Finite Group Databasebeta

← #29598336

Subgroups — #29598336

Order 15616.

540 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 111 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×#16839559C214yesmaximal, complemented
2#21445815C211yesmaximal, characteristic, complemented
3C4×#14519696C211yesmaximal, characteristic, complemented
4C61×#2548C211yesmaximal, characteristic
5#16839786—21no—
6C2×#14519696C2212yes—
7C2×#14519696C2212yes—
8C4×#11758194—21no—
9#16839559—28no—
10#16839531—21no—
11#16839531—21no—
12C2×#127×C61C411yescharacteristic
13C2×#124×C61C2212yes—
14C2×C4×Q8×C61C2211yescharacteristic
15C61×#394—21no—
16C61×#394—21no—
17C61×#390—21no—
18C61×#390—21no—
19C61×#390C412yes—
20C4×C4⋊C4×C61C411yescharacteristic
21C2×#11758194—22no—
22C2×#11758194—22no—
23C4⋊#92418—41no—
24C4⋊#92418—41no—
25#14519696—22no—
26#14519696—24no—
27#14519696D414yescomplemented
28#14519688—41no—
29C4×#92418—41no—
30C22×Q8×C61C2311yescharacteristic
31C22×D4×C61—21no—
32C22×D4×C61—21no—
33C22×D4×C61C2×C412yes—
34#127×C61—41no—
35#127×C61—41no—
36#124×C61—24no—
37#124×C61—24no—
38C4×Q8×C61—41no—
39C4×Q8×C61—41no—
40C2×C4⋊C4×C61—21no—
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C4⋊C4×C61D411yescharacteristic
43C2×C4⋊C4×C61D411yescharacteristic
44C2×C22⋊C4×C61—22no—
45C2×C22⋊C4×C61—22no—
46C2×C22⋊C4×C61—22no—
47C2×C22⋊C4×C61—22no—
48C2×C42×C61C2×C412yes—
49C2×C42×C61C2×C411yescharacteristic
50C2×C42×C61D411yescharacteristic
51C2×C42×C61D411yescharacteristic
52#95×C61—21no—
53C24×C61—22no—
54C2×Q8×C61—41no—
55C2×Q8×C61—41no—
56C2×Q8×C61C2×D412yes—
57C2×Q8×C61#4112yes—
58C2×D4×C61—24no—
59C2×D4×C61—42no—
60C2×D4×C61—42no—
61C2×D4×C61—42no—
62C2×D4×C61—42no—
63C2×D4×C61—42no—
64C2×D4×C61—41no—
65C2×D4×C61—41no—
66C2×D4×C61—82no—
67C22×C4×C61—22no—
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22×C411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C4⋊C4×C61—41no—
76C4⋊C4×C61—41no—
77C4⋊C4×C61—41no—
78C4⋊C4×C61—42no—
79C22⋊C4×C61—44no—
80C22⋊C4×C61—42no—
81C22⋊C4×C61—42no—
82C22⋊C4×C61—82no—
83C22⋊C4×C61—82no—
84C42×C61—21no—
85C42×C61—21no—
86C42×C61—21no—
87C42×C61—21no—
88C42×C61—21no—
89C42×C61—21no—
90C42×C61—24no—
91#11758194—24no—
92#11758194—42no—
93#11758194—42no—
94#11758194—42no—
95C2×#92418—42no—
96C2×#92418—42no—
97C23×C61—42no—
98C23×C61—42no—
99C23×C61—42no—
100C23×C61—42no—
101C23×C61—42no—
102C23×C61—42no—
103C23×C61—42no—
104C23×C61C2×C22⋊C411yescharacteristic
105Q8×C61—42no—
106Q8×C61—42no—
107D4×C61—84no—
108D4×C61—82no—
109D4×C61—84no—
110D4×C61—82no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—22no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61#9911yescharacteristic
128C2×C4×C61#9911yescharacteristic
129C2×C4×C61C4×D412yes—
130C2×C4×C61#12112yes—
131C2×C4×C61#12411yescharacteristic
132C2×C4×C61#12711yescharacteristic
133#92418—44no—
134#92418—82no—
135#13226—611nomaximal, Sylow
136C22×C61—21no—
137C22×C61—21no—
138C22×C61—82no—
139C22×C61—82no—
140C22×C61—82no—
141C22×C61—82no—
142C22×C61—82no—
143C22×C61—82no—
144C22×C61—82no—
145C22×C61#39011yescharacteristic
146C22×C61C2×#9911yescharacteristic
147C22×C61#41011yescharacteristic
148C4×C61—21no—
149C4×C61—22no—
150C4×C61—21no—
151C4×C61—22no—
152C4×C61—21no—
153C4×C61—41no—
154C4×C61—41no—
155C4×C61—41no—
156C4×C61—82no—
157C4×C61#35512yescomplemented
158#2548D6111yescharacteristic
159C2×#355—614no—
160#1801—611no—
161C4×#101—611no—
162C2×C61—21no—
163C2×C61—21no—
164C2×C61—162no—
165C2×C61#196411yescharacteristic
166C2×C61C2×#35511yescharacteristic
167C2×C61#213411yescharacteristic
168C2×#127Dic6111yescharacteristic
169C2×#124C2×D6112yes—
170C2×C4×Q8C2×D6111yescharacteristic
171#423—1221no—
172C2×#101—612no—
173C2×#101—612no—
174C4×M16—1221no—
175#394—21no—
176#394—21no—
177#390—21no—
178#390—21no—
179#390Dic6112yes—
180C4×C4⋊C4Dic6111yescharacteristic
181#355—1228no—
182#343—1221no—
183#343—1221no—
184C61#1322611yescharacteristic, complemented, Sylow
185C22×Q8C22×D6111yescharacteristic
186C22×D4—21no—
187C22×D4—21no—
188C22×D4C2×Dic6112yes—
189C2×M16—1222no—
190C2×M16—1222no—
191#127—41no—
192#127—41no—
193#124—24no—
194#124—24no—
195C4×Q8—41no—
196C4×Q8—41no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C2×Dic6111yescharacteristic
199C2×C4⋊C4C22⋊D6111yescharacteristic
200C2×C4⋊C4C22⋊D6111yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42C2×Dic6112yes—
206C2×C42C2×Dic6111yescharacteristic
207C2×C42C22⋊D6111yescharacteristic
208C2×C42C22⋊D6111yescharacteristic
209C4⋊C8—2441no—
210C4⋊C8—2441no—
211#101—614no—
212#101—1222no—
213#101—1224no—
214#97—2441no—
215C4×C8—2441no—
216#95—21no—
217C24—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8D4×D6112yes—
221C2×Q8#1175822412yes—
222C2×D4—24no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—82no—
231C22×C4—22no—
232C22×C4C22⋊Dic6111yescharacteristic
233C22×C4C22⋊Dic6111yescharacteristic
234C22×C4C22⋊Dic6111yescharacteristic
235C22×C4C22⋊Dic6111yescharacteristic
236C22×C4C22×Dic6111yescharacteristic
237C22×C4C2×C22⋊D6111yescharacteristic
238C22×C4C2×C22⋊D6111yescharacteristic
239M16—1224no—
240M16—2442no—
241M16—2442no—
242M16—2442no—
243C2×C8—2442no—
244C2×C8—2442no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—24no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23C2×C22⋊Dic6111yescharacteristic
269Q8—42no—
270Q8—42no—
271D4—84no—
272D4—82no—
273D4—84no—
274D4—82no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4#1451969211yescharacteristic
292C2×C4#1451969211yescharacteristic
293C2×C4D4×Dic6112yes—
294C2×C4#1451974812yes—
295C2×C4#1451976411yescharacteristic
296C2×C4#1451977311yescharacteristic
297C8—4882no—
298C8—2444no—
299C22—21no—
300C22—21no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22#1683966511yescharacteristic
309C22C2×#1451969211yescharacteristic
310C22#1683973611yescharacteristic
311C4—21no—
312C4—22no—
313C4—22no—
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—82no—
320C4#1683955912yescomplemented
321C2—21no—
322C2—21no—
323C2—162no—
324C2#2144665611yescharacteristic
325C2C2×#1683955911yescharacteristic
326C2#2144751111yescharacteristic
subgroupG/H
centerC22C2×#14519692
commutatorC22×C4×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#2548C2
socleC22×C61C2×#99
prime radicalC2×#127×C61C4
cosocleC22×Q8×C61C23
hypercenter#2548D61
hypocenterC61#13226
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. C2×#16839559C2semidirect
2. #21445815C2semidirect
3. C4×#14519696C2semidirect
27. #14519696D4semidirect
157. C4×C61#355semidirect
184. C61#13226semidirect
320. C4#16839559semidirect

Maximal subgroups

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

subgroupindex
C2×#168395592
#214458152
C4×#145196962
C61×#25482
#1322661