Finite Group Databasebeta

← C2×#21444878 #29655645

Subgroups — C2×#21444878 #29655645

Order 15616.

1190 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1190 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×#16840138C211yesmaximal, characteristic
2C2×#16839962C212yesmaximal, complemented
3#21444878C218yesmaximal, direct product factor
4C2×#16839506C211yesmaximal, characteristic, complemented
5C2×Q8⋊#92418C212yesmaximal, complemented
6C2×C61×#559C211yesmaximal, characteristic, complemented
7C22×#11758201—22no—
8C22×#11758199—22no—
9C2×#14519774C2211yescharacteristic
10#16840138—22no—
11#16840138C2214yes—
12#16839962—24no—
13#16839962C2218yescomplemented
14C2×C4⋊#92418C2212yes—
15C2×#14519697—22no—
16C2×C4×#92418C2211yescharacteristic
17#16839506C2214yescomplemented
18Q8⋊#92418C2218yescomplemented
19C61×#559—22no—
20C61×#559—24no—
21C61×#559C2214yescomplemented
22C2×#127×C61C2211yescharacteristic
23C2×#124×C61—22no—
24C2×#123×C61—22no—
25C2×C4×Q8×C61C2212yes—
26C2×C4×D4×C61—22no—
27C22×D244—22no—
28C22×D244—21no—
29C2×#11758201—28no—
30C2×#11758201—48no—
31C2×#11758199—28no—
32C2×#11758199—48no—
33C22×#92418—21no—
34C22×#92418—22no—
35#14519774—22no—
36#14519774—21no—
37#14519774C2312yes—
38C4⋊#92418—22no—
39C4⋊#92418C2314yes—
40#14519697—28no—
41C4×#92418—21no—
42C4×#92418C2312yes—
43C22×Q8×C61D412yes—
44C22×D4×C61—22no—
45C22×D4×C61D412yes—
46C23×C4×C61—22no—
47#127×C61—22no—
48#127×C61D412yes—
49#127×C61C2312yes—
50#124×C61—28no—
51#124×C61—28no—
52#123×C61—28no—
53#123×C61—28no—
54C4×Q8×C61—22no—
55C4×Q8×C61—22no—
56C4×Q8×C61D414yes—
57C4×Q8×C61C2314yes—
58C4×D4×C61—28no—
59C4×D4×C61—28no—
60C2×C4⋊C4×C61—22no—
61C2×C4⋊C4×C61D412yes—
62C2×C22⋊C4×C61—24no—
63C2×C22⋊C4×C61—22no—
64C2×C42×C61—22no—
65C2×C42×C61C2311yescharacteristic
66C24×C61—22no—
67C23×D61—41no—
68C2×Q8×C61—22no—
69C2×Q8×C61—24no—
70C2×Q8×C61—24no—
71C2×Q8×C61C2×D414yes—
72C2×D4×C61—24no—
73C2×D4×C61—22no—
74C2×D4×C61—24no—
75C2×D4×C61—44no—
76C2×D4×C61—44no—
77C2×D4×C61—44no—
78C2×D4×C61C2×D414yes—
79C22×C4×C61—22no—
80C22×C4×C61—22no—
81C22×C4×C61—28no—
82C22×C4×C61—84no—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D412yes—
85C2×D244—24no—
86C2×D244—22no—
87C2×D244—44no—
88C2×D244—44no—
89C2×D244—44no—
90C2×D244—41no—
91C2×D244—42no—
92C4⋊C4×C61—24no—
93C4⋊C4×C61—22no—
94C4⋊C4×C61—24no—
95C4⋊C4×C61C2×D414yes—
96C22⋊C4×C61—216no—
97C22⋊C4×C61—28no—
98C42×C61—24no—
99C42×C61—24no—
100C42×C61C2×D411yescharacteristic
101C42×C61C2×D412yes—
102C42×C61C2411yescharacteristic
103#11758201—416no—
104#11758199—416no—
105C2×#92418—24no—
106C2×#92418—22no—
107C2×#92418—24no—
108C2×#92418—44no—
109C23×C61—24no—
110C23×C61—48no—
111C23×C61—44no—
112C23×C61#12011yescharacteristic
113C22×D61—42no—
114C22×D61—84no—
115C22×D61—82no—
116Q8×C61—24no—
117Q8×C61—24no—
118Q8×C61—44no—
119D4×C61—44no—
120D4×C61—44no—
121D4×C61—44no—
122D4×C61—48no—
123D4×C61—82no—
124C2×C4×C61—22no—
125C2×C4×C61—24no—
126C2×C4×C61—24no—
127C2×C4×C61—44no—
128C2×C4×C61—44no—
129C2×C4×C61—88no—
130C2×C4×C61#12011yescharacteristic
131C2×C4×C61#12012yes—
132C2×C4×C61#13514yes—
133C2×C4×C61AGL(1,ℤ/8)14yes—
134C2×C4×C61C22×D411yescharacteristic
135C2×C4×C61C22×D412yes—
136D244—44no—
137D244—44no—
138D244—44no—
139D244—82no—
140D244—84no—
141#92418—24no—
142#92418—44no—
143C2×#1620—611nomaximal, Sylow
144C22×C61—48no—
145C22×C61—44no—
146C22×C61—88no—
147C22×C61#44914yes—
148C22×C61#45312yes—
149C22×C61C2×#12011yescharacteristic
150C2×D61—84no—
151C2×D61—82no—
152C2×D61—164no—
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—44no—
156C4×C61—44no—
157C4×C61#44914yes—
158C4×C61C2×#13512yes—
159C4×C61C2×AGL(1,ℤ/8)12yes—
160C2×#559D6111yescharacteristic, complemented
161C2×#493—611no—
162C2×#460—612no—
163#1620—618no—
164C2×#331—611no—
165C2×Q8⋊C8—612no—
166C2×C61—84no—
167C2×C61#162014yescomplemented
168C2×C61C2×#44912yes—
169C2×C61C2×#45311yescharacteristic
170D61—162no—
171C22×SD16—1222no—
172C22×D8—1222no—
173#559—22no—
174#559—24no—
175#559C2×D6114yescomplemented
176C2×#127—611no—
177C2×#127C2×D6111yescharacteristic
178C2×#124—22no—
179C2×#123—22no—
180C2×C4×Q8C2×D6112yes—
181C2×C4×D4—22no—
182#493—614no—
183#493—1222no—
184#460—618no—
185#460—1224no—
186C2×C4⋊C8—612no—
187C2×#102—1222no—
188C2×C4×C8—611no—
189#331—614no—
190Q8⋊C8—618no—
191C61C2×#162011yescharacteristic, complemented, Sylow
192C22×Q8C22⋊D6112yes—
193C22×D4—22no—
194C22×D4—1221no—
195C22×D4—1222no—
196C22×D4C22⋊D6112yes—
197C23×C4—22no—
198C2×SD16—1228no—
199C2×SD16—2448no—
200C2×D8—1228no—
201C2×D8—2448no—
202C22×C8—1221no—
203C22×C8—1222no—
204#127—22no—
205#127—612no—
206#127—1222no—
207#127—1221no—
208#127C22⋊D6112yes—
209#127C22×D6112yes—
210#124—28no—
211#124—28no—
212#123—28no—
213#123—28no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8C22⋊D6114yes—
217C4×Q8C22×D6114yes—
218C4×D4—28no—
219C4×D4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4C22⋊D6112yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C42—22no—
225C2×C42C22×D6111yescharacteristic
226C4⋊C8—614no—
227C4⋊C8—1222no—
228#102—1228no—
229C4×C8—612no—
230C4×C8—1221no—
231C24—22no—
232C24—2441no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8C2×C22⋊D6114yes—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—1224no—
244C2×D4—1222no—
245C2×D4—2444no—
246C2×D4—2441no—
247C2×D4—2442no—
248C2×D4—2444no—
249C2×D4—2444no—
250C2×D4C2×C22⋊D6114yes—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—84no—
255C22×C4C2×C22⋊D6111yescharacteristic
256C22×C4C2×C22⋊D6112yes—
257SD16—24416no—
258D8—24416no—
259C2×C8—1224no—
260C2×C8—1222no—
261C2×C8—1224no—
262C2×C8—2444no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4C2×C22⋊D6114yes—
267C22⋊C4—216no—
268C22⋊C4—28no—
269C42—24no—
270C42—24no—
271C42C2×C22⋊D6111yescharacteristic
272C42C2×C22⋊D6112yes—
273C42C23×D6111yescharacteristic
274C23—24no—
275C23—48no—
276C23—44no—
277C23—4884no—
278C23—4882no—
279C23—2442no—
280C23#1451974611yescharacteristic
281Q8—24no—
282Q8—24no—
283Q8—44no—
284D4—44no—
285D4—44no—
286D4—44no—
287D4—48no—
288D4—82no—
289D4—4882no—
290D4—4884no—
291D4—2444no—
292D4—2444no—
293D4—2444no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—88no—
300C2×C4#1451974611yescharacteristic
301C2×C4#1451974612yes—
302C2×C4#1451979714yes—
303C2×C4#1451980214yes—
304C2×C4C22×C22⋊D6111yescharacteristic
305C2×C4C22×C22⋊D6112yes—
306C8—1224no—
307C8—2444no—
308C22—44no—
309C22—48no—
310C22—88no—
311C22—9764no—
312C22—4882no—
313C22—4884no—
314C22#1683990714yes—
315C22#1683993012yes—
316C22C2×#1451974611yescharacteristic
317C4—21no—
318C4—21no—
319C4—44no—
320C4—44no—
321C4#1683990714yes—
322C4C2×#1451979712yes—
323C4C2×#1451980212yes—
324C2—84no—
325C2—9762no—
326C2#2144487814yesdirect product factor
327C2C2×#1683990712yes—
328C2C2×#1683993011yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#559C2
socleC23×C61#120
prime radicalC2×#16840138C2
hypercenterC2×#559D61
hypocenterC61C2×#1620
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. C2×#16839962C2semidirect
3. #21444878C2direct (N × H)
4. C2×#16839506C2semidirect
5. C2×Q8⋊#92418C2semidirect
6. C2×C61×#559C2semidirect
13. #16839962C22semidirect
17. #16839506C22semidirect
18. Q8⋊#92418C22semidirect
21. C61×#559C22semidirect
160. C2×#559D61semidirect
167. C2×C61#1620semidirect
175. #559C2×D61semidirect
191. C61C2×#1620semidirect
326. C2#21444878direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168401382
C2×#168399622
#214448782
C2×#168395062
C2×Q8⋊#924182
C2×C61×#5592
C2×#162061