Finite Group Databasebeta

← #29592178

Subgroups — #29592178

Order 15616.

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

normal subgroups only

This group has 474 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#21447764C212yesmaximal, complemented
2#21446886C211yesmaximal, characteristic, complemented
3#21443804C212yesmaximal, complemented
4#21443755C211yesmaximal, characteristic, complemented
5C61×#3192C211yesmaximal, characteristic, complemented
6#16840562C2211yescharacteristic, complemented
7#16840273—21no—
8#16840198—22no—
9#16839778—21no—
10#16839571—22no—
11#16839544C2212yescomplemented
12#16839519C2211yescharacteristic, complemented
13C2×#121×C61—21no—
14C61×#481—22no—
15C61×#469—21no—
16C61×#469C2212yes—
17C61×#447—22no—
18C61×#407C2211yescharacteristic
19C2×#11758222—21no—
20#14519810—41no—
21#14519805—41no—
22#14519799—22no—
23#14519760—41no—
24#14519756—41no—
25#14519751—41no—
26#14519746—41no—
27C4×C22⋊D61—42no—
28#14519736—21no—
29#14519721—42no—
30#14519718—42no—
31#14519717—22no—
32#14519703—42no—
33C22×D4×C61—21no—
34C22×D4×C61—41no—
35C23×C4×C61D411yescharacteristic
36C2×D8×C61—21no—
37C2×D8×C61D412yescomplemented
38C2×M16×C61D411yescharacteristic
39C2×M16×C61C2311yescharacteristic
40#121×C61—22no—
41#121×C61—22no—
42#121×C61—41no—
43#121×C61—41no—
44C2×C4⋊C4×C61—21no—
45C2×C22⋊C4×C61—41no—
46#106×C61—21no—
47#106×C61D412yescomplemented
48#102×C61—22no—
49#102×C61—22no—
50C22⋊C8×C61—22no—
51C24×C61—41no—
52C24×C61D811yescharacteristic
53C2×D4×C61—22no—
54C2×D4×C61—41no—
55C2×D4×C61—42no—
56C2×D4×C61—42no—
57C2×D4×C61—81no—
58C2×D4×C61—81no—
59C2×D4×C61—84no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—41no—
63C22×C4×C61—41no—
64C22×C4×C61—41no—
65C22×C4×C61D811yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C2×C22⋊D61—42no—
68C2×C22⋊D61—41no—
69#11758222—41no—
70#11758222—41no—
71#11758222—41no—
72#11758222—41no—
73C2×D244—21no—
74C2×C4×D61—41no—
75C2×Dic122—21no—
76D8×C61—42no—
77D8×C61—81no—
78M16×C61—41no—
79M16×C61—41no—
80C2×C8×C61—21no—
81C2×C8×C61C2×D412yes—
82C4⋊C4×C61—22no—
83C4⋊C4×C61—41no—
84C22⋊C4×C61—42no—
85C22⋊C4×C61—42no—
86C22⋊Dic61—42no—
87C22⋊Dic61—41no—
88#11758198—42no—
89C4⋊Dic61—41no—
90#11758196—42no—
91C4×Dic61—22no—
92Dic244—22no—
93D488—22no—
94#11758190—42no—
95#11758185—42no—
96C23×C61—21no—
97C23×C61—41no—
98C23×C61—42no—
99C23×C61—42no—
100C23×C61—82no—
101C23×C61—82no—
102C23×C61—82no—
103C23×C61C2×D811yescharacteristic
104C22×D61—41no—
105D4×C61—42no—
106D4×C61—42no—
107D4×C61—82no—
108D4×C61—84no—
109D4×C61—84no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—42no—
117C2×C4×C61—81no—
118C2×C4×C61—82no—
119C2×C4×C61#12011yescharacteristic
120C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
121C22⋊D61—81no—
122C22⋊D61—84no—
123C22⋊D61—81no—
124C2×Dic61—42no—
125C2×Dic61—41no—
126D244—41no—
127D244—41no—
128C4×D61—81no—
129C4×D61—81no—
130Dic122—41no—
131Dic122—41no—
132C8×C61—22no—
133C8×C61—41no—
134#12114—611nomaximal, Sylow
135C22×C61—22no—
136C22×C61—21no—
137C22×C61—41no—
138C22×C61—41no—
139C22×C61—42no—
140C22×C61—41no—
141C22×C61—82no—
142C22×C61—82no—
143C22×C61—81no—
144C22×C61—82no—
145C22×C61—164no—
146C22×C61#44711yescharacteristic
147C2×D61—82no—
148C2×D61—81no—
149C4×C61—21no—
150C4×C61—41no—
151C4×C61—82no—
152C4×C61#45311yescharacteristic
153C4×C61B(4,𝔽2)11yescharacteristic
154Dic61—82no—
155Dic61—81no—
156#3192D6111yescharacteristic, complemented
157#2190—612no—
158#2010—611no—
159#1339—612no—
160#1320—611no—
161C2×C61—21no—
162C2×C61—41no—
163C2×C61—42no—
164C2×C61—162no—
165C2×C61#159511yescharacteristic
166D61—161no—
167#575—611no—
168#525—1221no—
169C2×#121—21no—
170#509—1222no—
171#481—22no—
172#469—21no—
173#469C2×D6112yes—
174#447—22no—
175#421—1221no—
176#407C2×D6111yescharacteristic
177#360—1222no—
178#349—612no—
179H(3,ℤ/4)—611no—
180C61#1211411yescharacteristic, complemented, Sylow
181C2×#41—1221no—
182C22×D4—21no—
183C22×D4—41no—
184C23×C4C22⋊D6111yescharacteristic
185#137—2441no—
186AGL(1,ℤ/8)—2441no—
187#135—1222no—
188C2×D8—21no—
189C2×D8C22⋊D6112yescomplemented
190C2×M16C22⋊D6111yescharacteristic
191C2×M16C22×D6111yescharacteristic
192#123—2441no—
193C22⋊Q8—2441no—
194#121—22no—
195#121—22no—
196#121—41no—
197#121—41no—
198#121—2441no—
199#120—2441no—
200C4×D4—2442no—
201#117—1221no—
202C2×C4⋊C4—21no—
203C2×C22⋊C4—41no—
204SD32—2442no—
205D16—2442no—
206M32—1222no—
207#106—21no—
208#106C22⋊D6112yescomplemented
209#104—2442no—
210#102—22no—
211#102—22no—
212C22⋊C8—22no—
213C24—41no—
214C24#1175819911yescharacteristic
215#41—2441no—
216#41—2441no—
217#41—2441no—
218#41—2441no—
219C2×Q8—1221no—
220C2×D4—22no—
221C2×D4—41no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—81no—
225C2×D4—81no—
226C2×D4—84no—
227C2×D4—1221no—
228C2×D4—2442no—
229C2×D4—2441no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—2441no—
236C22×C4#1175819911yescharacteristic
237C22×C4C2×C22⋊D6111yescharacteristic
238Q16—1222no—
239SD16—2442no—
240D8—42no—
241D8—81no—
242D8—1222no—
243M16—41no—
244M16—41no—
245C2×C8—21no—
246C2×C8C2×C22⋊D6112yes—
247C4⋊C4—22no—
248C4⋊C4—41no—
249C4⋊C4—2441no—
250C4⋊C4—2442no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—2442no—
254C22⋊C4—2442no—
255C22⋊C4—2441no—
256C42—1222no—
257C16—2442no—
258C23—21no—
259C23—41no—
260C23—42no—
261C23—42no—
262C23—82no—
263C23—82no—
264C23—82no—
265C23—2441no—
266C23C2×#1175819911yescharacteristic
267Q8—2441no—
268Q8—2441no—
269D4—42no—
270D4—42no—
271D4—82no—
272D4—84no—
273D4—84no—
274D4—4881no—
275D4—4884no—
276D4—4881no—
277D4—2441no—
278D4—2441no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—4881no—
289C2×C4—4881no—
290C2×C4—2442no—
291C2×C4—2441no—
292C2×C4#1451974611yescharacteristic
293C2×C4#1451980311yescharacteristic
294C8—22no—
295C8—41no—
296C22—22no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—42no—
301C22—41no—
302C22—82no—
303C22—82no—
304C22—81no—
305C22—82no—
306C22—164no—
307C22—4882no—
308C22—4881no—
309C22#1683989611yescharacteristic
310C4—21no—
311C4—41no—
312C4—82no—
313C4—4882no—
314C4—4881no—
315C4#1683993011yescharacteristic
316C4#1683994911yescharacteristic
317C2—21no—
318C2—41no—
319C2—42no—
320C2—162no—
321C2—9761no—
322C2#2144473711yescharacteristic
subgroupG/H
centerC2#21444737
commutator, cosocleC2×M16×C61C23
FrattiniC2×M16C22×D61
FittingC61×#3192C2
socleC2×C61#1595
hypercenter#3192D61
hypocenterC61#12114
LCSG ▷ C2×M16×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3192

Complements

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

NH ≅ G/Ntype
1. #21447764C2semidirect
2. #21446886C2semidirect
3. #21443804C2semidirect
4. #21443755C2semidirect
5. C61×#3192C2semidirect
6. #16840562C22semidirect
11. #16839544C22semidirect
12. #16839519C22semidirect
37. C2×D8×C61D4semidirect
47. #106×C61D4semidirect
156. #3192D61semidirect
180. C61#12114semidirect
189. C2×D8C22⋊D61semidirect
208. #106C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214477642
#214468862
#214438042
#214437552
C61×#31922
#1211461