Finite Group Databasebeta

← #29661266

Subgroups — #29661266

Order 15616.

1066 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 155 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×#16840504C211yesmaximal, characteristic
2#21456818C212yesmaximal, complemented
3#21445124C218yesmaximal, complemented
4#21444443C211yesmaximal, characteristic, complemented
5#21444416C212yesmaximal, complemented
6C61×#3140C211yesmaximal, characteristic, complemented
7#16840504—24no—
8#16840504—22no—
9#16840504C2214yes—
10C2×#14519766—21no—
11C2×#14519765C2212yes—
12C2×#14519763—22no—
13C2×#14519747—22no—
14C2×#14519745—21no—
15#16840154C2214yescomplemented
16#16840124C2214yescomplemented
17#16839984—24no—
18#16839917—22no—
19#16839912—22no—
20#16839819C2212yescomplemented
21#16839502C2214yescomplemented
22#16839501C2218yescomplemented
23C61×#533C2211yescharacteristic, complemented
24C2×#124×C61C2211yescharacteristic
25C61×#489C2214yescomplemented
26C61×#488—22no—
27C61×#419—21no—
28C61×#418—21no—
29C4×M16×C61C2211yescharacteristic, complemented
30C24×D61—21no—
31C22×Dic122D412yes—
32C22×D244—21no—
33C22×D244D412yes—
34C22×C22⋊D61—22no—
35C2×#11758202—24no—
36C2×#11758201—24no—
37#14519766—24no—
38#14519766—24no—
39#14519765—22no—
40#14519765—22no—
41#14519765D414yes—
42#14519765C2314yes—
43#14519763—28no—
44#14519763—28no—
45#14519747—28no—
46#14519747—28no—
47#14519745—24no—
48#14519745—24no—
49C2×C22⋊Dic61—21no—
50C2×#11758198—22no—
51C2×#11758198—22no—
52C2×#11758198—22no—
53C2×C4×Dic61—21no—
54C4⋊#92418—22no—
55#14519700—24no—
56#14519697—24no—
57C22⋊#92418—22no—
58#14519688C2314yescomplemented
59C22×Q8×C61D411yescharacteristic
60C22×D4×C61D411yescharacteristic
61C2×M16×C61—21no—
62C4⋊Q8×C61C2312yescomplemented
63#125×C61—21no—
64#124×C61—22no—
65#124×C61D412yes—
66#124×C61C2312yes—
67C22⋊Q8×C61—22no—
68C4×Q8×C61—22no—
69#117×C61—21no—
70C2×C22⋊C4×C61—22no—
71C2×C42×C61C2311yescharacteristic
72#103×C61—24no—
73#102×C61—24no—
74#97×C61C2312yescomplemented
75C4×C8×C61—21no—
76C24×C61—21no—
77C23×D61—22no—
78C23×D61—24no—
79C23×D61—84no—
80C23×D61—82no—
81C2×Q8×C61—21no—
82C2×Q8×C61—22no—
83C2×Q8×C61—42no—
84C2×Q8×C61C2×D412yes—
85C2×D4×C61—21no—
86C2×D4×C61—42no—
87C2×D4×C61C2×D412yes—
88C22×C4×C61—21no—
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D412yes—
91C2×C22⋊D61—28no—
92C2×C22⋊D61—84no—
93C22×Dic61—22no—
94C2×D244—24no—
95C2×D244—22no—
96C2×D244—82no—
97C2×D244—82no—
98C2×D244C2×D414yes—
99C2×Dic122—22no—
100C2×Dic122—82no—
101C2×Dic122C2×D414yes—
102M16×C61—42no—
103C2×C8×C61—22no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—42no—
107C22⋊C4×C61—24no—
108C22⋊C4×C61—24no—
109C22⋊C4×C61—41no—
110C42×C61—41no—
111C42×C61C2×D412yes—
112C42×C61C2×D411yescharacteristic
113C42×C61C2411yescharacteristic
114C22⋊Dic61—24no—
115#11758202—84no—
116#11758201—84no—
117#11758198—28no—
118#11758198—24no—
119#11758198—24no—
120#11758198—24no—
121#11758198—24no—
122C4×Dic61—24no—
123C2×#92418—24no—
124C23×C61—22no—
125C23×C61—44no—
126C23×C61—42no—
127C23×C61#12011yescharacteristic
128C22×D61—24no—
129C22×D61—44no—
130C22×D61—44no—
131C22×D61—44no—
132C22×D61—84no—
133C22×D61—88no—
134C22×D61—816no—
135Q8×C61—42no—
136Q8×C61—44no—
137Q8×C61—81no—
138D4×C61—42no—
139D4×C61—81no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61#12011yescharacteristic
146C2×C4×C61#12012yes—
147C2×C4×C61C22×D412yes—
148C2×C4×C61C22×D411yescharacteristic
149C22⋊D61—88no—
150C2×Dic61—24no—
151C2×Dic61—44no—
152D244—44no—
153D244—82no—
154D244—84no—
155Dic122—44no—
156Dic122—82no—
157C8×C61—42no—
158#92418—44no—
159#18997—611nomaximal, Sylow
160C22×C61—22no—
161C22×C61—42no—
162C22×C61—44no—
163C22×C61—84no—
164C22×C61#45512yes—
165C22×C61C2×#12011yescharacteristic
166C2×D61—44no—
167C2×D61—44no—
168C2×D61—44no—
169C2×D61—88no—
170C2×D61—816no—
171C4×C61—21no—
172C4×C61—21no—
173C4×C61—22no—
174C4×C61—22no—
175C4×C61—42no—
176C4×C61—82no—
177Dic61—44no—
178C2×H(3,𝔽4)—611no—
179#3215—612no—
180#3140D6111yescharacteristic, complemented
181#1659—618no—
182#1533—611no—
183#1527—612no—
184C2×C61—22no—
185C2×C61—82no—
186C2×C61C2×#45511yescharacteristic
187C2×C61#301912yes—
188D61—84no—
189H(3,𝔽4)—614no—
190H(3,𝔽4)—1224no—
191H(3,𝔽4)—1222no—
192#533C2×D6111yescharacteristic, complemented
193C2×#124—612no—
194C2×#124—1222no—
195C2×#124—1221no—
196C2×#124C2×D6111yescharacteristic
197C2×#120—1222no—
198C2×#120—1221no—
199#497—614no—
200#489—614no—
201#489C2×D6114yescomplemented
202#488—22no—
203#464—1224no—
204#451—1222no—
205#450—1222no—
206#432—612no—
207#419—21no—
208#418—21no—
209C4×M16C2×D6111yescharacteristic, complemented
210#329—614no—
211#329—618no—
212C61#1899711yescharacteristic, complemented, Sylow
213C25—1221no—
214C22×Q8—612no—
215C22×Q8C22⋊D6111yescharacteristic
216C22×D4—612no—
217C22×D4—1221no—
218C22×D4—1222no—
219C22×D4C22⋊D6111yescharacteristic
220C2×Q16—1224no—
221C2×SD16—1224no—
222C2×M16—21no—
223C4⋊Q8C22×D6112yescomplemented
224#125—21no—
225#124—22no—
226#124—614no—
227#124—614no—
228#124—1228no—
229#124—1228no—
230#124—1224no—
231#124—1224no—
232#124—1222no—
233#124—1222no—
234#124C22⋊D6112yes—
235#124C22×D6112yes—
236C22⋊Q8—22no—
237#120—1228no—
238#120—1228no—
239#120—1224no—
240#120—1224no—
241C4×Q8—22no—
242#117—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—1222no—
245C2×C22⋊C4—1221no—
246C2×C22⋊C4—1222no—
247C2×C22⋊C4—1222no—
248C2×C42—1221no—
249C2×C42C22×D6111yescharacteristic
250C4⋊C8—1222no—
251#103—24no—
252#103—1224no—
253#102—24no—
254#102—1224no—
255C22⋊C8—1222no—
256#97—614no—
257#97C22×D6112yescomplemented
258C4×C8—21no—
259C24—21no—
260C24—1224no—
261C24—1222no—
262C24—4884no—
263C24—4882no—
264C2×Q8—21no—
265C2×Q8—22no—
266C2×Q8—42no—
267C2×Q8—614no—
268C2×Q8—1222no—
269C2×Q8—4882no—
270C2×Q8C2×C22⋊D6112yes—
271C2×D4—21no—
272C2×D4—42no—
273C2×D4—614no—
274C2×D4—1224no—
275C2×D4—1228no—
276C2×D4—1222no—
277C2×D4—4882no—
278C2×D4—4882no—
279C2×D4—4884no—
280C2×D4C2×C22⋊D6112yes—
281C22×C4—21no—
282C22×C4—1222no—
283C22×C4D4×D6112yes—
284C22×C4C2×C22⋊D6111yescharacteristic
285Q16—4884no—
286SD16—4884no—
287M16—42no—
288C2×C8—22no—
289C2×C8—1224no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—41no—
296C22⋊C4—1228no—
297C22⋊C4—1224no—
298C22⋊C4—1224no—
299C22⋊C4—1224no—
300C22⋊C4—1224no—
301C22⋊C4—1224no—
302C42—41no—
303C42—1224no—
304C42D4×D6112yes—
305C42C2×C22⋊D6111yescharacteristic
306C42C23×D6111yescharacteristic
307C23—22no—
308C23—44no—
309C23—42no—
310C23—1224no—
311C23—4884no—
312C23—4888no—
313C23—48816no—
314C23—2444no—
315C23—2444no—
316C23—2444no—
317C23#1451974511yescharacteristic
318Q8—42no—
319Q8—44no—
320Q8—81no—
321Q8—4882no—
322Q8—2444no—
323D4—42no—
324D4—81no—
325D4—4882no—
326D4—4884no—
327D4—4888no—
328D4—2444no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—1224no—
335C2×C4—2444no—
336C2×C4#1451974511yescharacteristic
337C2×C4#1451974512yes—
338C2×C4C2×D4×D6112yes—
339C2×C4C22×C22⋊D6111yescharacteristic
340C8—42no—
341C8—2444no—
342C22—22no—
343C22—42no—
344C22—44no—
345C22—84no—
346C22—4888no—
347C22—48816no—
348C22—2444no—
349C22—2444no—
350C22—2444no—
351C22#1683993912yes—
352C22C2×#1451974511yescharacteristic
353C4—21no—
354C4—21no—
355C4—22no—
356C4—22no—
357C4—42no—
358C4—82no—
359C4—2444no—
360C2—22no—
361C2—82no—
362C2—4884no—
363C2C2×#1683993911yescharacteristic
364C2#2145517412yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3140C2
socleC22×C61C2×#120
prime radicalC2×#16840504C2
hypercenter#3140D61
hypocenterC61#18997
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3140

Complements

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

NH ≅ G/Ntype
2. #21456818C2semidirect
3. #21445124C2semidirect
4. #21444443C2semidirect
5. #21444416C2semidirect
6. C61×#3140C2semidirect
15. #16840154C22semidirect
16. #16840124C22semidirect
20. #16839819C22semidirect
21. #16839502C22semidirect
22. #16839501C22semidirect
23. C61×#533C22semidirect
25. C61×#489C22semidirect
29. C4×M16×C61C22semidirect
58. #14519688C23semidirect
62. C4⋊Q8×C61C23semidirect
74. #97×C61C23semidirect
180. #3140D61semidirect
192. #533C2×D61semidirect
201. #489C2×D61semidirect
209. C4×M16C2×D61semidirect
212. C61#18997semidirect
223. C4⋊Q8C22×D61semidirect
257. #97C22×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168405042
#214568182
#214451242
#214444432
#214444162
C61×#31402
#1899761