Finite Group Databasebeta

← #29658186

Subgroups — #29658186

Order 15616.

1004 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 179 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
1#21456757C212yesmaximal, complemented
2#21456272C211yesmaximal, characteristic
3#21444897C214yesmaximal, complemented
4#21444878C214yesmaximal, complemented
5#21444481C211yesmaximal, characteristic, complemented
6#21444240C212yesmaximal, complemented
7C61×#3550C211yesmaximal, characteristic, complemented
8C2×#14519802—22no—
9C2×#14519774C2211yescharacteristic
10#16840152—22no—
11#16840138C2212yes—
12#16840136—22no—
13#16840136C2212yes—
14#16840110—22no—
15#16839962C2218yes—
16#16839912—24no—
17#16839815C2212yes—
18C4×#11758194C2211yescharacteristic
19#16839511C2212yescomplemented
20#16839506C2212yescomplemented
21Q8⋊#92418C2218yescomplemented
22C61×#563—22no—
23C61×#559C2212yescomplemented
24Q82×C61C2212yescomplemented
25C61×#556—22no—
26D4×Q8×C61—22no—
27C61×#542—24no—
28C61×#536—22no—
29C61×#533C2212yes—
30C61×#532C2211yescharacteristic
31C22×D244—21no—
32C22×D244—22no—
33#14519802—48no—
34C2×#11758201—24no—
35C2×#11758201—28no—
36C2×#11758199—24no—
37C2×#11758194—21no—
38#14519774—22no—
39#14519774—21no—
40#14519774C2312yes—
41C4⋊#92418C2314yes—
42#14519697—28no—
43C22⋊#92418—22no—
44#14519688—21no—
45C4×#92418—22no—
46C4×#92418C2312yes—
47C2×#41×C61D412yes—
48C22×Q8×C61—22no—
49C4⋊Q8×C61—24no—
50C4⋊Q8×C61—24no—
51C4⋊Q8×C61D412yes—
52C4⋊Q8×C61D412yes—
53C4⋊Q8×C61C2311yescharacteristic
54#127×C61C2311yescharacteristic
55#126×C61—28no—
56#125×C61—22no—
57#124×C61—24no—
58#124×C61—24no—
59#124×C61D412yes—
60#123×C61—24no—
61#123×C61—24no—
62C22⋊Q8×C61—24no—
63C22⋊Q8×C61—28no—
64C22⋊Q8×C61—22no—
65C22⋊Q8×C61D414yes—
66#121×C61—22no—
67C4×Q8×C61—24no—
68C4×Q8×C61—22no—
69C4×Q8×C61C2314yes—
70C4×D4×C61—24no—
71C4×D4×C61—22no—
72#117×C61—22no—
73#117×C61—22no—
74C2×C42×C61C2311yescharacteristic
75C23×D61—41no—
76#41×C61—44no—
77C2×Q8×C61—24no—
78C2×Q8×C61—22no—
79C2×Q8×C61—84no—
80C2×Q8×C61C2×D414yes—
81C2×Q8×C61C2×D412yes—
82C2×D4×C61—22no—
83C2×D4×C61—22no—
84C2×D4×C61C2×D412yes—
85C22×C4×C61—24no—
86C22×C4×C61—22no—
87C22×C4×C61C2×D412yes—
88C22×C4×C61C2×D411yescharacteristic
89C2×D244—24no—
90C2×D244—22no—
91C2×D244—44no—
92C2×D244—42no—
93C2×D244—41no—
94C2×D244—84no—
95C4⋊C4×C61—28no—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61—24no—
100C4⋊C4×C61—22no—
101C4⋊C4×C61—24no—
102C4⋊C4×C61C2×D414yes—
103C22⋊C4×C61—24no—
104C22⋊C4×C61—28no—
105C22⋊C4×C61—24no—
106C22⋊C4×C61—24no—
107C42×C61—24no—
108C42×C61—24no—
109C42×C61C2×D412yes—
110C42×C61C2×D411yescharacteristic
111C42×C61C2411yescharacteristic
112#11758201—48no—
113#11758201—88no—
114#11758199—48no—
115#11758194—24no—
116C2×#92418—22no—
117C2×#92418—24no—
118C23×C61—22no—
119C23×C61#12011yescharacteristic
120C22×D61—42no—
121C22×D61—82no—
122C22×D61—84no—
123Q8×C61—28no—
124Q8×C61—42no—
125Q8×C61—44no—
126Q8×C61—84no—
127D4×C61—42no—
128D4×C61—44no—
129D4×C61—82no—
130C2×C4×C61—24no—
131C2×C4×C61—22no—
132C2×C4×C61—24no—
133C2×C4×C61—22no—
134C2×C4×C61—24no—
135C2×C4×C61—44no—
136C2×C4×C61—84no—
137C2×C4×C61#12012yes—
138C2×C4×C61#12011yescharacteristic
139C2×C4×C61C22×D411yescharacteristic
140C2×C4×C61C22×D412yes—
141D244—44no—
142D244—48no—
143D244—84no—
144D244—82no—
145#92418—24no—
146#92418—44no—
147#18718—611nomaximal, Sylow
148C22×C61—22no—
149C22×C61—21no—
150C22×C61—44no—
151C22×C61—42no—
152C22×C61C2×#12011yescharacteristic
153C2×D61—84no—
154C2×D61—82no—
155C2×D61—164no—
156C4×C61—22no—
157C4×C61—21no—
158C4×C61—44no—
159C4×C61—42no—
160C4×C61—44no—
161C4×C61#45312yes—
162C4×C61#57714yes—
163#3550D6111yescharacteristic, complemented
164#3208—612no—
165#3151—611no—
166#1623—614no—
167#1620—614no—
168#1543—611no—
169#1485—612no—
170C2×C61—41no—
171C2×C61—82no—
172C2×C61#301012yes—
173C2×C61C2×#45311yescharacteristic
174D61—162no—
175C2×AGL(1,ℤ/8)—1222no—
176#563—22no—
177#559C2×D6112yescomplemented
178Q82C2×D6112yescomplemented
179#556—22no—
180D4×Q8—22no—
181#542—24no—
182#536—22no—
183#533C2×D6112yes—
184#532C2×D6111yescharacteristic
185C2×#127—611no—
186#496—1222no—
187#493—612no—
188#492—612no—
189#492—1222no—
190#486—1222no—
191#460—618no—
192#450—1224no—
193#431—612no—
194C4×M16—611no—
195#333—612no—
196#331—612no—
197Q8⋊C8—618no—
198C61#1871811yescharacteristic, complemented, Sylow
199C2×#41C22⋊D6112yes—
200C22×Q8—22no—
201C22×D4—1222no—
202C22×D4—1221no—
203AGL(1,ℤ/8)—2448no—
204C2×SD16—1228no—
205C2×SD16—1224no—
206C2×D8—1224no—
207C2×M16—1221no—
208C4⋊Q8—24no—
209C4⋊Q8—24no—
210C4⋊Q8C22⋊D6112yes—
211C4⋊Q8C22⋊D6112yes—
212C4⋊Q8C22×D6111yescharacteristic
213#127—612no—
214#127—1222no—
215#127—1221no—
216#127C22×D6111yescharacteristic
217#126—28no—
218#125—22no—
219#124—24no—
220#124—24no—
221#124C22⋊D6112yes—
222#123—24no—
223#123—24no—
224C22⋊Q8—24no—
225C22⋊Q8—28no—
226C22⋊Q8—22no—
227C22⋊Q8C22⋊D6114yes—
228#121—22no—
229C4×Q8—24no—
230C4×Q8—22no—
231C4×Q8C22×D6114yes—
232C4×D4—24no—
233C4×D4—22no—
234#117—22no—
235#117—22no—
236C2×C42C22×D6111yescharacteristic
237C4⋊C8—614no—
238#102—1228no—
239C22⋊C8—1222no—
240#97—1221no—
241C4×C8—612no—
242C4×C8—1222no—
243C24—2441no—
244#41—44no—
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—84no—
248C2×Q8C2×C22⋊D6114yes—
249C2×Q8C2×C22⋊D6112yes—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—1224no—
253C2×D4—1222no—
254C2×D4—4884no—
255C2×D4—2444no—
256C2×D4—2442no—
257C2×D4—2441no—
258C2×D4C2×C22⋊D6112yes—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4C2×C22⋊D6112yes—
262C22×C4C2×C22⋊D6111yescharacteristic
263SD16—4888no—
264SD16—2448no—
265D8—2448no—
266M16—1224no—
267C2×C8—1222no—
268C2×C8—1224no—
269C4⋊C4—28no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4C2×C22⋊D6114yes—
277C22⋊C4—24no—
278C22⋊C4—28no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C42—24no—
282C42—24no—
283C42C2×C22⋊D6112yes—
284C42C2×C22⋊D6111yescharacteristic
285C42C23×D6111yescharacteristic
286C23—22no—
287C23—4884no—
288C23—4882no—
289C23—2442no—
290C23#1451974611yescharacteristic
291Q8—28no—
292Q8—42no—
293Q8—44no—
294Q8—84no—
295D4—42no—
296D4—44no—
297D4—82no—
298D4—4884no—
299D4—4882no—
300D4—2444no—
301D4—2448no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—44no—
308C2×C4—84no—
309C2×C4#1451974612yes—
310C2×C4#1451974611yescharacteristic
311C2×C4C22×C22⋊D6111yescharacteristic
312C2×C4C22×C22⋊D6112yes—
313C8—1224no—
314C8—2444no—
315C22—22no—
316C22—21no—
317C22—44no—
318C22—42no—
319C22—9764no—
320C22—4884no—
321C22—4882no—
322C22C2×#1451974611yescharacteristic
323C4—22no—
324C4—21no—
325C4—44no—
326C4—42no—
327C4—44no—
328C4#1683993012yes—
329C4#1684057014yes—
330C2—41no—
331C2—82no—
332C2—9762no—
333C2#2145510512yes—
334C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3550C2
socleC22×C61C2×#120
prime radical#21456272C2
hypercenter#3550D61
hypocenterC61#18718
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3550

Complements

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

NH ≅ G/Ntype
1. #21456757C2semidirect
3. #21444897C2semidirect
4. #21444878C2semidirect
5. #21444481C2semidirect
6. #21444240C2semidirect
7. C61×#3550C2semidirect
19. #16839511C22semidirect
20. #16839506C22semidirect
21. Q8⋊#92418C22semidirect
23. C61×#559C22semidirect
24. Q82×C61C22semidirect
163. #3550D61semidirect
177. #559C2×D61semidirect
178. Q82C2×D61semidirect
198. C61#18718semidirect

Maximal subgroups

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

subgroupindex
#214567572
#214562722
#214448972
#214448782
#214444812
#214442402
C61×#35502
#1871861