Finite Group Databasebeta

← #30597574

Subgroups — #30597574

Order 15616.

486 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461605C211yesmaximal, characteristic
2#21461533C211yesmaximal, characteristic
3C2×Q8⋊#92420C212yesmaximal
4#21460655C212yesmaximal
5#21448294C211yesmaximal, characteristic
6C2×C4⋊#92420C2212yes—
7C2×#14519832—21no—
8C2×C4×#92420C2212yes—
9#16840628—21no—
10#16840628—21no—
11#16840628—21no—
12Q8⋊#92420C418yescomplemented
13C2×Q8×Dic61C2211yescharacteristic
14#16839669C411yescharacteristic
15#16839660C411yescharacteristic
16#16839658C411yescharacteristic
17#16839647—21no—
18#16839645—21no—
19C4⋊C4×Dic61C2211yescharacteristic
20#16839613—21no—
21#16839607—21no—
22C42×Dic61C2211yescharacteristic
23C61×#391C411yescharacteristic
24C22×#92420—22no—
25C22×#92420—22no—
26C4⋊#92420—22no—
27C4⋊#92420C2×C414yes—
28#14519832—42no—
29C4×#92420—22no—
30C4×#92420C2×C414yes—
31Q8×Dic61—22no—
32Q8×Dic61C2×C414yes—
33C2×C4⋊Dic61—21no—
34C2×C4⋊Dic61C2×C411yescharacteristic
35C2×C4⋊Dic61C2×C411yescharacteristic
36C2×#11758196—21no—
37C2×#11758196—21no—
38C2×C4×Dic61—21no—
39C2×C4×Dic61—21no—
40C2×C4×Dic61C2×C411yescharacteristic
41C2×C4×Dic61D411yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43C2×C4×Dic61D411yescharacteristic
44C2×C4×Dic61D411yescharacteristic
45C2×C4×Dic61C2311yescharacteristic
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51C22×Q8×C61C2×C411yescharacteristic
52C2×C4⋊C4×C61C2×C411yescharacteristic
53C2×C42×C61C2×C411yescharacteristic
54#95×C61—21no—
55#95×C61—21no—
56C2×Q8×C61—21no—
57C2×Q8×C61—22no—
58C2×Q8×C61—22no—
59C2×Q8×C61C4212yes—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C22×Dic61—21no—
66C22×Dic61—21no—
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C2×D411yescharacteristic
70C22×Dic61C2×D411yescharacteristic
71C2×#92420—24no—
72C2×#92420—24no—
73C2×#92420—42no—
74C2×#92420—42no—
75C2×#92420—42no—
76C2×#92420—42no—
77C4⋊C4×C61—22no—
78C42×C61—22no—
79C4⋊Dic61—21no—
80C4⋊Dic61—24no—
81C4⋊Dic61—22no—
82C4⋊Dic61C4212yes—
83#11758196—42no—
84#11758196—42no—
85C4×Dic61—22no—
86C4×Dic61—24no—
87C4×Dic61—21no—
88C4×Dic61—21no—
89C4×Dic61—22no—
90C4×Dic61—41no—
91C4×Dic61—41no—
92C4×Dic61—41no—
93C4×Dic61—41no—
94C4×Dic61C22×C412yes—
95C4×Dic61#4112yes—
96C23×C61C2×C22⋊C411yescharacteristic
97Q8×C61—24no—
98Q8×C61—42no—
99C2×C4×C61—22no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61C2×C4211yescharacteristic
112C2×C4×C61#11711yescharacteristic
113C2×C4×C61#13112yes—
114C2×C4×C61#13112yes—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—22no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61C4×D412yes—
130C2×Dic61C4×D412yes—
131C2×Dic61AGL(1,ℤ/8)11yescharacteristic
132C2×Dic61#13712yes—
133C2×Dic61#13711yescharacteristic
134#92420—44no—
135#92420—44no—
136#8178—611nomaximal, Sylow
137C22×C61C4×C22⋊C411yescharacteristic
138C22×C61#40811yescharacteristic
139C22×C61#40811yescharacteristic
140C22×C61#41711yescharacteristic
141C22×C61#41911yescharacteristic
142C22×C61#42111yescharacteristic
143C22×C61#42111yescharacteristic
144C4×C61—42no—
145C4×C61—41no—
146C4×C61—41no—
147C4×C61—81no—
148C4×C61#40512yes—
149C4×C61#43312yes—
150Dic61—22no—
151Dic61—41no—
152Dic61—41no—
153Dic61—42no—
154Dic61—81no—
155Dic61#44012yes—
156Dic61#44112yes—
157#2295—611no—
158#1767—611no—
159#1744—611no—
160C2×Q8⋊C8—612no—
161#1279—612no—
162C2×C61#149511yescharacteristic
163C2×C61#149511yescharacteristic
164C2×C61#149711yescharacteristic
165C2×C61#149711yescharacteristic
166C2×C61#175011yescharacteristic
167C2×C61#175911yescharacteristic
168C2×C61#176311yescharacteristic
169C2×C4×Q8—611no—
170C2×C4⋊C8—612no—
171C2×#97—1221no—
172C2×C4×C8—612no—
173#391—611no—
174#391#692311yescharacteristic
175#389—611no—
176#389—611no—
177#387—1221no—
178#387—1221no—
179C4×C4⋊C4—611no—
180C4×C4⋊C4—1221no—
181#376—1221no—
182C43—611no—
183#336—1221no—
184#336—1221no—
185#336—1221no—
186Q8⋊C8—618no—
187C61#817811yescharacteristic, complemented, Sylow
188C22×Q8C2×#692311yescharacteristic
189C22×C8—1222no—
190C22×C8—1222no—
191C4×Q8—614no—
192C4×Q8—1222no—
193C2×C4⋊C4—611no—
194C2×C4⋊C4—611no—
195C2×C4⋊C4—1221no—
196C2×C4⋊C4—1221no—
197C2×C4⋊C4—1221no—
198C2×C4⋊C4C2×#692311yescharacteristic
199C2×C42—611no—
200C2×C42—611no—
201C2×C42—611no—
202C2×C42—611no—
203C2×C42—611no—
204C2×C42—611no—
205C2×C42—1221no—
206C2×C42—1221no—
207C2×C42C2×#692311yescharacteristic
208C4⋊C8—614no—
209C4⋊C8—1222no—
210#97—2442no—
211C4×C8—614no—
212C4×C8—1222no—
213#95—21no—
214#95—21no—
215#95—1221no—
216#95—1221no—
217#95—1221no—
218#95—1221no—
219#95—1221no—
220C2×Q8—21no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8C4×#692312yes—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—611no—
227C22×C4—611no—
228C22×C4—611no—
229C22×C4—611no—
230C22×C4—1221no—
231C22×C4—1221no—
232C22×C4C22⋊#692311yescharacteristic
233C22×C4C22⋊#692311yescharacteristic
234C22×C4C22×#692311yescharacteristic
235C2×C8—1224no—
236C2×C8—1224no—
237C2×C8—2442no—
238C2×C8—2442no—
239C2×C8—2442no—
240C2×C8—2442no—
241C4⋊C4—22no—
242C4⋊C4—612no—
243C4⋊C4—1224no—
244C4⋊C4—1221no—
245C4⋊C4—1222no—
246C4⋊C4—2442no—
247C4⋊C4—2442no—
248C42—22no—
249C42—612no—
250C42—612no—
251C42—1221no—
252C42—1221no—
253C42—1222no—
254C42—1224no—
255C42—1222no—
256C42—2441no—
257C42—2441no—
258C42—2441no—
259C42—2441no—
260C23C2×C22⋊#692311yescharacteristic
261Q8—24no—
262Q8—42no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—611no—
276C2×C4—611no—
277C2×C4—612no—
278C2×C4—612no—
279C2×C4—612no—
280C2×C4—1221no—
281C2×C4—1221no—
282C2×C4—1221no—
283C2×C4—1221no—
284C2×C4—1221no—
285C2×C4—1221no—
286C2×C4—1221no—
287C2×C4—1221no—
288C2×C4—1222no—
289C2×C4—2441no—
290C2×C4—2441no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4C2×C4×#692311yescharacteristic
295C2×C4#1451985611yescharacteristic
296C2×C4#1451986412yes—
297C2×C4#1451986412yes—
298C8—2444no—
299C8—2444no—
300C22C4×C22⋊#692311yescharacteristic
301C22#1684073511yescharacteristic
302C22#1684073611yescharacteristic
303C22#1684075311yescharacteristic
304C22#1684075511yescharacteristic
305C22#1684075911yescharacteristic
306C22#1684076111yescharacteristic
307C4—42no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4—612no—
312C4—612no—
313C4—1222no—
314C4—4881no—
315C4—2442no—
316C4—2441no—
317C4—2441no—
318C4#1684072612yes—
319C4#1684078612yes—
320C2#2146117511yescharacteristic
321C2#2146117611yescharacteristic
322C2#2146118111yescharacteristic
323C2#2146118211yescharacteristic
324C2#2146154911yescharacteristic
325C2#2146158311yescharacteristic
326C2#2146159111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socle, prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#391#6923
hypocenterC61#8178
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
12. Q8⋊#92420C4semidirect
187. C61#8178semidirect

Maximal subgroups

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

subgroupindex
#214616052
#214615332
C2×Q8⋊#924202
#214606552
#214482942
#817861