Finite Group Databasebeta

← C2×#21445117 #29661153

Subgroups — C2×#21445117 #29661153

Order 15616.

1010 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 187 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×#16840152C211yesmaximal, characteristic
2C2×#16840128C212yesmaximal, complemented
3#21445117C218yesmaximal, direct product factor
4C2×#16839504C212yesmaximal, complemented
5C2×#16839502C211yesmaximal, characteristic, complemented
6C2×C61×#563C211yesmaximal, characteristic, complemented
7C22×#11758201—21no—
8C22×#11758199—21no—
9C2×#14519774C2211yescharacteristic
10#16840152—22no—
11#16840152C2214yes—
12#16840128—24no—
13#16840128C2218yescomplemented
14C2×#14519697—22no—
15C2×#14519697—22no—
16C2×#14519688C2212yes—
17C2×#14519688C2211yescharacteristic
18#16839504C2218yescomplemented
19#16839502C2214yescomplemented
20C61×#563—24no—
21C61×#563—22no—
22C61×#563C2214yescomplemented
23C2×C4⋊Q8×C61—21no—
24C2×#126×C61—22no—
25C2×#125×C61C2212yes—
26C2×#124×C61C2211yescharacteristic
27C2×#123×C61—21no—
28C2×C22⋊Q8×C61—22no—
29C22×D244—22no—
30C22×D244—21no—
31C2×#11758201—24no—
32C2×#11758201—44no—
33C2×#11758199—24no—
34C2×#11758199—44no—
35C22×#92418—21no—
36C22×#92418—22no—
37#14519774—22no—
38#14519774—21no—
39#14519774C2312yes—
40#14519697—28no—
41#14519697—28no—
42#14519688—22no—
43#14519688—21no—
44#14519688C2314yes—
45#14519688C2312yes—
46C22×Q8×C61—21no—
47C22×Q8×C61D411yescharacteristic
48C22×D4×C61D411yescharacteristic
49C23×C4×C61—21no—
50C4⋊Q8×C61—24no—
51C4⋊Q8×C61—24no—
52#126×C61—28no—
53#126×C61—28no—
54#125×C61—22no—
55#125×C61—22no—
56#125×C61D414yes—
57#125×C61C2314yes—
58#124×C61—22no—
59#124×C61D412yes—
60#124×C61C2312yes—
61#123×C61—24no—
62#123×C61—24no—
63C22⋊Q8×C61—28no—
64C22⋊Q8×C61—28no—
65C2×C4⋊C4×C61—22no—
66C2×C4⋊C4×C61—22no—
67C2×C4⋊C4×C61—22no—
68C2×C4⋊C4×C61D412yes—
69C2×C4⋊C4×C61D412yes—
70C2×C22⋊C4×C61—22no—
71C2×C22⋊C4×C61—21no—
72C2×C22⋊C4×C61—22no—
73C2×C42×C61—21no—
74C2×C42×C61C2311yescharacteristic
75C24×C61—21no—
76C23×D61—41no—
77C2×Q8×C61—24no—
78C2×Q8×C61—21no—
79C2×Q8×C61—42no—
80C2×Q8×C61—82no—
81C2×Q8×C61C2×D412yes—
82C2×D4×C61—21no—
83C2×D4×C61—42no—
84C2×D4×C61C2×D412yes—
85C22×C4×C61—22no—
86C22×C4×C61—21no—
87C22×C4×C61—22no—
88C22×C4×C61—24no—
89C22×C4×C61—82no—
90C22×C4×C61C2×D412yes—
91C22×C4×C61C2×D411yescharacteristic
92C2×D244—24no—
93C2×D244—22no—
94C2×D244—48no—
95C2×D244—42no—
96C2×D244—42no—
97C2×D244—42no—
98C2×D244—41no—
99C4⋊C4×C61—28no—
100C4⋊C4×C61—24no—
101C4⋊C4×C61—24no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—24no—
105C4⋊C4×C61—24no—
106C4⋊C4×C61C2×D414yes—
107C4⋊C4×C61C2×D414yes—
108C22⋊C4×C61—28no—
109C22⋊C4×C61—24no—
110C22⋊C4×C61—24no—
111C22⋊C4×C61—24no—
112C42×C61—24no—
113C42×C61C2×D412yes—
114C42×C61C2×D411yescharacteristic
115C42×C61C2411yescharacteristic
116#11758201—48no—
117#11758199—48no—
118C2×#92418—24no—
119C2×#92418—22no—
120C2×#92418—24no—
121C2×#92418—44no—
122C23×C61—22no—
123C23×C61—42no—
124C23×C61—44no—
125C23×C61#12011yescharacteristic
126C22×D61—42no—
127C22×D61—82no—
128C22×D61—84no—
129Q8×C61—42no—
130Q8×C61—81no—
131Q8×C61—84no—
132D4×C61—42no—
133D4×C61—81no—
134C2×C4×C61—22no—
135C2×C4×C61—24no—
136C2×C4×C61—24no—
137C2×C4×C61—22no—
138C2×C4×C61—24no—
139C2×C4×C61—44no—
140C2×C4×C61—42no—
141C2×C4×C61—44no—
142C2×C4×C61—84no—
143C2×C4×C61#12012yes—
144C2×C4×C61#12011yescharacteristic
145C2×C4×C61C22×D412yes—
146C2×C4×C61C22×D411yescharacteristic
147D244—42no—
148D244—42no—
149D244—48no—
150D244—82no—
151D244—84no—
152#92418—24no—
153#92418—44no—
154C2×#1657—611nomaximal, Sylow
155C22×C61—42no—
156C22×C61—44no—
157C22×C61—84no—
158C22×C61#45312yes—
159C22×C61#45414yes—
160C22×C61C2×#12011yescharacteristic
161C2×D61—82no—
162C2×D61—84no—
163C2×D61—164no—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—22no—
167C4×C61—22no—
168C4×C61—44no—
169C4×C61—42no—
170C4×C61—44no—
171C2×#563D6111yescharacteristic, complemented
172C2×#496—611no—
173C2×#490—612no—
174#1657—618no—
175C2×#330—612no—
176C2×#329—611no—
177C2×C61—82no—
178C2×C61#165714yescomplemented
179C2×C61C2×#45311yescharacteristic
180C2×C61C2×#45412yes—
181D61—162no—
182C22×SD16—1221no—
183C22×D8—1221no—
184#563—24no—
185#563—22no—
186#563C2×D6114yescomplemented
187C2×C4⋊Q8—21no—
188C2×#127—611no—
189C2×#126—22no—
190C2×#125C2×D6112yes—
191C2×#124C2×D6111yescharacteristic
192C2×#123—21no—
193C2×C22⋊Q8—22no—
194#496—614no—
195#496—1222no—
196#490—618no—
197#490—1224no—
198C2×#102—1222no—
199C2×#102—1222no—
200C2×#97—612no—
201C2×#97—611no—
202#330—618no—
203#329—614no—
204C61C2×#165711yescharacteristic, complemented, Sylow
205C22×Q8—21no—
206C22×Q8C22⋊D6111yescharacteristic
207C22×D4—1221no—
208C22×D4—1222no—
209C22×D4C22⋊D6111yescharacteristic
210C23×C4—21no—
211C2×SD16—1224no—
212C2×SD16—2444no—
213C2×D8—1224no—
214C2×D8—2444no—
215C22×C8—1221no—
216C22×C8—1222no—
217C4⋊Q8—24no—
218C4⋊Q8—24no—
219#127—612no—
220#127—1222no—
221#127—1221no—
222#126—28no—
223#126—28no—
224#125—22no—
225#125—22no—
226#125C22⋊D6114yes—
227#125C22×D6114yes—
228#124—22no—
229#124C22⋊D6112yes—
230#124C22×D6112yes—
231#123—24no—
232#123—24no—
233C22⋊Q8—28no—
234C22⋊Q8—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4C22⋊D6112yes—
239C2×C4⋊C4C22⋊D6112yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—22no—
243C2×C42—21no—
244C2×C42C22×D6111yescharacteristic
245#102—1228no—
246#102—1228no—
247#97—614no—
248#97—612no—
249#97—1222no—
250#97—1221no—
251C24—21no—
252C24—2441no—
253C2×Q8—24no—
254C2×Q8—21no—
255C2×Q8—42no—
256C2×Q8—82no—
257C2×Q8C2×C22⋊D6112yes—
258C2×D4—21no—
259C2×D4—42no—
260C2×D4—1224no—
261C2×D4—1222no—
262C2×D4—2442no—
263C2×D4—2442no—
264C2×D4—2448no—
265C2×D4—2442no—
266C2×D4—2441no—
267C2×D4C2×C22⋊D6112yes—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—82no—
273C22×C4C2×C22⋊D6112yes—
274C22×C4C2×C22⋊D6111yescharacteristic
275SD16—2448no—
276D8—2448no—
277C2×C8—1224no—
278C2×C8—1222no—
279C2×C8—1224no—
280C2×C8—2444no—
281C4⋊C4—28no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4C2×C22⋊D6114yes—
289C4⋊C4C2×C22⋊D6114yes—
290C22⋊C4—28no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C42—24no—
295C42C2×C22⋊D6112yes—
296C42C2×C22⋊D6111yescharacteristic
297C42C23×D6111yescharacteristic
298C23—22no—
299C23—42no—
300C23—44no—
301C23—4882no—
302C23—4884no—
303C23—2442no—
304C23#1451974611yescharacteristic
305Q8—42no—
306Q8—81no—
307Q8—84no—
308D4—42no—
309D4—81no—
310D4—4882no—
311D4—4884no—
312D4—2442no—
313D4—2442no—
314D4—2448no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—84no—
324C2×C4#1451974612yes—
325C2×C4#1451974611yescharacteristic
326C2×C4C22×C22⋊D6112yes—
327C2×C4C22×C22⋊D6111yescharacteristic
328C8—1224no—
329C8—2444no—
330C22—42no—
331C22—44no—
332C22—84no—
333C22—9764no—
334C22—4882no—
335C22—4884no—
336C22#1683993012yes—
337C22#1683993514yes—
338C22C2×#1451974611yescharacteristic
339C4—21no—
340C4—21no—
341C4—22no—
342C4—22no—
343C4—44no—
344C4—42no—
345C4—44no—
346C2—82no—
347C2—9762no—
348C2#2144511714yesdirect product factor
349C2C2×#1683993011yescharacteristic
350C2C2×#1683993512yes—
subgroupG/H
centerC23#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#563C2
socleC23×C61#120
prime radicalC2×#16840152C2
hypercenterC2×#563D61
hypocenterC61C2×#1657
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
2. C2×#16840128C2semidirect
3. #21445117C2direct (N × H)
4. C2×#16839504C2semidirect
5. C2×#16839502C2semidirect
6. C2×C61×#563C2semidirect
13. #16840128C22semidirect
18. #16839504C22semidirect
19. #16839502C22semidirect
22. C61×#563C22semidirect
171. C2×#563D61semidirect
178. C2×C61#1657semidirect
186. #563C2×D61semidirect
204. C61C2×#1657semidirect
348. C2#21445117direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168401522
C2×#168401282
#214451172
C2×#168395042
C2×#168395022
C2×C61×#5632
C2×#165761