Finite Group Databasebeta

← #29577967

Subgroups — #29577967

Order 15616.

682 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 107 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#21449184C211yesmaximal, characteristic
2#21445742C212yesmaximal, complemented
3#21444157C211yesmaximal, characteristic
4#21443724C212yesmaximal, complemented
5C61×#2975C211yesmaximal, characteristic
6C23×Dic122C2211yescharacteristic
7C2×#14519756—21no—
8C2×#14519695—24no—
9#16839726C2212yes—
10#16839682—21no—
11#16839674—21no—
12#16839668—21no—
13#16839642C412yes—
14#16839634C412yes—
15#16839622C2211yescharacteristic
16#16839488—21no—
17#16839486—21no—
18C2×C4×D4×C61C2211yescharacteristic
19C61×#436—24no—
20C61×#431—21no—
21C61×#407—21no—
22C61×#407C2212yes—
23C22×Dic122—22no—
24C22×Dic122—21no—
25C22×Dic122—21no—
26C22×Dic122—22no—
27C22×Dic122—44no—
28C23×Dic61C2×C412yes—
29C2×#11758194—22no—
30#14519756—44no—
31C2×C4⋊Dic61—21no—
32C2×#11758196—22no—
33C2×C22⋊Dic61—22no—
34C2×C22⋊Dic61C2×C412yes—
35#14519695—48no—
36C22⋊#92418—22no—
37C22⋊#92418—22no—
38#14519684—22no—
39#14519684—22no—
40#14519684—21no—
41C22×D4×C61C2×C411yescharacteristic
42C23×C4×C61D411yescharacteristic
43C23×C4×C61C2311yescharacteristic
44C2×M16×C61—22no—
45C2×M16×C61D414yes—
46C4×D4×C61—22no—
47C4×D4×C61—22no—
48C2×C4⋊C4×C61D411yescharacteristic
49C2×C22⋊C4×C61C2×C411yescharacteristic
50C2×C22⋊C4×C61D411yescharacteristic
51C2×C42×C61D411yescharacteristic
52C4⋊C8×C61—22no—
53C22⋊C8×C61—22no—
54C22⋊C8×C61—22no—
55C22⋊C8×C61—22no—
56C4×C8×C61—22no—
57C24×C61C22⋊C411yescharacteristic
58C24×C61C22×C411yescharacteristic
59C2×D4×C61—22no—
60C2×D4×C61—22no—
61C2×D4×C61—41no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—42no—
65C22×C4×C61—42no—
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C2×D412yes—
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61#4112yes—
73C22×Dic61—22no—
74C22×Dic61—24no—
75C22×Dic61—22no—
76C22×Dic61—44no—
77C2×Dic122—28no—
78C2×Dic122—24no—
79C2×Dic122—44no—
80C2×Dic122—48no—
81C2×Dic122—44no—
82C2×Dic122—42no—
83C2×Dic122—41no—
84C2×Dic122—42no—
85C2×Dic122—41no—
86M16×C61—44no—
87M16×C61—82no—
88C2×C8×C61—24no—
89C4⋊C4×C61—22no—
90C22⋊C4×C61—22no—
91C22⋊C4×C61—22no—
92C42×C61—21no—
93C42×C61—21no—
94C22⋊Dic61—44no—
95C22⋊Dic61—42no—
96C4⋊Dic61—42no—
97#11758196—44no—
98#11758194—44no—
99C2×#92418—42no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61—41no—
108C23×C61—41no—
109C23×C61C2×C22⋊C411yescharacteristic
110C23×C61C4×D412yes—
111D4×C61—44no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—24no—
115C2×C4×C61—22no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—41no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—81no—
124C2×C4×C61#10112yescomplemented
125C2×C4×C61#12011yescharacteristic
126C2×C4×C61#12112yes—
127C2×C4×C61#12311yescharacteristic
128C2×Dic61—24no—
129C2×Dic61—28no—
130C2×Dic61—42no—
131C2×Dic61—44no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61—42no—
136Dic122—48no—
137Dic122—44no—
138Dic122—44no—
139C8×C61—44no—
140#92418—82no—
141#9506—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—22no—
145C22×C61—22no—
146C22×C61—22no—
147C22×C61—42no—
148C22×C61—42no—
149C22×C61—42no—
150C22×C61—41no—
151C22×C61—41no—
152C22×C61—41no—
153C22×C61—81no—
154C22×C61#38611yescharacteristic
155C22×C61C2×#10111yescharacteristic
156C22×C61#41311yescharacteristic
157C4×C61—22no—
158C4×C61—21no—
159C4×C61—22no—
160C4×C61—41no—
161C4×C61—41no—
162C4×C61B(4,𝔽2)12yes—
163Dic61—44no—
164Dic61—82no—
165#2975D6111yescharacteristic
166#2409—611no—
167#1784—612no—
168#1465—611no—
169#1305—612no—
170C2×C61—21no—
171C2×C61—21no—
172C2×C61—42no—
173C2×C61—81no—
174C2×C61#188911yescharacteristic
175C2×C61#190111yescharacteristic
176C2×C61#191511yescharacteristic
177C23×Q8—611no—
178C2×C22⋊Q8—1221no—
179C2×C4×D4C2×D6111yescharacteristic
180#436—24no—
181#431—21no—
182C2×#101—1224no—
183#407—21no—
184#407—612no—
185#407C2×D6112yes—
186#395—1221no—
187#393—1221no—
188#391—1221no—
189#386—612no—
190#385—612no—
191#380—611no—
192#324—1221no—
193#323—1221no—
194C61#950611yescharacteristic, complemented, Sylow
195C22×Q8—1221no—
196C22×Q8—1222no—
197C22×Q8—1221no—
198C22×Q8—1222no—
199C22×Q8—2444no—
200C22×D4C4×D6111yescharacteristic
201C23×C4—612no—
202C23×C4C22⋊D6111yescharacteristic
203C23×C4C22×D6111yescharacteristic
204C2×M16—22no—
205C2×M16—1222no—
206C2×M16C22⋊D6114yes—
207C22⋊Q8—2444no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—1222no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4D24411yescharacteristic
213C2×C22⋊C4—612no—
214C2×C22⋊C4—1222no—
215C2×C22⋊C4C4×D6111yescharacteristic
216C2×C22⋊C4D24411yescharacteristic
217C2×C42C22⋊D6111yescharacteristic
218C4⋊C8—22no—
219#101—2448no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—1222no—
224C22⋊C8—1222no—
225C4×C8—22no—
226#95—1222no—
227#95—1222no—
228#95—1221no—
229C24#1175819811yescharacteristic
230C24C2×C4×D6111yescharacteristic
231C2×Q8—1228no—
232C2×Q8—1224no—
233C2×Q8—2442no—
234C2×Q8—2444no—
235C2×Q8—2444no—
236C2×Q8—2448no—
237C2×Q8—2441no—
238C2×Q8—2442no—
239C2×Q8—2441no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—1222no—
248C22×C4—1224no—
249C22×C4—1222no—
250C22×C4—2444no—
251C22×C4#1175819811yescharacteristic
252C22×C4#1175819811yescharacteristic
253C22×C4#1175819811yescharacteristic
254C22×C4C2×D24411yescharacteristic
255C22×C4#1175822212yes—
256C22×C4C2×C22⋊D6112yes—
257C22×C4C2×C22⋊D6111yescharacteristic
258M16—44no—
259M16—82no—
260M16—2444no—
261C2×C8—24no—
262C2×C8—2442no—
263C4⋊C4—22no—
264C4⋊C4—2442no—
265C4⋊C4—2444no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—2444no—
269C22⋊C4—2442no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23C2×#1175819811yescharacteristic
282C23C4×C22⋊D6112yes—
283Q8—2448no—
284Q8—2444no—
285Q8—2444no—
286D4—44no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—1224no—
300C2×C4—1228no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4—2444no—
306C2×C4—2442no—
307C2×C4#1451969512yescomplemented
308C2×C4#1451974611yescharacteristic
309C2×C4#1451975112yes—
310C2×C4#1451976011yescharacteristic
311C8—44no—
312C8—4882no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22#1683964111yescharacteristic
326C22C2×#1451969511yescharacteristic
327C22#1683974411yescharacteristic
328C4—22no—
329C4—21no—
330C4—22no—
331C4—41no—
332C4—41no—
333C4—4882no—
334C4—2444no—
335C4#1683994912yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—81no—
340C2#2144622711yescharacteristic
341C2#2144628711yescharacteristic
342C2#2144637111yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2975C2
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC23×C4×C61C23
hypercenter#2975D61
hypocenterC61#9506
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #21445742C2semidirect
4. #21443724C2semidirect
124. C2×C4×C61#101semidirect
194. C61#9506semidirect
307. C2×C4#14519695semidirect

Maximal subgroups

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

subgroupindex
#214491842
#214457422
#214441572
#214437242
C61×#29752
#950661