Finite Group Databasebeta

← #30603529

Subgroups — #30603529

Order 15616.

606 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 134 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#21462472C211yesmaximal, characteristic
2#21461517C212yesmaximal
3#21461504C212yesmaximal
4#21461487C211yesmaximal, characteristic
5H(3,ℤ/4)×D61C211yesmaximal, characteristic
6C22×D61⋊C8C2211yescharacteristic
7C2×#14519856C2212yes—
8C2×#14519856C2211yescharacteristic
9#16840773—24no—
10#16840772—22no—
11#16840639—24no—
12#16840635—24no—
13(C2×#6923)⋊C8—22no—
14C2×M16×D61C2211yescharacteristic
15#117×D61C2212yes—
16#16839519C411yescharacteristic, complemented
17#16839518C412yescomplemented
18C61×H(3,ℤ/4)C411yescharacteristic, complemented
19C2×D61⋊C8—22no—
20C2×D61⋊C8—22no—
21C2×D61⋊C8—21no—
22C2×D61⋊C8Q812yes—
23C22×#92420—21no—
24#14519856—22no—
25#14519856—22no—
26#14519856—22no—
27#14519856—22no—
28#14519856—22no—
29#14519856—22no—
30C2×C4⋊#6923—21no—
31C2×C4⋊#6923D414yes—
32C2×C22⋊#6923—22no—
33C2×C22⋊#6923D412yes—
34C2×C4×#6923—21no—
35C2×C4×#6923—22no—
36#14519846—24no—
37#14519845—24no—
38C22×C4×D61C2311yescharacteristic
39C2×#11758194C2×C411yescharacteristic
40C2×#11758189Q812yes—
41M16×D61—22no—
42M16×D61—22no—
43C2×C8×D61D412yes—
44#14519736C2×C412yes—
45#14519735—22no—
46#14519734—22no—
47#14519733—22no—
48C4⋊C4×D61—22no—
49C22⋊C4×D61—22no—
50C42×D61—22no—
51C2×M16×C61C2×C411yescharacteristic
52#117×C61C2×C412yes—
53C23×D61C2×D411yescharacteristic
54C22×#6923—22no—
55C22×#6923—24no—
56C22×C4×C61C22×C411yescharacteristic
57C22×Dic61#4111yescharacteristic
58C2×C4×D61—42no—
59C2×C4×D61C2×D412yes—
60C2×C4×D61C2×D411yescharacteristic
61C2×C4×D61C2×Q811yescharacteristic
62C2×C4×D61C2×Q811yescharacteristic
63C2×C4×D61#4112yes—
64C2×C4×D61#4112yes—
65C2×C4×D61#4111yescharacteristic
66C2×C4×D61#4112yes—
67C22⋊#6923—24no—
68C22⋊#6923—42no—
69C22⋊#6923—42no—
70C2×#92420—22no—
71C2×#92420—42no—
72C4⋊#6923—24no—
73C4⋊#6923—24no—
74C4⋊#6923—42no—
75C4×#6923—41no—
76C4×#6923—42no—
77C4×#6923—41no—
78C4×#6923—42no—
79D61⋊C8—24no—
80D61⋊C8—41no—
81D61⋊C8—41no—
82M16×C61—22no—
83C2×C8×C61C4⋊C412yes—
84C4⋊C4×C61—22no—
85C22⋊C4×C61—22no—
86C42×C61—22no—
87C22⋊Dic61—22no—
88#11758198—42no—
89C4⋊Dic61—22no—
90#11758196—42no—
91C4×Dic61—22no—
92C4×Dic61—42no—
93#11758194—22no—
94C2×#92418C4⋊C412yes—
95#11758189—24no—
96C8×D61—22no—
97C8×D61—22no—
98C23×C61C4×D411yescharacteristic
99C22×D61—42no—
100C22×D61#12111yescharacteristic
101C22×D61#12112yes—
102C22×D61#12312yes—
103C22×D61C4⋊Q811yescharacteristic
104C2×#6923—44no—
105C2×#6923—44no—
106C2×#6923—42no—
107C2×#6923—84no—
108C2×C4×C61—42no—
109C2×C4×C61C2×C4⋊C411yescharacteristic
110C2×C4×C61#11712yes—
111C2×C4×C61C4×D412yes—
112C2×C4×C61C4×Q811yescharacteristic
113C2×Dic61—42no—
114C2×Dic61C22⋊Q811yescharacteristic
115C2×Dic61#12412yes—
116C2×Dic61#12511yescharacteristic
117C2×Dic61#12612yes—
118C4×D61—84no—
119C4×D61#12011yescharacteristic
120C4×D61C22⋊Q812yes—
121C4×D61C22⋊Q812yes—
122C4×D61#12311yescharacteristic
123C4×D61#12312yes—
124C4×D61#12411yescharacteristic
125C4×D61#12411yescharacteristic
126C4×D61#12512yes—
127C4×D61#12512yes—
128C4×D61#12612yes—
129#92420—42no—
130C8×C61—22no—
131#92418—22no—
132#11285—611nomaximal, Sylow
133C22×C61—41no—
134C22×C61#38812yes—
135C22×C61#38911yescharacteristic
136C2×D61—24no—
137C2×D61—22no—
138C2×D61—42no—
139C2×D61—41no—
140C2×D61#39411yescharacteristic
141C2×D61#39711yescharacteristic
142C2×D61#39812yes—
143#6923—82no—
144#6923—84no—
145C4×C61—82no—
146C4×C61#38211yescharacteristic
147C4×C61#38511yescharacteristic
148C4×C61#38712yes—
149Dic61—82no—
150Dic61#39311yescharacteristic
151Dic61#40012yes—
152Dic61#40011yescharacteristic
153#2911—611no—
154#1740—612no—
155#1735—612no—
156C2×H(3,ℤ/4)—611no—
157#1729—611no—
158C2×C61—22no—
159C2×C61—21no—
160C2×C61#144211yescharacteristic
161D61—21no—
162D61—22no—
163D61#205312yes—
164C22×M16—611no—
165C23×C8—611no—
166C2×#117—611no—
167C2×#117—612no—
168C2×#117—612no—
169#428—1224no—
170#427—1222no—
171#342—1224no—
172#340—1224no—
173H(3,ℤ/4)—611no—
174H(3,ℤ/4)—612no—
175H(3,ℤ/4)#692311yescharacteristic, complemented
176#336—1222no—
177C61#1128511yescharacteristic, complemented, Sylow
178C23×C4—611no—
179C2×M16—611no—
180C2×M16—612no—
181C2×M16—1222no—
182C2×M16—1222no—
183C2×M16C2×#692311yescharacteristic
184C22×C8—612no—
185C22×C8—612no—
186C22×C8—1221no—
187C22×C8—1222no—
188C22×C8—1221no—
189C22×C8—1222no—
190#117—612no—
191#117—1222no—
192#117—1222no—
193#117—1222no—
194#117—1222no—
195#117—1222no—
196#117—1222no—
197#117—1222no—
198#117—1222no—
199#117—1222no—
200#117C2×#692312yes—
201C2×C4⋊C4—614no—
202C2×C4⋊C4—1222no—
203C2×C4⋊C4—1221no—
204C2×C22⋊C4—612no—
205C2×C22⋊C4—1222no—
206C2×C22⋊C4—1222no—
207C2×C42—1222no—
208C2×C42—1222no—
209C2×C42—1221no—
210#107—1224no—
211#106—1224no—
212C24—611no—
213C22×C4—611no—
214C22×C4—611no—
215C22×C4—611no—
216C22×C4—612no—
217C22×C4—612no—
218C22×C4—612no—
219C22×C4—612no—
220C22×C4—611no—
221C22×C4—611no—
222C22×C4—1224no—
223C22×C4—1222no—
224C22×C4—2442no—
225C22×C4C22×#692311yescharacteristic
226M16—22no—
227M16—1224no—
228M16—1222no—
229C2×C8—612no—
230C2×C8—1222no—
231C2×C8—1222no—
232C2×C8—1224no—
233C2×C8—1222no—
234C2×C8—2441no—
235C2×C8—2441no—
236C2×C8—2442no—
237C2×C8C4⋊#692312yes—
238C4⋊C4—22no—
239C4⋊C4—1224no—
240C4⋊C4—1224no—
241C4⋊C4—1222no—
242C4⋊C4—2442no—
243C4⋊C4—2442no—
244C22⋊C4—22no—
245C22⋊C4—1224no—
246C22⋊C4—1222no—
247C22⋊C4—2442no—
248C22⋊C4—2442no—
249C22⋊C4—2442no—
250C42—22no—
251C42—1222no—
252C42—2442no—
253C42—2442no—
254C42—2441no—
255C42—2441no—
256C42—2442no—
257C23—612no—
258C23—611no—
259C23—612no—
260C23—611no—
261C23—2442no—
262C23D4×#692311yescharacteristic
263C2×C4—42no—
264C2×C4—612no—
265C2×C4—612no—
266C2×C4—611no—
267C2×C4—611no—
268C2×C4—611no—
269C2×C4—611no—
270C2×C4—612no—
271C2×C4—611no—
272C2×C4—612no—
273C2×C4—611no—
274C2×C4—612no—
275C2×C4—612no—
276C2×C4—612no—
277C2×C4—612no—
278C2×C4—4884no—
279C2×C4—4884no—
280C2×C4—2442no—
281C2×C4—2444no—
282C2×C4—2444no—
283C2×C4—2442no—
284C2×C4C2×C4⋊#692311yescharacteristic
285C2×C4#1451985612yes—
286C2×C4D4×#692312yes—
287C2×C4Q8×#692311yescharacteristic
288C8—22no—
289C8—1222no—
290C8—2442no—
291C22—41no—
292C22—611no—
293C22—611no—
294C22—612no—
295C22—1224no—
296C22—1222no—
297C22—2442no—
298C22—2441no—
299C22#1684071212yes—
300C22#1684071311yescharacteristic
301C4—82no—
302C4—611no—
303C4—612no—
304C4—611no—
305C4—4882no—
306C4—4884no—
307C4—4882no—
308C4#1684070611yescharacteristic
309C4#1684070911yescharacteristic
310C4#1684071112yes—
311C2—22no—
312C2—21no—
313C2—612no—
314C2—1221no—
315C2—1222no—
316C2#2146100511yescharacteristic
subgroupG/H
centerC4#16840706
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×H(3,ℤ/4)C4
socleC2×C61#1442
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenterH(3,ℤ/4)#6923
hypocenterC61#11285
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
16. #16839519C4semidirect
17. #16839518C4semidirect
18. C61×H(3,ℤ/4)C4semidirect
175. H(3,ℤ/4)#6923semidirect
177. C61#11285semidirect

Maximal subgroups

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

subgroupindex
#214624722
#214615172
#214615042
#214614872
H(3,ℤ/4)×D612
#1128561