Finite Group Databasebeta

← #29593870

Subgroups — #29593870

Order 15616.

350 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 71 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#21448055C211yesmaximal, characteristic, complemented
2#21447971C211yesmaximal, characteristic, complemented
3#21445332C211yesmaximal, characteristic, complemented
4#21443862C211yesmaximal, characteristic, complemented
5#21443789C211yesmaximal, characteristic, complemented
6#21443777C211yesmaximal, characteristic, complemented
7C61×#3277C211yesmaximal, characteristic
8#16840167C2211yescharacteristic
9#16839967—21no—
10#16839587—21no—
11C4⋊#11758185C2211yescharacteristic
12#16839566—21no—
13#16839566—21no—
14#16839563—21no—
15#16839538C2211yescharacteristic
16#16839513C2211yescharacteristic
17#16839507—21no—
18C2×#137×C61—21no—
19C61×#548—21no—
20C61×#495C2211yescharacteristic
21C61×#468—21no—
22C61×#464—21no—
23C61×#451—21no—
24C61×#449—21no—
25C4×Q16×C61C2211yescharacteristic
26C61×#436C2211yescharacteristic
27C2×#11758200—41no—
28#14519777—21no—
29C2×#11758185—21no—
30C2×#11758185—21no—
31#14519711—21no—
32#14519711—21no—
33C4⋊#92418—21no—
34#14519700—41no—
35C2×#41×C61—21no—
36C22×Q8×C61—21no—
37#137×C61—42no—
38#137×C61—42no—
39C2×Q16×C61—21no—
40C2×Q16×C61D411yescharacteristic
41C2×Q16×C61D411yescharacteristic
42C2×SD16×C61—21no—
43C2×SD16×C61—21no—
44C2×D8×C61D411yescharacteristic
45C2×M16×C61—21no—
46#124×C61—21no—
47#124×C61—41no—
48C22⋊Q8×C61—21no—
49C22⋊Q8×C61—41no—
50#121×C61—21no—
51#121×C61—41no—
52C4×Q8×C61—21no—
53C4×D4×C61—41no—
54C4×D4×C61D411yescharacteristic
55#107×C61D411yescharacteristic
56C4⋊C8×C61D411yescharacteristic
57#103×C61—21no—
58#103×C61—21no—
59#102×C61—21no—
60C22⋊C8×C61—21no—
61C4×C8×C61C2311yescharacteristic
62#41×C61—42no—
63#41×C61—81no—
64C2×Q8×C61—21no—
65C2×Q8×C61—21no—
66C2×Q8×C61—41no—
67C2×Q8×C61—42no—
68C2×Q8×C61—81no—
69C2×D4×C61—21no—
70C2×D4×C61—41no—
71C2×D4×C61—41no—
72C2×D4×C61D811yescharacteristic
73C22×C4×C61—21no—
74C22×C4×C61—41no—
75C22×C4×C61—41no—
76C2×Dic122—41no—
77Q16×C61—21no—
78Q16×C61—41no—
79Q16×C61—42no—
80Q16×C61SD1612yescomplemented
81SD16×C61—42no—
82SD16×C61—41no—
83SD16×C61—41no—
84D8×C61—41no—
85M16×C61—41no—
86M16×C61—41no—
87C2×C8×C61—21no—
88C2×C8×C61C2×D411yescharacteristic
89C2×C8×C61C2×D411yescharacteristic
90C4⋊C4×C61—21no—
91C4⋊C4×C61—41no—
92C4⋊C4×C61—41no—
93C4⋊C4×C61D811yescharacteristic
94C22⋊C4×C61—21no—
95C22⋊C4×C61—41no—
96C22⋊C4×C61—41no—
97C22⋊C4×C61—41no—
98C22⋊C4×C61—41no—
99C42×C61—41no—
100C42×C61C2×D411yescharacteristic
101#11758200—82no—
102C4⋊Dic61—41no—
103C4⋊Dic61—41no—
104C2×#92418—41no—
105#11758185—41no—
106#11758185—41no—
107C23×C61—21no—
108C23×C61—41no—
109Q8×C61—22no—
110Q8×C61—41no—
111Q8×C61—41no—
112Q8×C61—41no—
113Q8×C61—82no—
114D4×C61—41no—
115D4×C61—41no—
116D4×C61—41no—
117D4×C61—82no—
118D4×C61—82no—
119C2×C4×C61—21no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—82no—
126C2×C4×C61—82no—
127C2×C4×C61#12011yescharacteristic
128C2×C4×C61C2×D811yescharacteristic
129C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
130C2×Dic61—81no—
131Dic122—81no—
132Dic122—81no—
133C8×C61—21no—
134C8×C61—41no—
135C8×C61C2×SD1611yescharacteristic
136C8×C61AGL(1,ℤ/8)11yescharacteristic
137#92418—81no—
138#12379—611nomaximal, Sylow
139C22×C61—41no—
140C22×C61—41no—
141C22×C61—41no—
142C22×C61—81no—
143C22×C61—81no—
144C22×C61—81no—
145C22×C61#44711yescharacteristic
146C4×C61—21no—
147C4×C61—21no—
148C4×C61—41no—
149C4×C61—42no—
150C4×C61—81no—
151C4×C61—81no—
152C4×C61#45011yescharacteristic
153C4×C61#45511yescharacteristic
154Dic61—161no—
155#3277D6111yescharacteristic
156#2243—611no—
157#2227—611no—
158#1693—611no—
159#1360—611no—
160#1334—611no—
161#1330—611no—
162C2×C61—81no—
163C2×C61—161no—
164C2×C61#163911yescharacteristic
165C2×C61#219011yescharacteristic
166C2×C61#219511yescharacteristic
167C2×#137—21no—
168#548—21no—
169#500—611no—
170#495C2×D6111yescharacteristic
171#468—21no—
172#464—21no—
173#461—1221no—
174#451—21no—
175#449—21no—
176C4×Q16C2×D6111yescharacteristic
177#436C2×D6111yescharacteristic
178#366—1221no—
179C4⋊C16—611no—
180#358—1221no—
181#358—1221no—
182#357—1221no—
183#346—611no—
184#334—611no—
185#332—1221no—
186C61#1237911yescharacteristic, complemented, Sylow
187C2×#41—21no—
188C22×Q8—21no—
189#137—42no—
190#137—42no—
191C2×Q16—21no—
192C2×Q16C22⋊D6111yescharacteristic
193C2×Q16C22⋊D6111yescharacteristic
194C2×SD16—21no—
195C2×SD16—21no—
196C2×SD16—2441no—
197C2×D8C22⋊D6111yescharacteristic
198C2×M16—21no—
199C4⋊Q8—1221no—
200#124—21no—
201#124—41no—
202C22⋊Q8—21no—
203C22⋊Q8—41no—
204#121—21no—
205#121—41no—
206C4×Q8—21no—
207C4×D4—41no—
208C4×D4C22⋊D6111yescharacteristic
209C2×C16—1221no—
210C2×C16—1221no—
211#107—1221no—
212#107—1221no—
213#107C22⋊D6111yescharacteristic
214C4⋊C8—1221no—
215C4⋊C8C22⋊D6111yescharacteristic
216#103—21no—
217#103—21no—
218#103—2441no—
219#102—21no—
220C22⋊C8—21no—
221C4×C8C22×D6111yescharacteristic
222#41—42no—
223#41—81no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—41no—
227C2×Q8—42no—
228C2×Q8—81no—
229C2×Q8—2441no—
230C2×D4—21no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4#1175819911yescharacteristic
234C22×C4—21no—
235C22×C4—41no—
236C22×C4—41no—
237Q16—21no—
238Q16—41no—
239Q16—42no—
240Q16#1175820012yescomplemented
241SD16—42no—
242SD16—41no—
243SD16—41no—
244SD16—4882no—
245D8—41no—
246M16—41no—
247M16—41no—
248C2×C8—21no—
249C2×C8—2441no—
250C2×C8C2×C22⋊D6111yescharacteristic
251C2×C8C2×C22⋊D6111yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—2441no—
256C4⋊C4—2441no—
257C4⋊C4#1175819911yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C42—41no—
264C42C2×C22⋊D6111yescharacteristic
265C16—2441no—
266C16—2441no—
267C23—21no—
268C23—41no—
269Q8—22no—
270Q8—41no—
271Q8—41no—
272Q8—41no—
273Q8—82no—
274Q8—4881no—
275Q8—4881no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—82no—
280D4—82no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—4881no—
290C2×C4#1451974611yescharacteristic
291C2×C4C2×#1175819911yescharacteristic
292C2×C4#1451980311yescharacteristic
293C8—21no—
294C8—41no—
295C8—4881no—
296C8C2×#1175820011yescharacteristic
297C8#1451980311yescharacteristic
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22#1683989611yescharacteristic
305C4—21no—
306C4—21no—
307C4—41no—
308C4—42no—
309C4—81no—
310C4—81no—
311C4—9761no—
312C4#1683991311yescharacteristic
313C4#1683994011yescharacteristic
314C2—81no—
315C2—161no—
316C2#2144499811yescharacteristic
317C2#2144776511yescharacteristic
318C2#2144779711yescharacteristic
subgroupG/H
centerC22#16839896
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
Fitting, prime radicalC61×#3277C2
socleC22×C61#447
hypercenter#3277D61
hypocenterC61#12379
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #436 ◁ #3277

Complements

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

NH ≅ G/Ntype
1. #21448055C2semidirect
2. #21447971C2semidirect
3. #21445332C2semidirect
4. #21443862C2semidirect
5. #21443789C2semidirect
6. #21443777C2semidirect
80. Q16×C61SD16semidirect
186. C61#12379semidirect
240. Q16#11758200semidirect

Maximal subgroups

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

subgroupindex
#214480552
#214479712
#214453322
#214438622
#214437892
#214437772
C61×#32772
#1237961