Finite Group Databasebeta

← #29599692

Subgroups — #29599692

Order 15616.

484 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 81 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#21446099C212yesmaximal
2#21446028C211yesmaximal, characteristic
3#21443944C211yesmaximal, characteristic
4#21443937C212yesmaximal
5C61×#2804C211yesmaximal, characteristic
6#16840237C2211yescharacteristic
7#16839802—21no—
8C2×#14519694C2211yescharacteristic
9C2×#14519692C2212yes—
10#16839635—22no—
11#16839528—22no—
12#16839522—21no—
13C61×#400—21no—
14C61×#393—22no—
15C61×#391—21no—
16C61×#390C2211yescharacteristic
17C61×#386C2212yes—
18C61×#385—22no—
19C2×#11758194—21no—
20D4×Dic61—41no—
21D4×Dic61—41no—
22#14519736—21no—
23C2×#11758196—42no—
24C2×C22⋊Dic61—22no—
25C2×C22⋊Dic61—22no—
26#14519710—41no—
27#14519708—41no—
28#14519694—21no—
29#14519694Q812yescomplemented
30#14519692—22no—
31#14519692Q814yescomplemented
32#14519684—42no—
33C22×Q8×C61—21no—
34C22×D4×C61C2311yescharacteristic
35C23×C4×C61D412yes—
36C2×C4⋊C4×C61—21no—
37C2×C4⋊C4×C61—21no—
38C2×C4⋊C4×C61D411yescharacteristic
39C2×C22⋊C4×C61—22no—
40C2×C22⋊C4×C61—22no—
41C2×C22⋊C4×C61D412yes—
42C2×C42×C61D411yescharacteristic
43#95×C61—22no—
44#95×C61—22no—
45#95×C61—21no—
46#95×C61—22no—
47C24×C61C2×D412yes—
48C2×Q8×C61—42no—
49C2×Q8×C61—81no—
50C2×Q8×C61—81no—
51C2×D4×C61—41no—
52C2×D4×C61—41no—
53C2×D4×C61C2×Q811yescharacteristic
54C2×D4×C61C2×Q812yes—
55C2×D4×C61#4111yescharacteristic
56C22×C4×C61—21no—
57C22×C4×C61—22no—
58C22×C4×C61—21no—
59C22×C4×C61—22no—
60C22×C4×C61—42no—
61C22×C4×C61—82no—
62C22×C4×C61C2×D411yescharacteristic
63C22×Dic61—22no—
64C22×Dic61—22no—
65C22×Dic61—42no—
66C4⋊C4×C61—41no—
67C4⋊C4×C61—81no—
68C4⋊C4×C61—81no—
69C22⋊C4×C61—44no—
70C22⋊C4×C61—42no—
71C22⋊C4×C61—82no—
72C42×C61—41no—
73C22⋊Dic61—24no—
74C22⋊Dic61—42no—
75C22⋊Dic61—42no—
76C22⋊Dic61—41no—
77C4⋊Dic61—21no—
78C4⋊Dic61—21no—
79#11758196—84no—
80C4×Dic61—41no—
81#11758194—22no—
82#11758194—41no—
83C2×#92418—41no—
84C23×C61—22no—
85C23×C61—22no—
86C23×C61—42no—
87C23×C61—42no—
88C23×C61#12011yescharacteristic
89C23×C61C22⋊Q812yes—
90C23×C61C4⋊Q812yes—
91Q8×C61—84no—
92D4×C61—41no—
93D4×C61—42no—
94D4×C61—41no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—22no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—42no—
101C2×C4×C61—42no—
102C2×C4×C61—42no—
103C2×C4×C61—42no—
104C2×C4×C61—42no—
105C2×C4×C61—44no—
106C2×C4×C61—41no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—82no—
110C2×C4×C61C22⋊Q811yescharacteristic
111C2×C4×C61C4⋊Q811yescharacteristic
112C2×Dic61—44no—
113C2×Dic61—41no—
114C2×Dic61—41no—
115C2×Dic61—82no—
116C2×Dic61—82no—
117C2×Dic61—82no—
118C2×Dic61—82no—
119#92418—42no—
120#13463—611nomaximal, Sylow
121C22×C61—21no—
122C22×C61—21no—
123C22×C61—22no—
124C22×C61—22no—
125C22×C61—42no—
126C22×C61—42no—
127C22×C61—42no—
128C22×C61—42no—
129C22×C61—82no—
130C22×C61#39511yescharacteristic
131C22×C61B(4,𝔽2)11yescharacteristic
132C22×C61#45811yescharacteristic
133C4×C61—21no—
134C4×C61—21no—
135C4×C61—42no—
136C4×C61—82no—
137C4×C61—81no—
138C4×C61—82no—
139Dic61—84no—
140Dic61—81no—
141Dic61—81no—
142#2804D6111yescharacteristic
143#1867—612no—
144#1856—611no—
145#1393—611no—
146#1390—612no—
147C2×C61—21no—
148C2×C61—21no—
149C2×C61—42no—
150C2×C61—42no—
151C2×C61#206211yescharacteristic
152C2×C61#219811yescharacteristic
153C2×C61#219911yescharacteristic
154#518—611no—
155#428—1221no—
156C2×#100—611no—
157C2×#99—612no—
158#400—21no—
159#393—22no—
160#391—21no—
161#390C2×D6111yescharacteristic
162#386C2×D6112yes—
163#385—22no—
164#385—1222no—
165#342—1222no—
166#339—1221no—
167C61#1346311yescharacteristic, complemented, Sylow
168C22×Q8—21no—
169C22×D4C22×D6111yescharacteristic
170C23×C4C22⋊D6112yes—
171C2×M16—1221no—
172C4×D4—2441no—
173C4×D4—2441no—
174#117—1221no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—2442no—
178C2×C4⋊C4C22⋊D6111yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—1222no—
182C2×C22⋊C4—1222no—
183C2×C22⋊C4C22⋊D6112yes—
184C2×C42C22⋊D6111yescharacteristic
185#107—2441no—
186#106—2441no—
187#100—612no—
188#100—1221no—
189#99—614no—
190#99—1222no—
191#95—22no—
192#95—22no—
193#95—21no—
194#95—22no—
195#95—2442no—
196C24C2×C22⋊D6112yes—
197C2×Q8—42no—
198C2×Q8—81no—
199C2×Q8—81no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4Q8×D6111yescharacteristic
203C2×D4Q8×D6112yes—
204C2×D4#1175822611yescharacteristic
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—82no—
211C22×C4—1222no—
212C22×C4—1222no—
213C22×C4—2442no—
214C22×C4C2×C22⋊D6111yescharacteristic
215M16—1222no—
216M16—2441no—
217C2×C8—2441no—
218C4⋊C4—41no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—1221no—
222C4⋊C4—1221no—
223C4⋊C4—4884no—
224C22⋊C4—44no—
225C22⋊C4—42no—
226C22⋊C4—82no—
227C22⋊C4—1224no—
228C22⋊C4—2442no—
229C22⋊C4—2442no—
230C22⋊C4—2441no—
231C42—41no—
232C42—2441no—
233C23—22no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23#1451974611yescharacteristic
238C23#1451975312yes—
239C23#1451977512yes—
240Q8—84no—
241D4—41no—
242D4—42no—
243D4—41no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—82no—
259C2×C4—4882no—
260C2×C4—4882no—
261C2×C4—4882no—
262C2×C4—4882no—
263C2×C4—2444no—
264C2×C4—2441no—
265C2×C4—2441no—
266C2×C4#1451975311yescharacteristic
267C2×C4#1451977511yescharacteristic
268C8—2442no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—82no—
278C22#1683968111yescharacteristic
279C22#1683994911yescharacteristic
280C22#1683995211yescharacteristic
281C4—21no—
282C4—21no—
283C4—42no—
284C4—82no—
285C4—81no—
286C4—82no—
287C4—4884no—
288C4—4881no—
289C4—4881no—
290C2—21no—
291C2—21no—
292C2—42no—
293C2—42no—
294C2#2144717111yescharacteristic
295C2#2144781511yescharacteristic
296C2#2144782011yescharacteristic
subgroupG/H
centerC22#16839949
commutator, prime radical, cosocleC22×D4×C61C23
FrattiniC22×D4C22×D61
FittingC61×#2804C2
socleC22×C61B(4,𝔽2)
hypercenter#2804D61
hypocenterC61#13463
LCSG ▷ C22×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2804

Complements

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

NH ≅ G/Ntype
29. #14519694Q8semidirect
31. #14519692Q8semidirect
167. C61#13463semidirect

Maximal subgroups

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

subgroupindex
#214460992
#214460282
#214439442
#214439372
C61×#28042
#1346361