Finite Group Databasebeta

← #29658154

Subgroups — #29658154

Order 15616.

1132 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1132 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#21456734C212yesmaximal, complemented
2#21456272C211yesmaximal, characteristic, complemented
3#21444902C214yesmaximal, complemented
4#21444873C214yesmaximal, complemented
5#21444481C211yesmaximal, characteristic, complemented
6#21444236C212yesmaximal, complemented
7C61×#3549C211yesmaximal, characteristic, complemented
8C2×#14519802—22no—
9C2×#14519774C2211yescharacteristic, complemented
10#16840152—22no—
11#16840138—22no—
12#16840138C2212yes—
13#16840136C2212yescomplemented
14#16840110—22no—
15#16839957C2218yescomplemented
16#16839895—24no—
17#16839815C2212yescomplemented
18C4×#11758194C2211yescharacteristic, complemented
19#16839511C2212yescomplemented
20#16839506C2212yescomplemented
21D4⋊#92418C2218yescomplemented
22C61×H(3,𝔽4)—22no—
23C61×#555—22no—
24C61×#555C2212yescomplemented
25D42×C61—22no—
26D42×C61C2212yescomplemented
27C61×#535—22no—
28C61×#535—24no—
29C61×#532C2211yescharacteristic
30C61×#532C2212yes—
31C22×D244—21no—
32C22×D244—22no—
33#14519802—48no—
34C2×#11758201—24no—
35C2×#11758199—24no—
36C2×#11758199—28no—
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×C61—22no—
48C2×#41×C61D412yes—
49C22×D4×C61—22no—
50C22×D4×C61—24no—
51C4⋊Q8×C61D412yes—
52C4⋊Q8×C61C2311yescharacteristic
53#127×C61—24no—
54#127×C61D412yes—
55#127×C61C2311yescharacteristic
56#124×C61—28no—
57#124×C61—24no—
58#124×C61—22no—
59#124×C61D412yes—
60#121×C61—24no—
61#121×C61—24no—
62#121×C61—22no—
63#121×C61—22no—
64#121×C61—24no—
65#121×C61D414yes—
66#120×C61—24no—
67#120×C61—28no—
68#120×C61—24no—
69C4×D4×C61—22no—
70C4×D4×C61—22no—
71C4×D4×C61C2314yes—
72#117×C61—22no—
73#117×C61—24no—
74C2×C42×C61C2311yescharacteristic
75C24×C61—24no—
76C23×D61—41no—
77#41×C61—44no—
78#41×C61—84no—
79C2×Q8×C61—22no—
80C2×Q8×C61C2×D412yes—
81C2×D4×C61—24no—
82C2×D4×C61—28no—
83C2×D4×C61—22no—
84C2×D4×C61—24no—
85C2×D4×C61—22no—
86C2×D4×C61—22no—
87C2×D4×C61—22no—
88C2×D4×C61—88no—
89C2×D4×C61—84no—
90C2×D4×C61C2×D414yes—
91C2×D4×C61C2×D412yes—
92C22×C4×C61—22no—
93C22×C4×C61—24no—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D412yes—
96C2×D244—24no—
97C2×D244—22no—
98C2×D244—44no—
99C2×D244—42no—
100C2×D244—41no—
101C2×D244—84no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61C2×D414yes—
105C22⋊C4×C61—28no—
106C22⋊C4×C61—24no—
107C22⋊C4×C61—24no—
108C22⋊C4×C61—24no—
109C22⋊C4×C61—24no—
110C42×C61—24no—
111C42×C61C2×D412yes—
112C42×C61C2×D411yescharacteristic
113C42×C61C2411yescharacteristic
114#11758201—48no—
115#11758199—48no—
116#11758199—88no—
117#11758194—24no—
118C2×#92418—22no—
119C2×#92418—24no—
120C23×C61—24no—
121C23×C61—22no—
122C23×C61—88no—
123C23×C61—84no—
124C23×C61#12011yescharacteristic
125C22×D61—42no—
126C22×D61—82no—
127C22×D61—84no—
128Q8×C61—42no—
129Q8×C61—82no—
130D4×C61—28no—
131D4×C61—42no—
132D4×C61—44no—
133D4×C61—44no—
134D4×C61—82no—
135D4×C61—82no—
136D4×C61—84no—
137D4×C61—88no—
138D4×C61—84no—
139C2×C4×C61—24no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—24no—
143C2×C4×C61—48no—
144C2×C4×C61—44no—
145C2×C4×C61#12012yes—
146C2×C4×C61#12011yescharacteristic
147C2×C4×C61C22×D411yescharacteristic
148C2×C4×C61C22×D412yes—
149D244—44no—
150D244—48no—
151D244—84no—
152D244—82no—
153#92418—24no—
154#92418—44no—
155#18714—611nomaximal, Sylow
156C22×C61—22no—
157C22×C61—21no—
158C22×C61—44no—
159C22×C61—44no—
160C22×C61—44no—
161C22×C61—42no—
162C22×C61—44no—
163C22×C61—168no—
164C22×C61C2×#12011yescharacteristic
165C2×D61—84no—
166C2×D61—82no—
167C2×D61—164no—
168C4×C61—22no—
169C4×C61—21no—
170C4×C61—44no—
171C4×C61—42no—
172C4×C61#45312yes—
173C4×C61#57614yes—
174#3549D6111yescharacteristic, complemented
175#3205—612no—
176#3151—611no—
177#1624—614no—
178#1619—614no—
179#1543—611no—
180#1484—612no—
181C2×C61—41no—
182C2×C61—84no—
183C2×C61—82no—
184C2×C61#300812yes—
185C2×C61C2×#45311yescharacteristic
186D61—162no—
187C2×AGL(1,ℤ/8)—1222no—
188H(3,𝔽4)—22no—
189#555—22no—
190#555C2×D6112yescomplemented
191D42—22no—
192D42C2×D6112yescomplemented
193#535—22no—
194#535—24no—
195#532C2×D6111yescharacteristic
196#532C2×D6112yes—
197C2×#127—611no—
198#496—1222no—
199#493—612no—
200#493—1222no—
201#492—612no—
202#486—1222no—
203#459—618no—
204#447—1224no—
205#431—612no—
206C4×M16—611no—
207#333—612no—
208#331—612no—
209D4⋊C8—618no—
210C61#1871411yescharacteristic, complemented, Sylow
211C2×#41—22no—
212C2×#41C22⋊D6112yes—
213C22×D4—22no—
214C22×D4—24no—
215C22×D4—1222no—
216C22×D4—1221no—
217AGL(1,ℤ/8)—2448no—
218C2×SD16—1224no—
219C2×D8—1228no—
220C2×D8—1224no—
221C2×M16—1221no—
222C4⋊Q8C22⋊D6112yes—
223C4⋊Q8C22×D6111yescharacteristic
224#127—24no—
225#127—612no—
226#127—1222no—
227#127—1221no—
228#127C22⋊D6112yes—
229#127C22×D6111yescharacteristic
230#124—28no—
231#124—24no—
232#124—22no—
233#124C22⋊D6112yes—
234#121—24no—
235#121—24no—
236#121—22no—
237#121—22no—
238#121—24no—
239#121C22⋊D6114yes—
240#120—24no—
241#120—28no—
242#120—24no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4C22×D6114yes—
246#117—22no—
247#117—24no—
248C2×C42C22×D6111yescharacteristic
249C4⋊C8—614no—
250#102—1228no—
251C22⋊C8—1222no—
252#97—1221no—
253C4×C8—612no—
254C4×C8—1222no—
255C24—24no—
256C24—2441no—
257#41—44no—
258#41—84no—
259C2×Q8—22no—
260C2×Q8C2×C22⋊D6112yes—
261C2×D4—24no—
262C2×D4—28no—
263C2×D4—22no—
264C2×D4—24no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—88no—
269C2×D4—84no—
270C2×D4—1224no—
271C2×D4—1222no—
272C2×D4—4884no—
273C2×D4—2444no—
274C2×D4—2442no—
275C2×D4—2441no—
276C2×D4C2×C22⋊D6114yes—
277C2×D4C2×C22⋊D6112yes—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4C2×C22⋊D6112yes—
281C22×C4C2×C22⋊D6111yescharacteristic
282SD16—2448no—
283D8—4888no—
284D8—2448no—
285M16—1224no—
286C2×C8—1222no—
287C2×C8—1224no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4C2×C22⋊D6114yes—
291C22⋊C4—28no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C42—24no—
297C42C2×C22⋊D6112yes—
298C42C2×C22⋊D6111yescharacteristic
299C42C23×D6111yescharacteristic
300C23—22no—
301C23—24no—
302C23—84no—
303C23—88no—
304C23—4884no—
305C23—4882no—
306C23—2442no—
307C23#1451974611yescharacteristic
308Q8—42no—
309Q8—82no—
310D4—28no—
311D4—42no—
312D4—44no—
313D4—44no—
314D4—82no—
315D4—82no—
316D4—84no—
317D4—84no—
318D4—88no—
319D4—4884no—
320D4—4882no—
321D4—2444no—
322D4—2448no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—48no—
328C2×C4—44no—
329C2×C4#1451974612yes—
330C2×C4#1451974611yescharacteristic
331C2×C4C22×C22⋊D6111yescharacteristic
332C2×C4C22×C22⋊D6112yes—
333C8—1224no—
334C8—2444no—
335C22—21no—
336C22—22no—
337C22—42no—
338C22—44no—
339C22—44no—
340C22—44no—
341C22—44no—
342C22—168no—
343C22—9764no—
344C22—4882no—
345C22—4884no—
346C22C2×#1451974611yescharacteristic
347C4—21no—
348C4—22no—
349C4—42no—
350C4—44no—
351C4#1683993012yes—
352C4#1684056514yes—
353C2—41no—
354C2—82no—
355C2—84no—
356C2—9762no—
357C2#2145508712yes—
358C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3549C2
socleC22×C61C2×#120
hypercenter#3549D61
hypocenterC61#18714
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3549

Complements

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

NH ≅ G/Ntype
1. #21456734C2semidirect
2. #21456272C2semidirect
3. #21444902C2semidirect
4. #21444873C2semidirect
5. #21444481C2semidirect
6. #21444236C2semidirect
7. C61×#3549C2semidirect
9. C2×#14519774C22semidirect
13. #16840136C22semidirect
15. #16839957C22semidirect
17. #16839815C22semidirect
18. C4×#11758194C22semidirect
19. #16839511C22semidirect
20. #16839506C22semidirect
21. D4⋊#92418C22semidirect
24. C61×#555C22semidirect
26. D42×C61C22semidirect
174. #3549D61semidirect
190. #555C2×D61semidirect
192. D42C2×D61semidirect
210. C61#18714semidirect

Maximal subgroups

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

subgroupindex
#214567342
#214562722
#214449022
#214448732
#214444812
#214442362
C61×#35492
#1871461