Finite Group Databasebeta

← #29590751

Subgroups — #29590751

Order 15616.

586 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 95 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#21445899C211yesmaximal, characteristic
2#21444149C211yesmaximal, characteristic
3#21443849C214yesmaximal, complemented
4C2×C61×#395C211yesmaximal, characteristic
5#16839726—21no—
6#16839726—21no—
7#16839531—21no—
8#16839531C2212yes—
9#16839488—21no—
10#16839488C2212yes—
11C23×Q8×C61C411yescharacteristic
12C22×C4⋊C4×C61—21no—
13C22×C4⋊C4×C61C2211yescharacteristic
14C61×#395—21no—
15C61×#395—21no—
16C61×#395C412yes—
17C61×#395C2212yes—
18C2×#95×C61—21no—
19C2×#95×C61C411yescharacteristic
20C2×#11758194—22no—
21C2×#11758194—41no—
22C22⋊#92418—22no—
23C22⋊#92418—41no—
24C22⋊#92418—41no—
25C22⋊#92418—41no—
26C22×Q8×C61—21no—
27C22×Q8×C61—44no—
28C22×Q8×C61—41no—
29C22×Q8×C61—41no—
30C22×Q8×C61C2×C412yes—
31C23×C4×C61—21no—
32C23×C4×C61—21no—
33C23×C4×C61C2×C411yescharacteristic
34C23×C4×C61D411yescharacteristic
35C23×C4×C61D411yescharacteristic
36C2×C4⋊C4×C61—22no—
37C2×C4⋊C4×C61—22no—
38C2×C4⋊C4×C61—41no—
39C2×C4⋊C4×C61—41no—
40C2×C4⋊C4×C61—44no—
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C4⋊C4×C61D411yescharacteristic
43C2×C4⋊C4×C61D411yescharacteristic
44C2×C4⋊C4×C61C2311yescharacteristic
45#95×C61—22no—
46#95×C61—21no—
47#95×C61—22no—
48#95×C61C2×C412yes—
49C24×C61C22⋊C411yescharacteristic
50C2×Q8×C61—44no—
51C2×Q8×C61—42no—
52C2×Q8×C61—44no—
53C2×Q8×C61—42no—
54C2×Q8×C61—48no—
55C2×Q8×C61—44no—
56C2×Q8×C61—44no—
57C22×C4×C61—24no—
58C22×C4×C61—24no—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—22no—
65C22×C4×C61—22no—
66C22×C4×C61—21no—
67C22×C4×C61—22no—
68C22×C4×C61—41no—
69C22×C4×C61—42no—
70C22×C4×C61—42no—
71C22×C4×C61—42no—
72C22×C4×C61—41no—
73C22×C4×C61—41no—
74C22×C4×C61—42no—
75C22×C4×C61—42no—
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22×C411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C4⋊C4×C61—41no—
83C4⋊C4×C61—41no—
84C4⋊C4×C61—41no—
85C4⋊C4×C61—41no—
86C4⋊C4×C61—44no—
87C4⋊C4×C61—44no—
88#11758194—42no—
89#11758194—42no—
90#11758194—82no—
91C2×#92418—42no—
92C2×#92418—42no—
93C23×C61—22no—
94C23×C61—22no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61#9911yescharacteristic
98C23×C61#9911yescharacteristic
99C23×C61C2×C22⋊C411yescharacteristic
100Q8×C61—48no—
101Q8×C61—44no—
102Q8×C61—44no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—24no—
106C2×C4×C61—24no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—42no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61—48no—
118C2×C4×C61—42no—
119C2×C4×C61—44no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—44no—
123C2×C4×C61—42no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—81no—
131C2×C4×C61#13712yes—
132#92418—82no—
133#92418—82no—
134#11815—611nomaximal, Sylow
135C22×C61—24no—
136C22×C61—24no—
137C22×C61—21no—
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—21no—
143C22×C61AΣL(1,𝔽16)12yes—
144C22×C61#35612yes—
145C22×C61C2×#9911yescharacteristic
146C22×C61#41911yescharacteristic
147C22×C61#42111yescharacteristic
148C4×C61—22no—
149C4×C61—44no—
150C4×C61—41no—
151C4×C61—44no—
152C4×C61—41no—
153C4×C61—41no—
154C4×C61—41no—
155C4×C61—41no—
156C2×#395D6111yescharacteristic
157#1821—611no—
158#1463—611no—
159#1354—614no—
160C2×C61—21no—
161C2×C61—22no—
162C2×C61—22no—
163C2×C61—21no—
164C2×C61#151111yescharacteristic
165C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
166C2×C61C2×#35611yescharacteristic
167C23×Q8Dic6111yescharacteristic
168C22×C4⋊C4—21no—
169C22×C4⋊C4C2×D6111yescharacteristic
170#407—1221no—
171#407—1221no—
172#395—21no—
173#395—21no—
174#395Dic6112yes—
175#395C2×D6112yes—
176C2×#95—21no—
177C2×#95Dic6111yescharacteristic
178#343—612no—
179#343—1221no—
180#324—612no—
181#324—1221no—
182C61#1181511yescharacteristic, complemented, Sylow
183C22×Q8—21no—
184C22×Q8—44no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8C2×Dic6112yes—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C2×Dic6111yescharacteristic
191C23×C4C22⋊D6111yescharacteristic
192C23×C4C22⋊D6111yescharacteristic
193C2×M16—1222no—
194C2×M16—2441no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4C2×Dic6111yescharacteristic
201C2×C4⋊C4C22⋊D6111yescharacteristic
202C2×C4⋊C4C22⋊D6111yescharacteristic
203C2×C4⋊C4C22×D6111yescharacteristic
204C22⋊C8—1222no—
205C22⋊C8—2441no—
206C22⋊C8—2441no—
207C22⋊C8—2441no—
208#95—22no—
209#95—21no—
210#95—22no—
211#95C2×Dic6112yes—
212C24C22⋊Dic6111yescharacteristic
213C2×Q8—44no—
214C2×Q8—44no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—48no—
218C2×Q8—44no—
219C2×Q8—44no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C22⋊Dic6111yescharacteristic
240C22×C4C22⋊Dic6111yescharacteristic
241C22×C4C22⋊Dic6111yescharacteristic
242C22×C4C22×Dic6111yescharacteristic
243C22×C4C2×C22⋊D6111yescharacteristic
244C22×C4C2×C22⋊D6111yescharacteristic
245M16—4882no—
246M16—2442no—
247M16—2442no—
248C2×C8—2442no—
249C2×C8—2442no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23#1451969211yescharacteristic
261C23#1451969211yescharacteristic
262C23C2×C22⋊Dic6111yescharacteristic
263Q8—44no—
264Q8—44no—
265Q8—48no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—48no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4#1451980812yes—
295C8—4882no—
296C8—4882no—
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#1683954912yes—
306C22#1683956112yes—
307C22C2×#1451969211yescharacteristic
308C22#1683977111yescharacteristic
309C22#1683977911yescharacteristic
310C4—22no—
311C4—44no—
312C4—41no—
313C4—41no—
314C4—44no—
315C4—41no—
316C4—41no—
317C4—41no—
318C2—21no—
319C2—21no—
320C2—22no—
321C2—22no—
322C2#2144435211yescharacteristic
323C2C2×#1683954911yescharacteristic
324C2C2×#1683956111yescharacteristic
subgroupG/H
centerC22C2×#14519692
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC2×C61×#395C2
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC2×C4⋊C4×C61C23
hypercenterC2×#395D61
hypocenterC61#11815
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
3. #21443849C2semidirect
182. C61#11815semidirect

Maximal subgroups

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

subgroupindex
#214458992
#214441492
#214438492
C2×C61×#3952
#1181561