Finite Group Databasebeta

← #29593237

Subgroups — #29593237

Order 15616.

344 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 63 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#21456590C211yesmaximal, characteristic
2#21448022C211yesmaximal, characteristic, complemented
3#21448019C211yesmaximal, characteristic, complemented
4#21447249C211yesmaximal, characteristic
5#21443830C211yesmaximal, characteristic, complemented
6#21443742C211yesmaximal, characteristic
7C61×#1348C211yesmaximal, characteristic, complemented
8C22⋊Q8×D61—21no—
9#16840099—21no—
10#16840096—21no—
11#16840003C2211yescharacteristic
12#16840001—21no—
13#16839919—21no—
14#16839916—21no—
15C2×#14519711C2211yescharacteristic
16C2×#14519689C2211yescharacteristic
17#16839619—21no—
18#16839588—21no—
19#16839587—21no—
20#16839567—21no—
21#16839566—21no—
22C22⋊#11758185C2211yescharacteristic
23#16839524—21no—
24C61×#467C2211yescharacteristic
25C61×C22⋊C16C2211yescharacteristic
26C61×#340C2211yescharacteristic
27C2×Q8×D61—21no—
28C22×C4×D61D411yescharacteristic
29C2×Dic244D411yescharacteristic
30C2×#11758202—21no—
31C22×#92418D411yescharacteristic
32#14519757—21no—
33#14519756—21no—
34#14519755—41no—
35#14519754—41no—
36#14519753—21no—
37C2×C4⋊Dic61—21no—
38C4⋊C4×D61—21no—
39C4⋊C4×D61—41no—
40C2×#11758198—41no—
41C22⋊C4×D61—41no—
42C2×#11758185—21no—
43#14519711—21no—
44#14519711—21no—
45#14519711D411yescharacteristic
46#14519710—21no—
47#14519702—21no—
48#14519701—21no—
49#14519700—21no—
50#14519689—21no—
51#14519689—21no—
52#14519684—41no—
53C2×Q16×C61D411yescharacteristic
54C22×C8×C61C2311yescharacteristic
55C22⋊Q8×C61—21no—
56C2×C4⋊C4×C61—21no—
57C2×C16×C61—21no—
58#107×C61D411yescharacteristic
59#103×C61—21no—
60C23×D61D811yescharacteristic
61C2×Q8×C61—21no—
62C22×C4×C61—41no—
63C22×C4×C61C2×D411yescharacteristic
64C22×Dic61—41no—
65C22×Dic61D811yescharacteristic
66Q8×D61—81no—
67Q8×D61—81no—
68C2×C4×D61—21no—
69C2×C4×D61—21no—
70C2×C4×D61—41no—
71C2×C4×D61—41no—
72C2×C4×D61—41no—
73C2×C4×D61—41no—
74C2×Dic122—21no—
75C2×Dic122—41no—
76Q16×C61—41no—
77C2×C8×C61—41no—
78C2×C8×C61C2×D411yescharacteristic
79C2×C8×C61C2×D411yescharacteristic
80C4⋊C4×C61—21no—
81C4⋊C4×C61—41no—
82C4⋊C4×C61—41no—
83C4⋊C4×C61—41no—
84C22⋊C4×C61—41no—
85C22⋊Dic61—41no—
86#11758202—81no—
87#11758198—41no—
88#11758198—41no—
89#11758198—82no—
90C4⋊Dic61—21no—
91C4⋊Dic61—41no—
92C4⋊Dic61—41no—
93C4⋊Dic61—41no—
94#11758196—41no—
95#11758196—41no—
96#11758196—41no—
97C2×#92418—21no—
98C2×#92418—41no—
99Dic244—41no—
100C16×C61—41no—
101#11758185—41no—
102C23×C61C2×D811yescharacteristic
103C22×D61—21no—
104C22×D61—21no—
105C22×D61—41no—
106C22×D61—41no—
107C22×D61Q3212yes—
108Q8×C61—41no—
109Q8×C61—41no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—81no—
115C2×C4×C61—81no—
116C2×C4×C61—81no—
117C2×C4×C61#12011yescharacteristic
118C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
119C2×Dic61—21no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—81no—
124C2×Dic61—81no—
125C2×Dic61—81no—
126C4×D61—41no—
127C4×D61—41no—
128C4×D61—81no—
129C4×D61—82no—
130C4×D61—82no—
131Dic122—41no—
132Dic122—41no—
133Dic122—82no—
134C8×C61—21no—
135C8×C61—21no—
136#92418—41no—
137#12282—611nomaximal, Sylow
138C22×C61—21no—
139C22×C61—21no—
140C22×C61#44711yescharacteristic
141C22×C61C2×Q3211yescharacteristic
142C22×C61#51011yescharacteristic
143C2×D61—22no—
144C2×D61—41no—
145C2×D61—41no—
146C2×D61—42no—
147C2×D61—81no—
148C4×C61—21no—
149C4×C61—81no—
150C4×C61—81no—
151C4×C61—161no—
152C4×C61#45511yescharacteristic
153C4×C61#45811yescharacteristic
154Dic61—41no—
155Dic61—81no—
156Dic61—81no—
157Dic61—161no—
158#3189—611no—
159#2236—611no—
160#2236—611no—
161#2076—611no—
162#1348—611no—
163#1348D6111yescharacteristic, complemented
164#1314—611no—
165C2×C61—21no—
166C2×C61—21no—
167C2×C61#160711yescharacteristic
168C2×C61#218811yescharacteristic
169C2×C61#219011yescharacteristic
170D61—42no—
171C2×C22⋊Q8—1221no—
172#484—1221no—
173#484—1221no—
174#467—611no—
175#467—1221no—
176#467C2×D6111yescharacteristic
177#451—1221no—
178#451—1221no—
179C2×#107—611no—
180C2×C22⋊C8—611no—
181#380—1221no—
182#366—1221no—
183#366—1221no—
184#358—1221no—
185#358—1221no—
186C22⋊C16—611no—
187C22⋊C16C2×D6111yescharacteristic
188#340—1221no—
189#340C2×D6111yescharacteristic
190C61#1228211yescharacteristic, complemented, Sylow
191C22×Q8—1221no—
192C23×C4—611no—
193C2×Q16—611no—
194C2×Q16—1221no—
195C2×Q16D24411yescharacteristic
196C22×C8—611no—
197C22×C8C22×D6111yescharacteristic
198C22⋊Q8—21no—
199C22⋊Q8—1221no—
200C22⋊Q8—1221no—
201C22⋊Q8—1221no—
202C22⋊Q8—2441no—
203C22⋊Q8—2441no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—1221no—
206C2×C4⋊C4—1221no—
207C2×C4⋊C4—2441no—
208C2×C22⋊C4—2441no—
209C2×C22⋊C4—2441no—
210C2×C16—21no—
211C2×C16—1221no—
212#107—611no—
213#107—1221no—
214#107—1221no—
215#107—1221no—
216#107D24411yescharacteristic
217#103—21no—
218#103—1221no—
219#103—1221no—
220#103—1221no—
221C22⋊C8—1221no—
222C22⋊C8—1221no—
223#95—2441no—
224C24—611no—
225C2×Q8—21no—
226C2×Q8—1221no—
227C2×Q8—4881no—
228C2×Q8—4881no—
229C2×Q8—2441no—
230C22×C4—41no—
231C22×C4—611no—
232C22×C4—1221no—
233C22×C4—1221no—
234C22×C4—2441no—
235C22×C4—2441no—
236C22×C4—2441no—
237C22×C4—2441no—
238C22×C4—2441no—
239C22×C4D4×D6111yescharacteristic
240Q16—41no—
241Q16—4881no—
242Q16—2441no—
243C2×C8—41no—
244C2×C8—1221no—
245C2×C8—2441no—
246C2×C8C2×D24411yescharacteristic
247C2×C8D4×D6111yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—1221no—
253C4⋊C4—2441no—
254C4⋊C4—2441no—
255C4⋊C4—2441no—
256C4⋊C4—2441no—
257C4⋊C4—2441no—
258C4⋊C4—2441no—
259C22⋊C4—41no—
260C22⋊C4—4882no—
261C22⋊C4—2441no—
262C22⋊C4—2441no—
263C22⋊C4—2441no—
264C16—41no—
265C16—2441no—
266C23—612no—
267C23—1221no—
268C23—1221no—
269C23—2441no—
270C23—2441no—
271C23D8×D6111yescharacteristic
272Q8—41no—
273Q8—41no—
274Q8—4882no—
275Q8—2441no—
276Q8—2441no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—1221no—
285C2×C4—4881no—
286C2×C4—4882no—
287C2×C4—4882no—
288C2×C4—4881no—
289C2×C4—4881no—
290C2×C4—4881no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4—2441no—
296C2×C4#1451974711yescharacteristic
297C2×C4#1451980411yescharacteristic
298C8—21no—
299C8—21no—
300C8—2441no—
301C22—21no—
302C22—21no—
303C22—1222no—
304C22—4881no—
305C22—2442no—
306C22—2441no—
307C22—2441no—
308C22#1683989711yescharacteristic
309C22Q32×D6111yescharacteristic
310C22#1684020111yescharacteristic
311C4—21no—
312C4—81no—
313C4—81no—
314C4—161no—
315C4—9761no—
316C4—4881no—
317C4—4881no—
318C4—2441no—
319C4#1683994111yescharacteristic
320C4#1683995311yescharacteristic
321C2—21no—
322C2—21no—
323C2—2442no—
324C2#2144480611yescharacteristic
325C2#2144775411yescharacteristic
326C2#2144776611yescharacteristic
subgroupG/H
centerC22#16839897
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#1348C2
socleC22×C61#447
prime radicalC23×D61D8
hypercenter#1348D61
hypocenterC61#12282
LCSG ▷ C22×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1348

Complements

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

NH ≅ G/Ntype
2. #21448022C2semidirect
3. #21448019C2semidirect
5. #21443830C2semidirect
7. C61×#1348C2semidirect
163. #1348D61semidirect
190. C61#12282semidirect

Maximal subgroups

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

subgroupindex
#214565902
#214480222
#214480192
#214472492
#214438302
#214437422
C61×#13482
#1228261