2×#92418—44no—
114C23×C61—24no—
115C23×C61—48no—
116C23×C61—44no—
117C23×C61#12011yescharacteristic
118C22×D61—24no—
119C22×D61—28no—
120C22×D61—432no—
121C22×D61—44no—
122C22×D61—44no—
123C22×D61—88no—
124C22×D61—832no—
125Q8×C61—44no—
126Q8×C61—82no—
127D4×C61—44no—
128D4×C61—44no—
129D4×C61—44no—
130D4×C61—48no—
131D4×C61—82no—
132C2×C4×C61—22no—
133C2×C4×C61—42no—
134C2×C4×C61—82no—
135C2×C4×C61#12011yescharacteristic
136C2×C4×C61#12012yes—
137C2×C4×C61C2×SD1614yes—
138C2×C4×C61AGL(1,ℤ/8)14yes—
139C2×C4×C61C22×D411yescharacteristic
140C2×C4×C61C22×D412yes—
141C22⋊D61—432no—
142C2×Dic61—24no—
143C2×Dic61—44no—
144D244—24no—
145D244—28no—
146D244C2×SD1618yescomplemented
147C4×D61—44no—
148C4×D61—44no—
149C4×D61—48no—
150C8×C61—24no—
151#92418—44no—
152C2×#1629—611nomaximal, Sylow
153C22×C61—48no—
154C22×C61—44no—
155C22×C61—88no—
156C22×C61#45014yes—
157C22×C61#45312yes—
158C22×C61C2×#12011yescharacteristic
159C2×D61—28no—
160C2×D61—416no—
161C2×D61—44no—
162C2×D61—44no—
163C2×D61—832no—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—82no—
167C4×C61#45014yes—
168C4×C61C22×SD1612yes—
169C4×C61C2×AGL(1,ℤ/8)12yes—
170Dic61—44no—
171C2×D42—611no—
172C2×#492D6111yescharacteristic, complemented
173C2×#476—612no—
174#1629—618no—
175C2×#331—611no—
176C2×D4⋊C8—612no—
177C2×C61—84no—
178C2×C61#162914yescomplemented
179C2×C61C2×#45012yes—
180C2×C61C2×#45311yescharacteristic
181D61—48no—
182C23×D4—1222no—
183C22×SD16—22no—
184D42—614no—
185D42—1224no—
186D42—1222no—
187C2×C4⋊Q8C2×D6111yescharacteristic, complemented
188C2×#127C2×D6111yescharacteristic
189C2×#121—1222no—
190C2×#120—1222no—
191C2×C4×D4—612no—
192#492—22no—
193#492C2×D6114yescomplemented
194#476—618no—
195#476—1224no—
196C2×#106—1222no—
197C2×C4⋊C8—612no—
198C2×#102—1222no—
199C2×C4×C8C2×D6111yescharacteristic, complemented
200#331—614no—
201D4⋊C8—618no—
202C61C2×#162911yescharacteristic, complemented, Sylow
203C25—1222no—
204C22×Q8—21no—
205C22×D4—22no—
206C22×D4—612no—
207C22×D4—1228no—
208C22×D4—1228no—
209C22×D4—1224no—
210C22×D4—2448no—
211C22×D4—2448no—
212C22×D4C22⋊D6112yes—
213C23×C4—1222no—
214C2×SD16—28no—
215C2×SD16—48no—
216C22×C8—21no—
217C22×C8—1222no—
218C4⋊Q8—22no—
219C4⋊Q8—21no—
220C4⋊Q8C22×D6112yescomplemented
221#127—22no—
222#127C22⋊D6112yes—
223#127C22×D6112yes—
224#121—1228no—
225#121—1228no—
226#120—1228no—
227#120—1228no—
228C4×D4—614no—
229C4×D4—614no—
230C4×D4—1222no—
231C4×D4—1222no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—612no—
234C2×C22⋊C4—1222no—
235C2×C22⋊C4—1222no—
236C2×C42C22×D6111yescharacteristic
237#106—1228no—
238C4⋊C8—614no—
239C4⋊C8—1222no—
240#102—1228no—
241C4×C8—21no—
242C4×C8C22×D6112yescomplemented
243C24—22no—
244C24—1228no—
245C24—1222no—
246C24—4884no—
247C24—24416no—
248C2×Q8—22no—
249C2×Q8—44no—
250C2×Q8—41no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—614no—
258C2×D4—618no—
259C2×D4—12216no—
260C2×D4—1222no—
261C2×D4—1224no—
262C2×D4—24416no—
263C2×D4—24416no—
264C2×D4—24432no—
265C2×D4—24416no—
266C2×D4C2×C22⋊D6114yes—
267C22×C4—41no—
268C22×C4—1222no—
269C22×C4—1224no—
270C22×C4—1224no—
271C22×C4—2444no—
272C22×C4—2444no—
273C22×C4D4×D6112yes—
274C22×C4C2×C22⋊D6111yescharacteristic
275SD16—416no—
276C2×C8—24no—
277C2×C8—22no—
278C2×C8—1224no—
279C2×C8—2444no—
280C4⋊C4—24no—
281C4⋊C4—42no—
282C4⋊C4—614no—
283C4⋊C4—1222no—
284C22⋊C4—1228no—
285C22⋊C4—1224no—
286C22⋊C4—1224no—
287C42D4×D6112yes—
288C42C2×C22⋊D6111yescharacteristic
289C42C23×D6111yescharacteristic
290C23—24no—
291C23—48no—
292C23—44no—
293C23—1224no—
294C23—1228no—
295C23—4888no—
296C23—48832no—
297C23—24432no—
298C23—2444no—
299C23—2444no—
300C23#1451974511yescharacteristic
301Q8—44no—
302Q8—82no—
303D4—44no—
304D4—44no—
305D4—44no—
306D4—48no—
307D4—82no—
308D4—618no—
309D4—1224no—
310D4—1228no—
311D4—24432no—
312C2×C4—22no—
313C2×C4—42no—
314C2×C4—82no—
315C2×C4—1224no—
316C2×C4—2444no—
317C2×C4—2444no—
318C2×C4—2444no—
319C2×C4—2448no—
320C2×C4#1451974511yescharacteristic
321C2×C4#1451974512yes—
322C2×C4SD16×D6114yes—
323C2×C4#1451980414yes—
324C2×C4C2×D4×D6112yes—
325C2×C4C22×C22⋊D6111yescharacteristic
326C8—24no—
327C8—2444no—
328C22—44no—
329C22—48no—
330C22—88no—
331C22—1228no—
332C22—48832no—
333C22—2444no—
334C22—24416no—
335C22—2444no—
336C22#1683991114yes—
337C22#1683992912yes—
338C22C2×#1451974511yescharacteristic
339C4—21no—
340C4—21no—
341C4—82no—
342C4—2444no—
343C4#1683991114yes—
344C4C2×SD16×D6112yes—
345C4C2×#1451980412yes—
346C2—84no—
347C2—2448no—
348C2#2144493314yesdirect product factor
349C2C2×#1683991112yes—
350C2C2×#1683992911yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#492C2
socleC23×C61#120
prime radicalC2×#16840397C2
hypercenterC2×#492D61
hypocenterC61C2×#1629
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
2. C2×#16840049C2semidirect
3. #21444933C2direct (N × H)
4. C2×#16839506C2semidirect
5. C2×#16839491C2semidirect
6. C2×C61×#492C2semidirect
15. #16840049C22semidirect
17. C2×C4⋊#92418C22semidirect
19. #16839506C22semidirect
20. #16839491C22semidirect
22. C2×C4⋊Q8×C61C22semidirect
25. C61×#492C22semidirect
26. C2×C4×C8×C61C22semidirect
49. C4⋊#92418C23semidirect
59. C4⋊Q8×C61C23semidirect
66. C4×C8×C61C23semidirect
93. C2×D244SD16semidirect
146. D244C2×SD16semidirect
172. C2×#492D61semidirect
178. C2×C61#1629semidirect
187. C2×C4⋊Q8C2×D61semidirect
193. #492C2×D61semidirect
199. C2×C4×C8C2×D61semidirect
202. C61C2×#1629semidirect
220. C4⋊Q8C22×D61semidirect
242. C4×C8C22×D61semidirect
348. C2#21444933direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403972
C2×#168400492
#214449332
C2×#168395062
C2×#168394912
C2×C61×#4922
C2×#162961