Finite Group Databasebeta

← #30601616

Subgroups — #30601616

Order 15616.

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

normal subgroups only

This group has 582 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#21461752C211yesmaximal, characteristic, complemented
2#21461533C211yesmaximal, characteristic, complemented
3C2×Q8⋊#92420C212yesmaximal, complemented
4C2×#16840610C212yesmaximal
5#21448362C211yesmaximal, characteristic
6C2×#14519844—22no—
7C2×C4⋊#92420C2212yes—
8C2×C4×#92420C2212yes—
9#16840628—22no—
10Q8⋊#92420—24no—
11#16840610—24no—
12C2×#14519766C411yescharacteristic
13C2×Q8×Dic61C2211yescharacteristic
14C2×#14519739C2211yescharacteristic
15#16839664C411yescharacteristic
16#16839654—22no—
17#16839627C411yescharacteristic
18#16839609—22no—
19C42×Dic61C2211yescharacteristic
20C61×#391C411yescharacteristic
21C22×#92420—22no—
22C22×#92420—22no—
23#14519844—44no—
24C4⋊#92420—22no—
25C4⋊#92420—22no—
26C4×#92420—22no—
27C4×#92420D414yes—
28C22×D244C2×C411yescharacteristic
29C22×C4×D61—21no—
30#14519766—22no—
31#14519766C814yescomplemented
32Q8×Dic61—22no—
33Q8×Dic61—22no—
34#14519739—22no—
35#14519739—22no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758198—22no—
39C2×#11758198—22no—
40C2×#11758198—21no—
41C2×C4×Dic61—21no—
42C2×C4×Dic61C2×C411yescharacteristic
43C2×C4×Dic61D412yes—
44C2×C4×Dic61D412yes—
45C2×C4×Dic61C2311yescharacteristic
46#14519684—22no—
47C22×Q8×C61C2×C411yescharacteristic
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C42×C61C2×C411yescharacteristic
50#95×C61—22no—
51C23×D61—21no—
52C2×Q8×C61—21no—
53C2×Q8×C61—22no—
54C2×Q8×C61—22no—
55C2×Q8×C61C2×C812yes—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61C22⋊C412yes—
59C22×C4×C61C22×C411yescharacteristic
60C22×Dic61—21no—
61C22×Dic61C22⋊C412yes—
62C22×Dic61C2×D411yescharacteristic
63C22×Dic61C2×D411yescharacteristic
64C2×D244—22no—
65C2×D244—22no—
66C2×D244—21no—
67C2×D244C2×C812yes—
68C2×C4×D61—41no—
69C2×C4×D61—41no—
70C2×C4×D61—42no—
71C2×C4×D61—42no—
72C2×#92420—24no—
73C2×#92420—44no—
74C2×#92420—42no—
75C2×#92420—44no—
76C4⋊C4×C61—22no—
77C42×C61—22no—
78#11758198—28no—
79#11758198—44no—
80#11758198—42no—
81C4⋊Dic61—22no—
82C4⋊Dic61—42no—
83C4×Dic61—22no—
84C4×Dic61—22no—
85C4×Dic61—42no—
86C4×Dic61—41no—
87C4×Dic61—41no—
88C4×Dic61C2×C812yes—
89C4×Dic61M1612yes—
90C4×Dic61C2×D412yes—
91C4×Dic61#4112yes—
92C23×C61C2×C22⋊C411yescharacteristic
93C22×D61—22no—
94C22×D61—41no—
95C22×D61—41no—
96C22×D61—42no—
97C22×D61—42no—
98Q8×C61—24no—
99Q8×C61—42no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—42no—
108C2×C4×C61C4×D412yes—
109C2×C4×C61C22×C811yescharacteristic
110C2×C4×C61C2×M1611yescharacteristic
111C2×C4×C61#13112yes—
112C2×Dic61—22no—
113C2×Dic61—22no—
114C2×Dic61—22no—
115C2×Dic61—41no—
116C2×Dic61—42no—
117C2×Dic61C22⋊C814yes—
118C2×Dic61#12112yes—
119C2×Dic61#12411yescharacteristic
120C2×Dic61#12711yescharacteristic
121C2×Dic61AGL(1,ℤ/8)12yes—
122C2×Dic61#13712yes—
123D244—24no—
124D244—42no—
125C4×D61—82no—
126C4×D61—81no—
127C4×D61—81no—
128#92420—44no—
129#92420—82no—
130#9640—611nomaximal, Sylow
131C22×C61#39011yescharacteristic
132C22×C61C2×C22⋊C811yescharacteristic
133C22×C61#40811yescharacteristic
134C22×C61#41712yes—
135C22×C61#42112yes—
136C2×D61—42no—
137C2×D61—44no—
138C2×D61—81no—
139C2×D61—81no—
140C2×D61—82no—
141C4×C61—42no—
142C4×C61—42no—
143C4×C61—81no—
144C4×C61C8×D412yes—
145C4×C61#43612yes—
146Dic61—22no—
147Dic61—24no—
148Dic61—81no—
149Dic61#48912yes—
150Dic61#49611yescharacteristic
151Dic61#49711yescharacteristic
152C4×#124—611no—
153#1842—611no—
154#1744—611no—
155C2×Q8⋊C8—612no—
156C2×D4⋊C8—612no—
157C2×C61#147712yes—
158C2×C61#149512yes—
159C2×C61#195411yescharacteristic
160C2×C61#195911yescharacteristic
161C2×C61#196311yescharacteristic
162D61—82no—
163C2×#124—611no—
164C2×C4×Q8—611no—
165C2×C4×D4—611no—
166C2×C4⋊C8—612no—
167C2×C4⋊C8—1222no—
168C2×C4×C8—612no—
169#391#692311yescharacteristic
170#390—611no—
171#388—1222no—
172#382—611no—
173C4×C22⋊C4—1222no—
174C43—611no—
175#336—1222no—
176Q8⋊C8—1224no—
177D4⋊C8—1224no—
178C61#964011yescharacteristic, complemented, Sylow
179C22×Q8C2×#692311yescharacteristic
180C22×D4—611no—
181C23×C4—1221no—
182C22×C8—1222no—
183C22×C8—1222no—
184#124—614no—
185#124—1222no—
186C4×Q8—1222no—
187C4×Q8—1222no—
188C4×D4—1222no—
189C4×D4—1222no—
190C2×C4⋊C4—611no—
191C2×C4⋊C4—1221no—
192C2×C4⋊C4C2×#692311yescharacteristic
193C2×C22⋊C4—1222no—
194C2×C22⋊C4—1221no—
195C2×C22⋊C4—1222no—
196C2×C42—611no—
197C2×C42—611no—
198C2×C42—612no—
199C2×C42—612no—
200C2×C42—1221no—
201C2×C42C2×#692311yescharacteristic
202C4⋊C8—1222no—
203C4⋊C8—1222no—
204C4⋊C8—2444no—
205C4×C8—614no—
206C4×C8—1222no—
207#95—22no—
208#95—1222no—
209C24—1221no—
210C2×Q8—21no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8D61⋊C812yes—
214C2×D4—612no—
215C2×D4—1222no—
216C2×D4—1221no—
217C2×D4—1222no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—611no—
221C22×C4—612no—
222C22×C4—611no—
223C22×C4—1221no—
224C22×C4—2441no—
225C22×C4—2441no—
226C22×C4—2442no—
227C22×C4—2442no—
228C22×C4C22⋊#692312yes—
229C22×C4C22×#692311yescharacteristic
230C2×C8—1224no—
231C2×C8—2442no—
232C2×C8—2444no—
233C2×C8—2444no—
234C4⋊C4—22no—
235C4⋊C4—1222no—
236C4⋊C4—2442no—
237C22⋊C4—1228no—
238C22⋊C4—2444no—
239C22⋊C4—2442no—
240C42—22no—
241C42—612no—
242C42—612no—
243C42—612no—
244C42—612no—
245C42—1222no—
246C42—1222no—
247C42—2442no—
248C42—2441no—
249C42—2441no—
250C23—1222no—
251C23—2441no—
252C23—2441no—
253C23—2442no—
254C23—2442no—
255C23C2×C22⋊#692311yescharacteristic
256Q8—24no—
257Q8—42no—
258D4—1224no—
259D4—2442no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—611no—
269C2×C4—611no—
270C2×C4—612no—
271C2×C4—612no—
272C2×C4—612no—
273C2×C4—614no—
274C2×C4—1222no—
275C2×C4—1222no—
276C2×C4—1222no—
277C2×C4—4882no—
278C2×C4—4881no—
279C2×C4—4881no—
280C2×C4—2441no—
281C2×C4—2442no—
282C2×C4D4×#692312yes—
283C2×C4C2×D61⋊C811yescharacteristic
284C2×C4D61⋊M1611yescharacteristic
285C2×C4#1451986412yes—
286C8—4882no—
287C8—2444no—
288C22—4882no—
289C22—4881no—
290C22—4881no—
291C22—2444no—
292C22—2442no—
293C22#1684071411yescharacteristic
294C22D61⋊(C22⋊C8)11yescharacteristic
295C22#1684073511yescharacteristic
296C22#1684075312yes—
297C22#1684076112yes—
298C4—42no—
299C4—42no—
300C4—81no—
301C4—612no—
302C4—611no—
303C4—611no—
304C4—1224no—
305C4—1222no—
306C4—4881no—
307C4#1684078912yes—
308C4#1684079512yes—
309C2—4882no—
310C2#2146111712yes—
311C2#2146117612yes—
312C2#2146194211yescharacteristic
313C2#2146194911yescharacteristic
314C2#2146195511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC2×C4×Dic61C23
hypercenter#391#6923
hypocenterC61#9640
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #21461752C2semidirect
2. #21461533C2semidirect
3. C2×Q8⋊#92420C2semidirect
31. #14519766C8semidirect
178. C61#9640semidirect

Maximal subgroups

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

subgroupindex
#214617522
#214615332
C2×Q8⋊#924202
C2×#168406102
#214483622
#964061