Finite Group Databasebeta

← #29592200

Subgroups — #29592200

Order 15616.

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

normal subgroups only

This group has 392 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#21447769C212yesmaximal, complemented
2#21446923C211yesmaximal, characteristic
3#21443813C212yesmaximal, complemented
4#21443757C211yesmaximal, characteristic, complemented
5C61×#3202C211yesmaximal, characteristic, complemented
6#16840562C2211yescharacteristic
7#16840560—21no—
8#16840203—22no—
9#16839728—21no—
10#16839571—22no—
11#16839544C2212yes—
12#16839520C2211yescharacteristic
13C61×#536—21no—
14C61×#485—22no—
15C61×#470C2212yes—
16C61×#467—21no—
17C61×#451—22no—
18C61×#408C2211yescharacteristic
19C2×#11758222—21no—
20#14519810—41no—
21#14519808—41no—
22#14519807—41no—
23#14519805—41no—
24#14519803—41no—
25#14519802—41no—
26#14519799—22no—
27#14519797—42no—
28C2×#11758194—21no—
29#14519723—42no—
30#14519720—42no—
31#14519717—22no—
32#14519689—42no—
33C2×#41×C61D411yescharacteristic
34C22×Q8×C61—21no—
35C2×Q16×C61—21no—
36C2×Q16×C61D412yes—
37C2×M16×C61C2311yescharacteristic
38C22×C8×C61D411yescharacteristic
39C4⋊Q8×C61—41no—
40#124×C61—41no—
41#123×C61—41no—
42#123×C61—41no—
43C22⋊Q8×C61—22no—
44C22⋊Q8×C61—22no—
45#117×C61—21no—
46#107×C61—21no—
47#106×C61D412yes—
48#103×C61—22no—
49#103×C61—22no—
50C22⋊C8×C61—22no—
51#41×C61—41no—
52#41×C61—41no—
53C2×Q8×C61—22no—
54C2×Q8×C61—41no—
55C2×Q8×C61—81no—
56C2×Q8×C61—81no—
57C2×Q8×C61D811yescharacteristic
58C2×D4×C61—21no—
59C2×D4×C61D811yescharacteristic
60C22×C4×C61—21no—
61C22×C4×C61—41no—
62C22×C4×C61C2×D411yescharacteristic
63C2×C22⋊D61—41no—
64#11758222—41no—
65#11758222—41no—
66#11758222—41no—
67#11758222—41no—
68C2×D244—21no—
69C2×C4×D61—41no—
70C2×Dic122—21no—
71Q16×C61—42no—
72Q16×C61—81no—
73M16×C61—41no—
74C2×C8×C61—21no—
75C2×C8×C61—41no—
76C2×C8×C61C2×D412yes—
77C4⋊C4×C61—22no—
78C4⋊C4×C61—42no—
79C4⋊C4×C61—42no—
80C22⋊C4×C61—41no—
81C22⋊C4×C61—42no—
82C22⋊C4×C61—42no—
83C42×C61—41no—
84#11758202—42no—
85#11758201—42no—
86#11758200—42no—
87#11758199—42no—
88#11758194—41no—
89#11758194—41no—
90C2×#92418—22no—
91Dic244—22no—
92D488—22no—
93#11758190—42no—
94#11758185—42no—
95C23×C61—21no—
96C23×C61AGL(1,ℤ/8)11yescharacteristic
97C22×D61—41no—
98Q8×C61—41no—
99Q8×C61—42no—
100Q8×C61—42no—
101Q8×C61—82no—
102D4×C61—41no—
103D4×C61—81no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61—82no—
112C2×C4×C61#12011yescharacteristic
113C2×C4×C61C2×D811yescharacteristic
114C22⋊D61—81no—
115C22⋊D61—81no—
116C2×Dic61—41no—
117D244—41no—
118D244—41no—
119C4×D61—81no—
120C4×D61—81no—
121Dic122—41no—
122Dic122—41no—
123C8×C61—22no—
124C8×C61—41no—
125#92418—42no—
126#12118—611nomaximal, Sylow
127C22×C61—21no—
128C22×C61—41no—
129C22×C61—42no—
130C22×C61—41no—
131C22×C61#44711yescharacteristic
132C2×D61—82no—
133C2×D61—81no—
134C4×C61—21no—
135C4×C61—41no—
136C4×C61—82no—
137C4×C61—82no—
138C4×C61#45311yescharacteristic
139C4×C61B(4,𝔽2)11yescharacteristic
140Dic61—81no—
141#3202D6111yescharacteristic, complemented
142#2191—612no—
143#2017—611no—
144#1342—612no—
145#1321—611no—
146C2×C61—21no—
147C2×C61—41no—
148C2×C61—81no—
149C2×C61#159511yescharacteristic
150D61—161no—
151#575—611no—
152#575—1221no—
153#536—21no—
154#510—1222no—
155#485—22no—
156#470C2×D6112yes—
157#467—21no—
158#451—22no—
159#408—1221no—
160#408C2×D6111yescharacteristic
161#360—1222no—
162#349—612no—
163#338—611no—
164C61#1211811yescharacteristic, complemented, Sylow
165C2×#41—1221no—
166C2×#41C22⋊D6111yescharacteristic
167C22×Q8—21no—
168#137—2441no—
169#137—2441no—
170#137—2441no—
171AGL(1,ℤ/8)—2441no—
172AGL(1,ℤ/8)—2441no—
173AGL(1,ℤ/8)—2441no—
174#135—1222no—
175#135—2442no—
176C2×Q16—21no—
177C2×Q16C22⋊D6112yes—
178C2×M16—1221no—
179C2×M16C22×D6111yescharacteristic
180C22×C8C22⋊D6111yescharacteristic
181C4⋊Q8—41no—
182#124—41no—
183#123—41no—
184#123—41no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187#117—21no—
188Q32—2442no—
189SD32—2442no—
190M32—1222no—
191#107—21no—
192#106C22⋊D6112yes—
193#103—22no—
194#103—22no—
195C22⋊C8—22no—
196C22⋊C8—2442no—
197#41—41no—
198#41—41no—
199#41—2441no—
200#41—2441no—
201#41—2441no—
202#41—2441no—
203C2×Q8—22no—
204C2×Q8—41no—
205C2×Q8—81no—
206C2×Q8—81no—
207C2×Q8—1221no—
208C2×Q8#1175819911yescharacteristic
209C2×D4—21no—
210C2×D4—1221no—
211C2×D4—2441no—
212C2×D4#1175819911yescharacteristic
213C22×C4—21no—
214C22×C4—41no—
215C22×C4—2441no—
216C22×C4C2×C22⋊D6111yescharacteristic
217Q16—42no—
218Q16—81no—
219Q16—1222no—
220Q16—2442no—
221SD16—2442no—
222SD16—2442no—
223SD16—2442no—
224D8—1222no—
225D8—2442no—
226M16—41no—
227M16—2441no—
228M16—2441no—
229C2×C8—21no—
230C2×C8—41no—
231C2×C8—1222no—
232C2×C8C2×C22⋊D6112yes—
233C4⋊C4—22no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C22⋊C4—41no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C42—41no—
240C16—2442no—
241C23—21no—
242C23—2441no—
243C23#1451980311yescharacteristic
244Q8—41no—
245Q8—42no—
246Q8—42no—
247Q8—82no—
248Q8—2441no—
249Q8—2441no—
250D4—41no—
251D4—81no—
252D4—4881no—
253D4—4881no—
254D4—2441no—
255D4—2441no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—82no—
264C2×C4—4881no—
265C2×C4—4881no—
266C2×C4—2441no—
267C2×C4#1451974611yescharacteristic
268C2×C4C2×#1175819911yescharacteristic
269C8—22no—
270C8—41no—
271C8—2442no—
272C22—21no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—4882no—
277C22—4881no—
278C22#1683989611yescharacteristic
279C4—21no—
280C4—41no—
281C4—82no—
282C4—82no—
283C4—4881no—
284C4#1683993011yescharacteristic
285C4#1683994911yescharacteristic
286C2—21no—
287C2—41no—
288C2—81no—
289C2—9761no—
290C2#2144473711yescharacteristic
subgroupG/H
centerC2#21444737
commutator, cosocleC2×M16×C61C23
FrattiniC2×M16C22×D61
FittingC61×#3202C2
socleC2×C61#1595
prime radical#21446923C2
hypercenter#3202D61
hypocenterC61#12118
LCSG ▷ C2×M16×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3202

Complements

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

NH ≅ G/Ntype
1. #21447769C2semidirect
3. #21443813C2semidirect
4. #21443757C2semidirect
5. C61×#3202C2semidirect
141. #3202D61semidirect
164. C61#12118semidirect

Maximal subgroups

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

subgroupindex
#214477692
#214469232
#214438132
#214437572
C61×#32022
#1211861