Finite Group Databasebeta

← #29596822

Subgroups — #29596822

Order 15616.

516 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#16839557C211yesmaximal, characteristic
2#21446208C211yesmaximal, characteristic
3#21446099C212yesmaximal
4#21443937C212yesmaximal
5C61×#2379C211yesmaximal, characteristic
6#16840237C2211yescharacteristic
7#16839771—21no—
8C2×#14519694C2211yescharacteristic
9C2×#14519692C2212yes—
10#16839635—22no—
11#16839557C414yescomplemented
12#16839528—22no—
13#16839522—21no—
14C2×#124×C61C2211yescharacteristic
15C61×#391—21no—
16C61×#388—22no—
17C61×#386C2212yes—
18C61×#385—22no—
19C61×#382—21no—
20C2×#11758194—21no—
21D4×Dic61—41no—
22D4×Dic61—41no—
23#14519736—21no—
24C2×#11758196—42no—
25C2×C22⋊Dic61—22no—
26C2×C22⋊Dic61—22no—
27#14519701—41no—
28#14519701—41no—
29#14519694—22no—
30#14519692—22no—
31#14519692C2×C414yes—
32#14519684—42no—
33C22×Q8×C61D411yescharacteristic
34C22×D4×C61C2311yescharacteristic
35C23×C4×C61D412yes—
36#124×C61—22no—
37#124×C61C2×C412yes—
38#124×C61Q812yes—
39C2×C4⋊C4×C61—21no—
40C2×C4⋊C4×C61—21no—
41C2×C4⋊C4×C61—21no—
42C2×C22⋊C4×C61—22no—
43C2×C22⋊C4×C61—22no—
44C2×C22⋊C4×C61D412yes—
45C2×C42×C61—21no—
46C2×C42×C61D411yescharacteristic
47#95×C61—22no—
48#95×C61—22no—
49#95×C61—22no—
50C24×C61C2×D412yes—
51C2×Q8×C61—21no—
52C2×Q8×C61—41no—
53C2×Q8×C61—41no—
54C2×Q8×C61C4⋊C412yescomplemented
55C2×D4×C61—41no—
56C2×D4×C61—41no—
57C2×D4×C61C22×C411yescharacteristic
58C2×D4×C61C2×Q811yescharacteristic
59C2×D4×C61#4112yes—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—22no—
63C22×C4×C61—22no—
64C22×C4×C61—42no—
65C22×C4×C61—82no—
66C22×C4×C61C2×D411yescharacteristic
67C22×Dic61—22no—
68C22×Dic61—22no—
69C22×Dic61—42no—
70C4⋊C4×C61—81no—
71C4⋊C4×C61—81no—
72C4⋊C4×C61—81no—
73C22⋊C4×C61—24no—
74C22⋊C4×C61—24no—
75C22⋊C4×C61—42no—
76C22⋊C4×C61—82no—
77C42×C61—21no—
78C42×C61—81no—
79C42×C61C4⋊C412yescomplemented
80C22⋊Dic61—24no—
81C22⋊Dic61—42no—
82C22⋊Dic61—42no—
83C22⋊Dic61—41no—
84C4⋊Dic61—21no—
85C4⋊Dic61—21no—
86#11758196—84no—
87C4×Dic61—41no—
88#11758194—41no—
89#11758194—41no—
90C2×#92418—41no—
91C23×C61—22no—
92C23×C61—22no—
93C23×C61—42no—
94C23×C61—42no—
95C23×C61C4×D412yes—
96C23×C61#12011yescharacteristic
97C23×C61C22⋊Q812yes—
98Q8×C61—42no—
99Q8×C61—81no—
100D4×C61—41no—
101D4×C61—41no—
102D4×C61—42no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—22no—
106C2×C4×C61—22no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—42no—
112C2×C4×C61—42no—
113C2×C4×C61—42no—
114C2×C4×C61—42no—
115C2×C4×C61—44no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—82no—
119C2×C4×C61C2×C4⋊C411yescharacteristic
120C2×C4×C61#12311yescharacteristic
121C2×Dic61—44no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—82no—
125C2×Dic61—82no—
126C2×Dic61—82no—
127C2×Dic61—82no—
128#92418—81no—
129#12967—611nomaximal, Sylow
130C22×C61—21no—
131C22×C61—21no—
132C22×C61—22no—
133C22×C61—22no—
134C22×C61—42no—
135C22×C61—42no—
136C22×C61—42no—
137C22×C61—42no—
138C22×C61—82no—
139C22×C61#38511yescharacteristic
140C22×C61B(4,𝔽2)11yescharacteristic
141C22×C61#45811yescharacteristic
142C4×C61—21no—
143C4×C61—21no—
144C4×C61—42no—
145C4×C61—42no—
146C4×C61—82no—
147C4×C61—82no—
148Dic61—84no—
149Dic61—81no—
150Dic61—81no—
151#2379D6111yescharacteristic
152C2×#354—611no—
153#1885—611no—
154#1867—612no—
155#1390—612no—
156C2×C61—21no—
157C2×C61—21no—
158C2×C61—42no—
159C2×C61—42no—
160C2×C61#186711yescharacteristic
161C2×C61#219911yescharacteristic
162C2×C61#220111yescharacteristic
163C2×#124C2×D6111yescharacteristic
164#518—611no—
165#419—1221no—
166C2×#100—611no—
167C2×#99—612no—
168#391—21no—
169#388—22no—
170#386C2×D6112yes—
171#385—22no—
172#385—1222no—
173#382—21no—
174#354—614no—
175#342—1222no—
176#339—1221no—
177C61#1296711yescharacteristic, complemented, Sylow
178C22×Q8C22⋊D6111yescharacteristic
179C22×D4C22×D6111yescharacteristic
180C23×C4C22⋊D6112yes—
181C2×M16—1221no—
182#124—22no—
183#124Dic12212yes—
184#124C4×D6112yes—
185C4×D4—2441no—
186C4×D4—2441no—
187#117—1221no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—2442no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—1222no—
195C2×C22⋊C4—1222no—
196C2×C22⋊C4C22⋊D6112yes—
197C2×C42—21no—
198C2×C42C22⋊D6111yescharacteristic
199#103—2441no—
200#103—2441no—
201#100—1222no—
202#99—614no—
203#99—1222no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—2442no—
208C24C2×C22⋊D6112yes—
209C2×Q8—21no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8#1175819612yescomplemented
213C2×D4—41no—
214C2×D4—41no—
215C2×D4C2×Dic12211yescharacteristic
216C2×D4C2×C4×D6111yescharacteristic
217C2×D4#1175822212yes—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—82no—
224C22×C4—1222no—
225C22×C4—1222no—
226C22×C4—2442no—
227C22×C4C2×C22⋊D6111yescharacteristic
228M16—2441no—
229M16—2441no—
230C2×C8—2441no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C4⋊C4—1221no—
235C4⋊C4—1221no—
236C4⋊C4—4884no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—42no—
240C22⋊C4—82no—
241C22⋊C4—1224no—
242C22⋊C4—2442no—
243C22⋊C4—2442no—
244C22⋊C4—2441no—
245C42—21no—
246C42—81no—
247C42—2441no—
248C42#1175819612yescomplemented
249C23—22no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23C4×C22⋊D6112yes—
254C23#1451974611yescharacteristic
255C23#1451975612yes—
256Q8—42no—
257Q8—81no—
258D4—41no—
259D4—41no—
260D4—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—82no—
277C2×C4—4882no—
278C2×C4—4882no—
279C2×C4—4882no—
280C2×C4—4882no—
281C2×C4—2444no—
282C2×C4—2441no—
283C2×C4—2441no—
284C2×C4C2×#1175819611yescharacteristic
285C2×C4#1451976011yescharacteristic
286C8—4881no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—82no—
296C22#1683963511yescharacteristic
297C22#1683994911yescharacteristic
298C22#1683995211yescharacteristic
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—82no—
304C4—82no—
305C4—4884no—
306C4—4881no—
307C4—4881no—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#2144610011yescharacteristic
313C2#2144782011yescharacteristic
314C2#2144783211yescharacteristic
subgroupG/H
centerC22#16839949
commutatorC2×D4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#2379C2
socleC22×C61B(4,𝔽2)
prime radical, cosocleC22×D4×C61C23
hypercenter#2379D61
hypocenterC61#12967
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2379

Complements

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

NH ≅ G/Ntype
11. #16839557C4semidirect
54. C2×Q8×C61C4⋊C4semidirect
79. C42×C61C4⋊C4semidirect
177. C61#12967semidirect
212. C2×Q8#11758196semidirect
248. C42#11758196semidirect

Maximal subgroups

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

subgroupindex
C2×#168395572
#214462082
#214460992
#214439372
C61×#23792
#1296761