Finite Group Databasebeta

← #30600138

Subgroups — #30600138

Order 15616.

570 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 106 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#21462352C211yesmaximal, characteristic
2#21461997C212yesmaximal
3D61⋊#333C211yesmaximal, characteristic
4#21460654C212yesmaximal
5#494×D61C211yesmaximal, characteristic
6D61⋊(C4⋊C8)C2212yes—
7(C2×#6923)⋊Q8—22no—
8#16840713—22no—
9C4⋊C4×#6923—22no—
10C4×C4⋊#6923C2212yes—
11#16840635—22no—
12#16840635—22no—
13#16840622—22no—
14C2×Q16×D61—22no—
15C4⋊Q8×D61C2211yescharacteristic
16C4⋊Q8×D61C2211yescharacteristic
17#16840142C411yescharacteristic, complemented
18#16840141C411yescharacteristic, complemented
19#16840141C411yescharacteristic, complemented
20#16840140—22no—
21C4×C8×D61C2211yescharacteristic
22C61×#494C411yescharacteristic, complemented
23C2×D61⋊C8—22no—
24C2×C4⋊#6923—22no—
25C2×C4⋊#6923D412yes—
26C2×C4⋊#6923D412yes—
27C2×C4×#6923—22no—
28C2×C4×#6923—22no—
29#14519844—22no—
30C4⋊#92420—22no—
31(C2×#6923)⋊C4—24no—
32(C2×#6923)⋊C4—22no—
33C2×Q8×D61—21no—
34C2×Q8×D61D412yes—
35C2×Dic244—22no—
36C2×#11758202—22no—
37C2×#11758202—22no—
38Q16×D61—44no—
39Q16×D61—44no—
40C2×C8×D61—21no—
41#14519777C2×C411yescharacteristic
42#14519777C2×C411yescharacteristic
43#14519776—22no—
44#14519776—22no—
45#14519775—21no—
46#14519775D412yes—
47C4⋊C4×D61—22no—
48C4⋊C4×D61—22no—
49C42×D61C2311yescharacteristic
50C4×#92418C2×C411yescharacteristic
51C8×Dic61—21no—
52C2×Q16×C61—22no—
53C4⋊Q8×C61C2×C411yescharacteristic
54C4⋊Q8×C61C2×C411yescharacteristic
55C4×C8×C61C2×C411yescharacteristic
56C22×#6923—22no—
57C22×#6923—22no—
58C2×Q8×C61—21no—
59C2×Q8×C61C22⋊C412yes—
60Q8×D61—42no—
61Q8×D61—42no—
62Q8×D61—42no—
63Q8×D61—42no—
64C2×C4×D61—22no—
65C2×C4×D61—41no—
66C2×C4×D61C2×D411yescharacteristic
67C2×C4×D61C2×D412yes—
68C2×Dic122—21no—
69C2×Dic122—22no—
70C2×Dic122—41no—
71C2×Dic122C22⋊C412yes—
72C2×#92420—42no—
73C4⋊#6923—22no—
74C4⋊#6923—22no—
75C4⋊#6923—22no—
76C4⋊#6923—82no—
77C4⋊#6923Q1614yescomplemented
78C4×#6923—42no—
79C4×#6923—42no—
80C4×#6923—82no—
81D61⋊C8—42no—
82D61⋊C8—42no—
83Q16×C61—44no—
84C2×C8×C61—21no—
85C4⋊C4×C61—22no—
86C4⋊C4×C61—22no—
87C42×C61C22×C411yescharacteristic
88#11758202—44no—
89#11758202—44no—
90C4⋊Dic61—22no—
91C4⋊Dic61—22no—
92#11758196—24no—
93#11758196—42no—
94C4×Dic61C2×D411yescharacteristic
95C4×Dic61#4112yes—
96C2×#92418—21no—
97Dic244—44no—
98C8×D61—22no—
99C8×D61—22no—
100C22×D61#12011yescharacteristic
101C2×#6923—24no—
102C2×#6923—42no—
103C2×#6923—42no—
104C2×#6923—42no—
105C2×#6923—42no—
106C2×#6923—42no—
107Q8×C61—42no—
108Q8×C61—42no—
109C2×C4×C61—22no—
110C2×C4×C61—41no—
111C2×C4×C61C2×C22⋊C411yescharacteristic
112C2×C4×C61C4×D412yes—
113C2×Dic61—22no—
114C2×Dic61—41no—
115C2×Dic61#12112yes—
116C2×Dic61#12311yescharacteristic
117C4×D61—21no—
118C4×D61—21no—
119C4×D61—42no—
120C4×D61—42no—
121C4×D61—81no—
122C4×D61—81no—
123C4×D61C2×Q1612yes—
124C4×D61#13512yes—
125C4×D61#13712yes—
126C4×D61#13712yes—
127Dic122—42no—
128Dic122—42no—
129Dic122—82no—
130Dic122—82no—
131#92420—82no—
132C8×C61—22no—
133#92418—22no—
134#9066—611nomaximal, Sylow
135C22×C61#38611yescharacteristic
136C2×D61#45112yes—
137C2×D61#45212yes—
138C2×D61#45311yescharacteristic
139C2×D61#45311yescharacteristic
140#6923—44no—
141#6923—82no—
142C4×C61—21no—
143C4×C61—42no—
144C4×C61—81no—
145C4×C61C4×Q1612yes—
146C4×C61#44112yes—
147Dic61—21no—
148Dic61—42no—
149Dic61—81no—
150Dic61#46212yes—
151Dic61#46412yes—
152C2×#494—611no—
153#2369—611no—
154#1987—612no—
155C2×#333—611no—
156#1279—612no—
157C2×C61#187812yes—
158C2×C61#188711yescharacteristic
159D61#163212yes—
160D61#163311yescharacteristic
161D61#163411yescharacteristic
162C22×Q16—1222no—
163C2×C4⋊Q8—611no—
164C2×C4⋊Q8—611no—
165#494—611no—
166#494—611no—
167#494—611no—
168#494—1222no—
169#494#692311yescharacteristic, complemented
170C2×C4⋊C8—612no—
171C2×C4×C8—611no—
172#391—1222no—
173#389—1222no—
174C4×C4⋊C4—612no—
175C4×C4⋊C4—1222no—
176#340—1222no—
177#340—1222no—
178#333—1222no—
179C61#906611yescharacteristic, complemented, Sylow
180C22×Q8—612no—
181C22×Q8—1221no—
182C2×Q16—22no—
183C2×Q16—1222no—
184C2×Q16—1222no—
185C2×Q16—1222no—
186C2×Q16—2444no—
187C2×Q16—2444no—
188C22×C8—1221no—
189C22×C8—1222no—
190C4⋊Q8—612no—
191C4⋊Q8—611no—
192C4⋊Q8—611no—
193C4⋊Q8—1222no—
194C4⋊Q8—1222no—
195C4⋊Q8—1221no—
196C4⋊Q8C2×#692311yescharacteristic
197C4⋊Q8C2×#692311yescharacteristic
198C2×C4⋊C4—612no—
199C2×C4⋊C4—612no—
200C2×C4⋊C4—1222no—
201C2×C4⋊C4—1222no—
202C2×C4⋊C4—1222no—
203C2×C42—611no—
204C2×C42—1222no—
205C2×C42—1222no—
206C4⋊C8—1222no—
207C4⋊C8—1222no—
208C4×C8—611no—
209C4×C8—1221no—
210C4×C8C2×#692311yescharacteristic
211#95—1222no—
212#95—1224no—
213C2×Q8—21no—
214C2×Q8—612no—
215C2×Q8—1222no—
216C2×Q8—1221no—
217C2×Q8—2441no—
218C2×Q8—2442no—
219C2×Q8—2442no—
220C2×Q8—2442no—
221C2×Q8—2442no—
222C2×Q8C22⋊#692312yes—
223C22×C4—612no—
224C22×C4—611no—
225C22×C4—1222no—
226C22×C4—1222no—
227C22×C4—1222no—
228C22×C4—2441no—
229Q16—44no—
230Q16—2444no—
231Q16—2444no—
232Q16—2444no—
233C2×C8—21no—
234C2×C8—1222no—
235C2×C8—1222no—
236C2×C8—1221no—
237C2×C8—2442no—
238C2×C8—2442no—
239C2×C8—2442no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—614no—
243C4⋊C4—1224no—
244C4⋊C4—1222no—
245C4⋊C4—1222no—
246C4⋊C4—1222no—
247C4⋊C4—1222no—
248C4⋊C4—1222no—
249C4⋊C4—4882no—
250C4⋊C4—2442no—
251C42—611no—
252C42—612no—
253C42—4882no—
254C42—2442no—
255C42—2442no—
256C42C22×#692311yescharacteristic
257C23—611no—
258Q8—42no—
259Q8—42no—
260Q8—4882no—
261Q8—4882no—
262Q8—2442no—
263Q8—2442no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—612no—
267C2×C4—612no—
268C2×C4—612no—
269C2×C4—612no—
270C2×C4—612no—
271C2×C4—611no—
272C2×C4—1221no—
273C2×C4—1221no—
274C2×C4—1224no—
275C2×C4—1222no—
276C2×C4—4881no—
277C2×C4—4881no—
278C2×C4—2442no—
279C2×C4—2442no—
280C2×C4—2442no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4—2441no—
284C2×C4—2442no—
285C2×C4—2442no—
286C2×C4C2×C22⋊#692311yescharacteristic
287C2×C4D4×#692312yes—
288C8—22no—
289C8—1222no—
290C8—4882no—
291C22—612no—
292C22—612no—
293C22—611no—
294C22—611no—
295C22#1684071011yescharacteristic
296C4—21no—
297C4—42no—
298C4—81no—
299C4—612no—
300C4—612no—
301C4—1221no—
302C4—4882no—
303C4—4881no—
304C4—2444no—
305C4—2442no—
306C4Q16×#692312yes—
307C4#1684080212yes—
308C2—611no—
309C2—612no—
310C2—611no—
311C2#2146181312yes—
312C2#2146182611yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#494C4
socleC22×C61#386
prime radicalC22×D61#120
cosocleC42×D61C23
hypercenter#494#6923
hypocenterC61#9066
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
17. #16840142C4semidirect
18. #16840141C4semidirect
19. #16840141C4semidirect
22. C61×#494C4semidirect
77. C4⋊#6923Q16semidirect
169. #494#6923semidirect
179. C61#9066semidirect

Maximal subgroups

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

subgroupindex
#214623522
#214619972
D61⋊#3332
#214606542
#494×D612
#906661