Finite Group Databasebeta

← #29571575

Subgroups — #29571575

Order 15616.

686 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 143 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#21459629C211yesmaximal, characteristic
2#21445815C211yesmaximal, characteristic
3#21444507C212yesmaximal
4#21444275C212yesmaximal
5C61×#1984C211yesmaximal, characteristic
6Q8×Dic122—22no—
7#16840491—22no—
8#16840488—24no—
9#16840393—21no—
10#16840393C414yes—
11#16840319—22no—
12#16840319C2212yes—
13#16840240—21no—
14#16839786C2212yes—
15C2×#14519696—21no—
16#16839531—21no—
17#16839510—22no—
18#16839509—22no—
19#16839494—24no—
20C2×C4×Q8×C61C2211yescharacteristic
21C2×C4⋊C8×C61C2211yescharacteristic
22C2×C4×C8×C61C2211yescharacteristic
23C61×#336—22no—
24C2×#11758194—22no—
25#14519777—22no—
26#14519777—22no—
27#14519777D412yes—
28#14519777D412yes—
29#14519776—24no—
30#14519776—28no—
31#14519769—24no—
32#14519769—22no—
33#14519768—24no—
34#14519768—28no—
35#14519756—24no—
36#14519756—22no—
37#14519756C2×C414yes—
38Q8×Dic61—24no—
39Q8×Dic61—24no—
40C4×Dic122—24no—
41C4×Dic122—24no—
42C4×Dic122—22no—
43C4×Dic122D414yes—
44#14519736—24no—
45#14519736—22no—
46C4⋊#92418—22no—
47C4⋊#92418—22no—
48#14519696—42no—
49C22×Q8×C61D411yescharacteristic
50C22×C8×C61—21no—
51C22×C8×C61—21no—
52C4×Q8×C61—22no—
53C4×Q8×C61—22no—
54C2×C4⋊C4×C61C2×C412yes—
55C2×C4⋊C4×C61D411yescharacteristic
56C2×C42×C61D412yes—
57C2×C42×C61C2311yescharacteristic
58C4⋊C8×C61—22no—
59C4×C8×C61—22no—
60C2×Q8×C61—22no—
61C2×Q8×C61—22no—
62C2×Q8×C61SD1612yes—
63C2×Q8×C61Q1612yes—
64C22×C4×C61C22⋊C412yes—
65C22×C4×C61C22⋊C412yes—
66C22×C4×C61C22×C411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C2×Dic122—22no—
70C2×Dic122—24no—
71C2×Dic122C22⋊C414yes—
72C2×C8×C61—22no—
73C2×C8×C61—41no—
74C2×C8×C61—41no—
75C2×C8×C61—42no—
76C2×C8×C61—41no—
77C4⋊C4×C61—24no—
78C4⋊C4×C61—22no—
79C42×C61—22no—
80C42×C61—22no—
81C42×C61C2×D412yes—
82C42×C61C2×D412yes—
83C22⋊Dic61—24no—
84C4⋊Dic61—24no—
85C4⋊Dic61—24no—
86C4⋊Dic61—24no—
87C4⋊Dic61—22no—
88C4⋊Dic61—22no—
89C4⋊Dic61—22no—
90C4⋊Dic61C22⋊C414yes—
91#11758196—28no—
92#11758196—24no—
93#11758196—24no—
94C4×Dic61—28no—
95C4×Dic61—24no—
96#11758194—42no—
97#11758194—42no—
98C2×#92418—42no—
99C23×C61C2×C22⋊C411yescharacteristic
100Q8×C61—24no—
101Q8×C61—24no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—22no—
107C2×C4×C61—41no—
108C2×C4×C61—42no—
109C2×C4×C61#10314yes—
110C2×C4×C61C2×C22⋊C412yes—
111C2×C4×C61#12011yescharacteristic
112C2×C4×C61#12012yes—
113C2×C4×C61#12011yescharacteristic
114C2×C4×C61C2×SD1611yescharacteristic
115C2×C4×C61C2×Q1611yescharacteristic
116C2×C4×C61#13712yes—
117C2×Dic61—24no—
118C2×Dic61—24no—
119Dic122—42no—
120Dic122—42no—
121Dic122—44no—
122Dic122—84no—
123C8×C61—42no—
124C8×C61—81no—
125#92418—82no—
126#8356—611nomaximal, Sylow
127C22×C61—21no—
128C22×C61—21no—
129C22×C61#37911yescharacteristic
130C22×C61C2×#10311yescharacteristic
131C22×C61#41911yescharacteristic
132C4×C61—21no—
133C4×C61—24no—
134C4×C61—22no—
135C4×C61—42no—
136C4×C61#44812yes—
137C4×C61#45112yes—
138C4×C61#45411yescharacteristic
139C4×C61#45611yescharacteristic
140Dic61—44no—
141Dic61—44no—
142#3502—611no—
143#1984D6111yescharacteristic
144#1801—611no—
145#1550—612no—
146#1493—612no—
147C2×C61—21no—
148C2×C61—21no—
149C2×C61#178711yescharacteristic
150C2×C61#179111yescharacteristic
151C2×C61#179311yescharacteristic
152Q82—1222no—
153#557—1224no—
154#557—1222no—
155#544—614no—
156#544—1221no—
157#533—612no—
158#533—1222no—
159#519—1221no—
160C2×C4×Q8C2×D6111yescharacteristic
161#423—612no—
162C2×C4⋊C8C2×D6111yescharacteristic
163C2×#101—1221no—
164C2×C4×C8C2×D6111yescharacteristic
165#343—1221no—
166#336—22no—
167#333—1222no—
168#332—1222no—
169Q8⋊C8—1224no—
170C61#835611yescharacteristic, complemented, Sylow
171C22×Q8D24411yescharacteristic
172C2×M16—1222no—
173C22×C8—21no—
174C22×C8—21no—
175C4⋊Q8—612no—
176C4⋊Q8—612no—
177C4⋊Q8—1222no—
178C4⋊Q8—1222no—
179C4⋊Q8—1228no—
180C4⋊Q8—1224no—
181#125—1224no—
182#125—1228no—
183#125—1224no—
184#125—1222no—
185C22⋊Q8—614no—
186C22⋊Q8—1224no—
187C22⋊Q8—1222no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—614no—
191C4×Q8—1224no—
192C4×Q8—1224no—
193C4×Q8—1224no—
194C4×Q8—1224no—
195C4×Q8—1222no—
196#117—1224no—
197#117—1222no—
198C2×C4⋊C4C4×D6112yes—
199C2×C4⋊C4D24411yescharacteristic
200C2×C42C22⋊D6112yes—
201C2×C42C22×D6111yescharacteristic
202C4⋊C8—22no—
203C4⋊C8—1222no—
204C4⋊C8—1222no—
205#101—2442no—
206C4×C8—22no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—614no—
210C2×Q8—1224no—
211C2×Q8—1222no—
212C2×Q8#1175819012yes—
213C2×Q8Dic24412yes—
214C22×C4#1175819812yes—
215C22×C4#1175819812yes—
216C22×C4C2×C4×D6111yescharacteristic
217C22×C4C2×D24411yescharacteristic
218C22×C4C2×C22⋊D6111yescharacteristic
219M16—2442no—
220M16—2442no—
221C2×C8—22no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—42no—
225C2×C8—41no—
226C2×C8—2442no—
227C4⋊C4—24no—
228C4⋊C4—22no—
229C4⋊C4—614no—
230C4⋊C4—1228no—
231C4⋊C4—1224no—
232C4⋊C4—1224no—
233C4⋊C4—1224no—
234C4⋊C4—1224no—
235C4⋊C4—1224no—
236C4⋊C4—1222no—
237C4⋊C4—1222no—
238C4⋊C4—1222no—
239C22⋊C4—1224no—
240C42—22no—
241C42—22no—
242C42—1228no—
243C42—1224no—
244C42D4×D6112yes—
245C42D4×D6112yes—
246C23C2×#1175819811yescharacteristic
247Q8—24no—
248Q8—24no—
249Q8—4884no—
250Q8—2444no—
251Q8—2442no—
252Q8—2442no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—1224no—
261C2×C4—1224no—
262C2×C4#1451970214yes—
263C2×C4C22⋊C4×D6112yes—
264C2×C4#1451974511yescharacteristic
265C2×C4#1451974511yescharacteristic
266C2×C4#1451974712yes—
267C2×C4C2×#1175819011yescharacteristic
268C2×C4C2×Dic24411yescharacteristic
269C2×C4#1451981012yes—
270C8—42no—
271C8—81no—
272C8—4882no—
273C22—21no—
274C22—21no—
275C22#1683961711yescharacteristic
276C22C2×#1451970211yescharacteristic
277C22#1683977211yescharacteristic
278C4—24no—
279C4—22no—
280C4—21no—
281C4—42no—
282C4—2444no—
283C4—2444no—
284C4#1683990312yes—
285C4#1683992012yes—
286C4#1683993411yescharacteristic
287C4#1683994411yescharacteristic
288C2—21no—
289C2—21no—
290C2#2144575511yescharacteristic
291C2#2144576911yescharacteristic
292C2#2144577711yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1984C2
socleC22×C61#379
prime radicalC23×C61C2×C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1984D61
hypocenterC61#8356
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1984

Complements

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

NH ≅ G/Ntype
170. C61#8356semidirect

Maximal subgroups

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

subgroupindex
#214596292
#214458152
#214445072
#214442752
C61×#19842
#835661