Finite Group Databasebeta

← #29593265

Subgroups — #29593265

Order 15616.

306 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 59 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#21456714C211yesmaximal, characteristic
2#21448022C211yesmaximal, characteristic, complemented
3#21448017C211yesmaximal, characteristic, complemented
4#21447259C211yesmaximal, characteristic
5#21443838C211yesmaximal, characteristic, complemented
6#21443838C211yesmaximal, characteristic, complemented
7C61×#1315C211yesmaximal, characteristic
8#16840333—21no—
9#16840104—21no—
10#16840104—21no—
11#16840020—21no—
12#16840003C2211yescharacteristic
13#16840003C2211yescharacteristic
14#16839920—21no—
15#16839920—21no—
16C2×#14519711C2211yescharacteristic
17#16839763—21no—
18#16839590—21no—
19#16839588—21no—
20#16839567—21no—
21#16839567—21no—
22#16839530—21no—
23#16839527C2211yescharacteristic
24C61×#408C2211yescharacteristic
25C61×C22⋊C16C2211yescharacteristic
26C61×C22⋊C16C2211yescharacteristic
27C22×Dic122—21no—
28C2×Dic244—21no—
29C2×Dic244D411yescharacteristic
30C2×Dic244D411yescharacteristic
31C2×#11758194—21no—
32#14519775—41no—
33#14519764—41no—
34#14519758—41no—
35#14519758—41no—
36#14519756—21no—
37#14519756—21no—
38#14519756—21no—
39#14519756—21no—
40#14519736—21no—
41C2×C4⋊Dic61—21no—
42#14519711—21no—
43#14519711—21no—
44#14519711D411yescharacteristic
45#14519711D411yescharacteristic
46#14519709—21no—
47#14519702—21no—
48#14519702—21no—
49#14519702—21no—
50#14519702—21no—
51#14519701—41no—
52#14519698—41no—
53C2×#41×C61D411yescharacteristic
54C2×M16×C61D411yescharacteristic
55C22×C8×C61C2311yescharacteristic
56C2×C16×C61—21no—
57C2×C16×C61—21no—
58C22⋊C8×C61—21no—
59C22⋊C8×C61—21no—
60#41×C61—41no—
61#41×C61—41no—
62C2×Q8×C61D811yescharacteristic
63C2×D4×C61—21no—
64C2×D4×C61D811yescharacteristic
65C22×C4×C61—21no—
66C22×C4×C61C2×D411yescharacteristic
67C22×Dic61—41no—
68C22×Dic61—41no—
69C2×Dic122—21no—
70C2×Dic122—21no—
71C2×Dic122—41no—
72C2×Dic122—81no—
73C2×Dic122—81no—
74M16×C61—41no—
75C2×C8×C61—21no—
76C2×C8×C61—41no—
77C2×C8×C61C2×D411yescharacteristic
78C2×C8×C61C2×D411yescharacteristic
79C22⋊Dic61—41no—
80C22⋊Dic61—41no—
81C22⋊Dic61—41no—
82C22⋊Dic61—41no—
83C22⋊Dic61—41no—
84C4⋊Dic61—21no—
85C4⋊Dic61—21no—
86C4⋊Dic61—41no—
87C4⋊Dic61—41no—
88#11758196—41no—
89#11758196—41no—
90#11758196—41no—
91#11758196—41no—
92C4×Dic61—41no—
93#11758194—41no—
94#11758194—41no—
95C2×#92418—41no—
96Dic244—41no—
97Dic244—41no—
98Dic244—81no—
99C16×C61—41no—
100C16×C61—41no—
101C23×C61—21no—
102C23×C61AGL(1,ℤ/8)11yescharacteristic
103Q8×C61—41no—
104D4×C61—41no—
105D4×C61—81no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61#12011yescharacteristic
112C2×C4×C61C2×D811yescharacteristic
113C2×Dic61—41no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—41no—
117C2×Dic61—82no—
118C2×Dic61—81no—
119C2×Dic61—161no—
120Dic122—41no—
121Dic122—41no—
122Dic122—41no—
123Dic122—41no—
124Dic122—82no—
125C8×C61—21no—
126C8×C61—21no—
127C8×C61—41no—
128#92418—81no—
129#12286—611nomaximal, Sylow
130C22×C61—21no—
131C22×C61—21no—
132C22×C61—21no—
133C22×C61—41no—
134C22×C61—41no—
135C22×C61—41no—
136C22×C61#44711yescharacteristic
137C4×C61—21no—
138C4×C61—41no—
139C4×C61#45511yescharacteristic
140C4×C61#45811yescharacteristic
141Dic61—81no—
142Dic61—81no—
143Dic61—81no—
144Dic61—81no—
145Dic61—161no—
146#3202—611no—
147#2236—611no—
148#2235—611no—
149#2078—611no—
150#1350—611no—
151#1350—611no—
152#1315D6111yescharacteristic
153C2×C61—41no—
154C2×C61—81no—
155C2×C61#160711yescharacteristic
156C2×C61#219311yescharacteristic
157C2×C61#219511yescharacteristic
158#536—1221no—
159#485—1221no—
160#485—1221no—
161#470—1221no—
162#467—611no—
163#467—611no—
164#451—1221no—
165#451—1221no—
166C2×#107—611no—
167#417—1221no—
168#408C2×D6111yescharacteristic
169#367—1221no—
170#366—1221no—
171#358—1221no—
172#358—1221no—
173C22⋊C16C2×D6111yescharacteristic
174C22⋊C16C2×D6111yescharacteristic
175#343—1221no—
176#341—611no—
177C61#1228611yescharacteristic, complemented, Sylow
178C2×#41D24411yescharacteristic
179C22×Q8—1221no—
180C2×Q16—611no—
181C2×Q16—611no—
182C2×Q16—1221no—
183C2×M16—1221no—
184C2×M16D24411yescharacteristic
185C22×C8C22×D6111yescharacteristic
186C4⋊Q8—2441no—
187#124—2441no—
188#123—2441no—
189#123—2441no—
190C22⋊Q8—1221no—
191C22⋊Q8—1221no—
192C22⋊Q8—1221no—
193C22⋊Q8—1221no—
194#117—1221no—
195C2×C4⋊C4—1221no—
196C2×C16—21no—
197C2×C16—21no—
198#107—611no—
199#107—611no—
200#107—1221no—
201#107—1221no—
202#106—1221no—
203#103—1221no—
204#103—1221no—
205#103—1221no—
206#103—1221no—
207#103—2441no—
208#102—2441no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211#41—41no—
212#41—41no—
213C2×Q8—1221no—
214C2×Q8—1221no—
215C2×Q8—4881no—
216C2×Q8—4881no—
217C2×Q8—2441no—
218C2×Q8D48811yescharacteristic
219C2×D4—21no—
220C2×D4D48811yescharacteristic
221C22×C4—21no—
222C22×C4—2441no—
223C22×C4—2441no—
224C22×C4C2×D24411yescharacteristic
225Q16—4881no—
226Q16—2441no—
227Q16—2441no—
228M16—41no—
229M16—2441no—
230M16—2441no—
231C2×C8—21no—
232C2×C8—41no—
233C2×C8—2441no—
234C2×C8D4×D6111yescharacteristic
235C2×C8D4×D6111yescharacteristic
236C4⋊C4—1221no—
237C4⋊C4—1221no—
238C4⋊C4—2441no—
239C4⋊C4—2441no—
240C4⋊C4—2441no—
241C4⋊C4—2441no—
242C4⋊C4—2441no—
243C4⋊C4—2441no—
244C22⋊C4—2441no—
245C22⋊C4—2441no—
246C22⋊C4—2441no—
247C22⋊C4—2441no—
248C22⋊C4—2441no—
249C42—2441no—
250C16—41no—
251C16—41no—
252C23—21no—
253C23#1451980511yescharacteristic
254Q8—41no—
255Q8—4882no—
256Q8—2441no—
257Q8—2441no—
258Q8—2441no—
259Q8—2441no—
260D4—41no—
261D4—81no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—9761no—
268C2×C4—4881no—
269C2×C4—4882no—
270C2×C4—2441no—
271C2×C4—2441no—
272C2×C4—2441no—
273C2×C4—2441no—
274C2×C4#1451974711yescharacteristic
275C2×C4C2×D48811yescharacteristic
276C8—21no—
277C8—21no—
278C8—41no—
279C8—4881no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22#1683989811yescharacteristic
287C4—21no—
288C4—41no—
289C4—9761no—
290C4—4881no—
291C4—4881no—
292C4—4881no—
293C4—4881no—
294C4#1683994211yescharacteristic
295C4#1683995311yescharacteristic
296C2—41no—
297C2—81no—
298C2#2144480711yescharacteristic
299C2#2144778511yescharacteristic
300C2#2144779911yescharacteristic
subgroupG/H
centerC22#16839898
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#1315C2
socleC22×C61#447
prime radicalC2×#41×C61D4
hypercenter#1315D61
hypocenterC61#12286
LCSG ▷ C22×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #408 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #21448022C2semidirect
3. #21448017C2semidirect
5. #21443838C2semidirect
6. #21443838C2semidirect
177. C61#12286semidirect

Maximal subgroups

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

subgroupindex
#214567142
#214480222
#214480172
#214472592
#214438382
C61×#13152
#1228661