Finite Group Databasebeta

← #29577868

Subgroups — #29577868

Order 15616.

542 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 135 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#21454900C211yesmaximal, characteristic, complemented
2#21445815C211yesmaximal, characteristic, complemented
3#21444244C212yesmaximal
4#21443677C212yesmaximal, complemented
5C61×#2536C211yesmaximal, characteristic
6Q8×#92418—24no—
7#16839815—22no—
8#16839815C2212yes—
9#16839786C2212yes—
10C2×#14519696—21no—
11#16839531—21no—
12D4⋊#92418—24no—
13C2×C22⋊Q8×C61—21no—
14C2×C4×Q8×C61C2211yescharacteristic
15C2×C4×D4×C61C2211yescharacteristic
16C61×#398C412yes—
17C61×#394C412yes—
18C61×#388—22no—
19C61×#382C2211yescharacteristic
20C61×#380—21no—
21C2×#11758194—22no—
22C4⋊#92418—24no—
23C4⋊#92418—22no—
24C4⋊#92418—22no—
25C4⋊#92418—22no—
26#14519696—42no—
27C22⋊#92418—22no—
28C4×#92418—24no—
29C4×#92418D414yes—
30C22×Q8×C61D411yescharacteristic
31C22×D4×C61C2×C411yescharacteristic
32C23×C4×C61—21no—
33C22⋊Q8×C61—42no—
34C22⋊Q8×C61—42no—
35C4×Q8×C61—22no—
36C4×Q8×C61—22no—
37C4×D4×C61—22no—
38C4×D4×C61—22no—
39C2×C4⋊C4×C61—22no—
40C2×C4⋊C4×C61—21no—
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C4⋊C4×C61C2×C412yes—
43C2×C4⋊C4×C61D411yescharacteristic
44C2×C22⋊C4×C61—22no—
45C2×C22⋊C4×C61—22no—
46C2×C22⋊C4×C61—21no—
47C2×C42×C61D412yes—
48C2×C42×C61C2311yescharacteristic
49#95×C61—22no—
50#95×C61—21no—
51#95×C61C2×C412yes—
52C24×C61—21no—
53C2×Q8×C61—22no—
54C2×Q8×C61—22no—
55C2×Q8×C61D812yes—
56C2×Q8×C61SD1612yes—
57C2×D4×C61—22no—
58C2×D4×C61—42no—
59C22×C4×C61—22no—
60C22×C4×C61—21no—
61C22×C4×C61—42no—
62C22×C4×C61—41no—
63C22×C4×C61—41no—
64C22×C4×C61—42no—
65C22×C4×C61C22⋊C412yes—
66C22×C4×C61C22⋊C412yes—
67C22×C4×C61C22×C411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C4⋊C4×C61—24no—
71C4⋊C4×C61—22no—
72C4⋊C4×C61—22no—
73C4⋊C4×C61—44no—
74C4⋊C4×C61—41no—
75C4⋊C4×C61—41no—
76C22⋊C4×C61—44no—
77C22⋊C4×C61—44no—
78C22⋊C4×C61—41no—
79C22⋊C4×C61—41no—
80C42×C61—22no—
81C42×C61—22no—
82C42×C61C2×D412yes—
83C42×C61#4112yes—
84#11758194—42no—
85#11758194—42no—
86C2×#92418—24no—
87C2×#92418—42no—
88C23×C61—22no—
89C23×C61—41no—
90C23×C61—41no—
91C23×C61—41no—
92C23×C61—41no—
93C23×C61—41no—
94C23×C61—41no—
95C23×C61C2×C22⋊C411yescharacteristic
96Q8×C61—24no—
97Q8×C61—24no—
98D4×C61—24no—
99D4×C61—42no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61—41no—
109C2×C4×C61—42no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—82no—
113C2×C4×C61—82no—
114C2×C4×C61#10214yes—
115C2×C4×C61C4×D412yes—
116C2×C4×C61#12011yescharacteristic
117C2×C4×C61#12112yes—
118C2×C4×C61#12311yescharacteristic
119C2×C4×C61C2×D811yescharacteristic
120C2×C4×C61C2×SD1611yescharacteristic
121C2×C4×C61#13512yes—
122#92418—44no—
123#92418—82no—
124#9486—611nomaximal, Sylow
125C22×C61—21no—
126C22×C61—21no—
127C22×C61—42no—
128C22×C61—42no—
129C22×C61—42no—
130C22×C61—81no—
131C22×C61—81no—
132C22×C61—81no—
133C22×C61—81no—
134C22×C61#38611yescharacteristic
135C22×C61C2×#10211yescharacteristic
136C22×C61#41611yescharacteristic
137C4×C61—22no—
138C4×C61—24no—
139C4×C61—21no—
140C4×C61—21no—
141C4×C61—42no—
142C4×C61—82no—
143C4×C61—81no—
144C4×C61#46512yes—
145C4×C61#46612yes—
146#2982—611no—
147#2536D6111yescharacteristic
148#1801—611no—
149#1486—612no—
150#1286—612no—
151C2×C61—21no—
152C2×C61—21no—
153C2×C61—82no—
154C2×C61#189111yescharacteristic
155C2×C61#189311yescharacteristic
156C2×C61#191211yescharacteristic
157C2×C22⋊Q8—21no—
158C2×C4×Q8C2×D6111yescharacteristic
159C2×C4×D4C2×D6111yescharacteristic
160C8×Q8—1224no—
161#431—612no—
162#431—1222no—
163#423—612no—
164C2×#101—1221no—
165#398Dic6112yes—
166#394Dic6112yes—
167#388—22no—
168#382C2×D6111yescharacteristic
169#380—21no—
170#343—1221no—
171D4⋊C8—1224no—
172C61#948611yescharacteristic, complemented, Sylow
173C22×Q8C22⋊D6111yescharacteristic
174C22×D4C2×Dic6111yescharacteristic
175C23×C4—21no—
176C2×M16—1222no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—22no—
182C4×D4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C2×Dic6111yescharacteristic
186C2×C4⋊C4C2×Dic6112yes—
187C2×C4⋊C4C22⋊D6111yescharacteristic
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C42C22⋊D6112yes—
192C2×C42C22×D6111yescharacteristic
193C4⋊C8—1224no—
194C4⋊C8—1222no—
195C4⋊C8—1222no—
196C4⋊C8—1222no—
197#101—2442no—
198C22⋊C8—1222no—
199C4×C8—614no—
200C4×C8—1224no—
201#95—22no—
202#95—21no—
203#95C2×Dic6112yes—
204C24—21no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8#1175819912yes—
208C2×Q8#1175820012yes—
209C2×D4—22no—
210C2×D4—42no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4C22⋊Dic6112yes—
218C22×C4C22⋊Dic6112yes—
219C22×C4C22×Dic6111yescharacteristic
220C22×C4C2×C22⋊D6111yescharacteristic
221C22×C4C2×C22⋊D6111yescharacteristic
222M16—2442no—
223M16—2442no—
224C2×C8—1224no—
225C2×C8—2442no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—44no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C42—22no—
237C42—22no—
238C42D4×D6112yes—
239C42#1175822412yes—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23C2×C22⋊Dic6111yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—24no—
251D4—42no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4#1451969814yes—
267C2×C4D4×Dic6112yes—
268C2×C4#1451974511yescharacteristic
269C2×C4#1451974912yes—
270C2×C4#1451975811yescharacteristic
271C2×C4C2×#1175819911yescharacteristic
272C2×C4C2×#1175820011yescharacteristic
273C2×C4#1451979712yes—
274C8—4882no—
275C8—2444no—
276C22—21no—
277C22—21no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22#1683964211yescharacteristic
286C22C2×#1451969811yescharacteristic
287C22#1683975811yescharacteristic
288C4—22no—
289C4—24no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—82no—
294C4—81no—
295C4#1683998912yes—
296C4#1683999612yes—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2144623811yescharacteristic
301C2#2144624811yescharacteristic
302C2#2144635711yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2536C2
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC2×C42×C61C23
hypercenter#2536D61
hypocenterC61#9486
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. #21454900C2semidirect
2. #21445815C2semidirect
4. #21443677C2semidirect
172. C61#9486semidirect

Maximal subgroups

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

subgroupindex
#214549002
#214458152
#214442442
#214436772
C61×#25362
#948661