Finite Group Databasebeta

← #29571789

Subgroups — #29571789

Order 15616.

782 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16840472C211yesmaximal, characteristic
2#21445815C211yesmaximal, characteristic, complemented
3#21444503C212yesmaximal, complemented
4#21444245C212yesmaximal
5C61×#1461C211yesmaximal, characteristic, complemented
6#16840472—24no—
7#16840472—24no—
8C2×#14519777C2211yescharacteristic
9C2×#14519769C412yes—
10C2×#14519755—22no—
11C2×#14519753—21no—
12C2×C4×D244C412yes—
13C2×C4×D244C2211yescharacteristic
14C2×C4⋊C4×D61—21no—
15#16839786C2212yes—
16C2×#14519696—21no—
17#16839531—21no—
18#16839509—24no—
19#16839491—24no—
20C2×C4×Q8×C61C2211yescharacteristic
21C61×#431C2212yes—
22C61×#324—22no—
23C22×Dic122—21no—
24C22×D244C2×C411yescharacteristic
25C22×C4×D61—21no—
26C22×C4×D61—22no—
27C2×#11758194—22no—
28#14519777—22no—
29#14519777D414yes—
30#14519769—24no—
31#14519769—24no—
32#14519755—216no—
33#14519753—28no—
34C4×D244—24no—
35C4×D244—22no—
36C4×D244—24no—
37C4×D244D414yes—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61C2×C412yes—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×#11758196—22no—
42C2×#11758196—22no—
43C4⋊C4×D61—28no—
44C2×#11758198—22no—
45C2×#11758198—21no—
46C4⋊#92418—22no—
47C4⋊#92418—22no—
48#14519696—42no—
49C22×Q8×C61D411yescharacteristic
50C4×Q8×C61—22no—
51C4×Q8×C61—22no—
52C2×C4⋊C4×C61D411yescharacteristic
53C2×C4⋊C4×C61D412yes—
54C2×C42×C61C2×C412yes—
55C2×C42×C61C2311yescharacteristic
56C4⋊C8×C61—22no—
57C22⋊C8×C61—22no—
58C4×C8×C61—22no—
59C23×D61—21no—
60C2×Q8×C61—22no—
61C2×Q8×C61—22no—
62C2×Q8×C61—21no—
63C2×Q8×C61—21no—
64C22×C4×C61C22⋊C412yes—
65C22×C4×C61C22⋊C412yes—
66C22×C4×C61C22×C411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×Dic61—22no—
70C22×Dic61—21no—
71C2×D244—21no—
72C2×D244—24no—
73C2×D244C22⋊C412yes—
74C2×C4×D61—24no—
75C2×C4×D61—28no—
76C2×C4×D61—42no—
77C2×C4×D61—41no—
78C2×C4×D61—41no—
79C2×C4×D61—84no—
80C2×Dic122—24no—
81C2×Dic122—82no—
82C2×C8×C61—42no—
83C4⋊C4×C61—22no—
84C4⋊C4×C61—24no—
85C42×C61—22no—
86C42×C61—22no—
87C42×C61C2×D412yes—
88C42×C61C2×D412yes—
89#11758198—28no—
90#11758198—24no—
91C4⋊Dic61—21no—
92C4⋊Dic61—22no—
93C4⋊Dic61—24no—
94C4⋊Dic61C22⋊C414yes—
95C4⋊Dic61C22⋊C412yes—
96#11758196—28no—
97#11758196—28no—
98#11758194—42no—
99#11758194—42no—
100C2×#92418—42no—
101C23×C61C2×C22⋊C411yescharacteristic
102C22×D61—24no—
103C22×D61—22no—
104C22×D61—42no—
105C22×D61—41no—
106C22×D61—41no—
107Q8×C61—42no—
108Q8×C61—42no—
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—22no—
112C2×C4×C61—22no—
113C2×C4×C61—22no—
114C2×C4×C61—41no—
115C2×C4×C61—42no—
116C2×C4×C61C2×C22⋊C411yescharacteristic
117C2×C4×C61C2×C22⋊C411yescharacteristic
118C2×C4×C61#12012yes—
119C2×C4×C61#12012yes—
120C2×C4×C61#13512yes—
121C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C61#13711yescharacteristic
123C2×Dic61—22no—
124C2×Dic61—24no—
125C2×Dic61—44no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128D244—24no—
129D244—22no—
130D244#10114yescomplemented
131C4×D61—44no—
132C4×D61—44no—
133C4×D61—88no—
134Dic122—84no—
135C8×C61—82no—
136#92418—82no—
137#8398—611nomaximal, Sylow
138C22×C61—21no—
139C22×C61—21no—
140C22×C61#37911yescharacteristic
141C22×C61#41611yescharacteristic
142C22×C61#41711yescharacteristic
143C2×D61—24no—
144C2×D61—44no—
145C2×D61—44no—
146C2×D61—42no—
147C2×D61—42no—
148C4×C61—22no—
149C4×C61—21no—
150C4×C61—42no—
151C4×C61—42no—
152C4×C61C2×#10112yes—
153C4×C61#44912yes—
154C4×C61#45212yes—
155Dic61—44no—
156Dic61—42no—
157C2×#554—611no—
158#1801—611no—
159#1549—612no—
160#1486—612no—
161#1461D6111yescharacteristic, complemented
162C2×C61—21no—
163C2×C61—21no—
164C2×C61#178411yescharacteristic
165C2×C61#178811yescharacteristic
166C2×C61#179511yescharacteristic
167D61—44no—
168#554—1224no—
169#554—1224no—
170C2×C4⋊Q8—611no—
171C2×#125—612no—
172C2×C22⋊Q8—1221no—
173C2×C22⋊Q8—1222no—
174C2×C4×Q8C2×D6111yescharacteristic
175C2×C4×D4—612no—
176C2×C4×D4—611no—
177C22×C4⋊C4—1221no—
178#431C2×D6112yes—
179#423—612no—
180C2×#101—1221no—
181#343—1221no—
182#332—1224no—
183D4⋊C8—1224no—
184#324—22no—
185C61#839811yescharacteristic, complemented, Sylow
186C22×Q8—1221no—
187C22×Q8D24411yescharacteristic
188C22×D4—611no—
189C23×C4—1222no—
190C23×C4—1221no—
191C2×M16—1222no—
192C4⋊Q8—614no—
193C4⋊Q8—1222no—
194#125—1224no—
195#125—1224no—
196C22⋊Q8—12216no—
197C22⋊Q8—1228no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×D4—614no—
201C4×D4—1224no—
202C4×D4—1224no—
203C4×D4—1222no—
204C2×C4⋊C4—611no—
205C2×C4⋊C4—612no—
206C2×C4⋊C4—1228no—
207C2×C4⋊C4—1222no—
208C2×C4⋊C4—1222no—
209C2×C4⋊C4—1221no—
210C2×C4⋊C4D24411yescharacteristic
211C2×C4⋊C4C22⋊D6112yes—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4—1221no—
214C2×C42C4×D6112yes—
215C2×C42C22×D6111yescharacteristic
216C4⋊C8—22no—
217C4⋊C8—1222no—
218C4⋊C8—1222no—
219#101—2442no—
220C22⋊C8—22no—
221C4×C8—22no—
222C24—1221no—
223C2×Q8—21no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—21no—
227C2×Q8—1224no—
228C2×Q8—4882no—
229C2×D4—612no—
230C2×D4—1224no—
231C2×D4—1221no—
232C22×C4—1228no—
233C22×C4—1221no—
234C22×C4—1222no—
235C22×C4—1224no—
236C22×C4—4884no—
237C22×C4—2441no—
238C22×C4—2442no—
239C22×C4—2441no—
240C22×C4#1175819812yes—
241C22×C4#1175819812yes—
242C22×C4C2×C4×D6111yescharacteristic
243C22×C4C2×D24411yescharacteristic
244C22×C4C2×C22⋊D6111yescharacteristic
245M16—2442no—
246M16—2442no—
247C2×C8—42no—
248C2×C8—2442no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—612no—
252C4⋊C4—614no—
253C4⋊C4—1228no—
254C4⋊C4—1228no—
255C4⋊C4—1221no—
256C4⋊C4—1222no—
257C4⋊C4—1224no—
258C22⋊C4—1228no—
259C22⋊C4—1224no—
260C42—22no—
261C42—22no—
262C42D4×D6112yes—
263C42D4×D6112yes—
264C23—1224no—
265C23—1222no—
266C23—2442no—
267C23—2441no—
268C23—2441no—
269C23C2×#1175819811yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—4884no—
273D4—614no—
274D4—1224no—
275D4—1222no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—1222no—
284C2×C4—1224no—
285C2×C4—4888no—
286C2×C4—2444no—
287C2×C4—2441no—
288C2×C4—2441no—
289C2×C4—2444no—
290C2×C4—2444no—
291C2×C4C22⋊C4×D6111yescharacteristic
292C2×C4C22⋊C4×D6111yescharacteristic
293C2×C4#1451974512yes—
294C2×C4#1451974712yes—
295C2×C4#1451979912yes—
296C2×C4#1451980511yescharacteristic
297C2×C4#1451981011yescharacteristic
298C8—82no—
299C8—4882no—
300C22—21no—
301C22—21no—
302C22—1224no—
303C22—2444no—
304C22—2444no—
305C22—2442no—
306C22—2442no—
307C22#1683961711yescharacteristic
308C22#1683975911yescharacteristic
309C22#1683976411yescharacteristic
310C4—21no—
311C4—22no—
312C4—42no—
313C4—42no—
314C4—2442no—
315C4—2444no—
316C4#101×D6112yes—
317C4#1683990912yes—
318C4#1683992612yes—
319C2—21no—
320C2—21no—
321C2—2444no—
322C2#2144574111yescharacteristic
323C2#2144576011yescharacteristic
324C2#2144578711yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1461C2
socleC22×C61#379
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1461D61
hypocenterC61#8398
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #21445815C2semidirect
3. #21444503C2semidirect
5. C61×#1461C2semidirect
130. D244#101semidirect
161. #1461D61semidirect
185. C61#8398semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
#214458152
#214445032
#214442452
C61×#14612
#839861