Finite Group Databasebeta

← #29571342

Subgroups — #29571342

Order 15616.

1502 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1502 conjugacy classes of subgroups — too many to draw. Only its 207 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
1C23×#11758196C211yesmaximal, characteristic
2#21445729C211yesmaximal, characteristic
3C2×#16839487C214yesmaximal, complemented
4C61×#1779C211yesmaximal, characteristic
5C24×Dic61C411yescharacteristic
6C24×Dic61C411yescharacteristic
7C22×#11758196—28no—
8C22×#11758196—22no—
9C22×#11758196—22no—
10C22×#11758196C412yes—
11C22×#11758196C2212yes—
12C2×C22⋊#92418—22no—
13C2×C22⋊#92418C2212yes—
14#16839487—28no—
15C24×C4×C61C2211yescharacteristic
16C2×C22⋊C8×C61—22no—
17C2×C22⋊C8×C61C2212yes—
18C23×Dic61—22no—
19C23×Dic61—22no—
20C23×Dic61—24no—
21C23×Dic61—24no—
22C23×Dic61—22no—
23C23×Dic61—24no—
24C23×Dic61—44no—
25C23×Dic61C814yescomplemented
26C23×Dic61C2×C412yes—
27C23×Dic61C2×C412yes—
28C22×#92418—22no—
29C2×#11758196—216no—
30C2×#11758196—216no—
31C2×#11758196—24no—
32C2×#11758196—22no—
33C2×#11758196—24no—
34C2×#11758196—22no—
35C2×#11758196—24no—
36C2×#11758196—24no—
37C2×#11758196D418yes—
38C22⋊#92418—24no—
39C22⋊#92418—44no—
40C25×C61C2×C411yescharacteristic
41C23×C4×C61—24no—
42C23×C4×C61—44no—
43C23×C4×C61C2×C411yescharacteristic
44C23×C4×C61D412yes—
45C23×C4×C61D412yes—
46C23×C4×C61C2311yescharacteristic
47C22×C8×C61—22no—
48C22⋊C8×C61—24no—
49C22⋊C8×C61—44no—
50C24×C61—24no—
51C24×C61—24no—
52C24×C61—22no—
53C24×C61—22no—
54C24×C61C22⋊C412yes—
55C24×C61C2×C812yes—
56C24×C61M1612yes—
57C24×C61C22×C411yescharacteristic
58C22×C4×C61—24no—
59C22×C4×C61—24no—
60C22×C4×C61—22no—
61C22×C4×C61—24no—
62C22×C4×C61—24no—
63C22×C4×C61—42no—
64C22×C4×C61—42no—
65C22×C4×C61—42no—
66C22×C4×C61—42no—
67C22×C4×C61—416no—
68C22×C4×C61C22⋊C412yes—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D414yes—
72C22×Dic61—216no—
73C22×Dic61—28no—
74C22×Dic61—28no—
75C22×Dic61—28no—
76C22×Dic61—22no—
77C22×Dic61—22no—
78C22×Dic61—22no—
79C22×Dic61—22no—
80C22×Dic61—216no—
81C22×Dic61—28no—
82C22×Dic61—22no—
83C22×Dic61—22no—
84C22×Dic61—416no—
85C22×Dic61—44no—
86C22×Dic61—44no—
87C22×Dic61—44no—
88C22×Dic61C22⋊C414yes—
89C22×Dic61C22⋊C414yes—
90C22×Dic61C2×C814yes—
91C22×Dic61M1614yes—
92C2×C8×C61—44no—
93C2×C8×C61—42no—
94#11758196—216no—
95#11758196—28no—
96#11758196—28no—
97C2×#92418—44no—
98C2×#92418—42no—
99C23×C61—28no—
100C23×C61—216no—
101C23×C61—28no—
102C23×C61—28no—
103C23×C61—28no—
104C23×C61—22no—
105C23×C61—24no—
106C23×C61—22no—
107C23×C61—24no—
108C23×C61—22no—
109C23×C61—22no—
110C23×C61—22no—
111C23×C61—22no—
112C23×C61C22⋊C814yes—
113C23×C61#9914yes—
114C23×C61#10114yes—
115C23×C61C2×C22⋊C412yes—
116C23×C61C2×C22⋊C411yescharacteristic
117C23×C61C22×C811yescharacteristic
118C23×C61C2×M1612yes—
119C23×C61C2×M1611yescharacteristic
120C2×C4×C61—22no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—22no—
125C2×C4×C61—44no—
126C2×C4×C61—48no—
127C2×C4×C61—48no—
128C2×C4×C61—44no—
129C2×C4×C61#12012yes—
130C2×C4×C61#12012yes—
131C2×Dic61—232no—
132C2×Dic61—24no—
133C2×Dic61—24no—
134C2×Dic61—24no—
135C2×Dic61—24no—
136C2×Dic61—24no—
137C2×Dic61—24no—
138C2×Dic61—24no—
139C2×Dic61—416no—
140C2×Dic61—44no—
141C2×Dic61—44no—
142C2×Dic61—44no—
143C2×Dic61C22⋊C818yescomplemented
144C8×C61—82no—
145#92418—82no—
146#8299—611nomaximal, Sylow
147C22×C61—216no—
148C22×C61—216no—
149C22×C61—24no—
150C22×C61—22no—
151C22×C61—22no—
152C22×C61—24no—
153C22×C61—24no—
154C22×C61—24no—
155C22×C61—24no—
156C22×C61—24no—
157C22×C61—22no—
158C22×C61—24no—
159C22×C61—22no—
160C22×C61#32418yescomplemented
161C22×C61#37911yescharacteristic
162C22×C61C2×C22⋊C812yes—
163C22×C61C2×C22⋊C811yescharacteristic
164C22×C61#40712yes—
165C22×C61#40711yescharacteristic
166C22×C61C2×#9912yes—
167C22×C61C2×#10112yes—
168C4×C61—22no—
169C4×C61—22no—
170C4×C61—42no—
171Dic61—28no—
172Dic61—44no—
173C23×C4⋊C4—611no—
174#1779—611no—
175#1779D6111yescharacteristic
176C2×#324—614no—
177C2×C61—28no—
178C2×C61—22no—
179C2×C61—22no—
180C2×C61C2×#32412yes—
181C2×C61#146312yes—
182C2×C61#177911yescharacteristic
183C2×C61#178111yescharacteristic
184C2×C61#178411yescharacteristic
185C24×C4—611no—
186C24×C4—611no—
187C24×C4C2×D6111yescharacteristic
188C22×C4⋊C4—612no—
189C22×C4⋊C4—612no—
190C22×C4⋊C4—1228no—
191C22×C4⋊C4—1222no—
192C22×C4⋊C4—1222no—
193C2×C22⋊C8—22no—
194C2×C22⋊C8—612no—
195C2×C22⋊C8—1222no—
196C2×C22⋊C8C2×D6112yes—
197#324—1228no—
198C61#829911yescharacteristic, complemented, Sylow
199C25C4×D6111yescharacteristic
200C23×C4—24no—
201C23×C4—44no—
202C23×C4—612no—
203C23×C4—614no—
204C23×C4—612no—
205C23×C4—1224no—
206C23×C4—1224no—
207C23×C4—1224no—
208C23×C4—1222no—
209C23×C4—1222no—
210C23×C4—1222no—
211C23×C4—2444no—
212C23×C4C4×D6111yescharacteristic
213C23×C4D24412yes—
214C23×C4C22⋊D6112yes—
215C23×C4C22×D6111yescharacteristic
216C22×C8—22no—
217C22×C8—1222no—
218C2×C4⋊C4—618no—
219C2×C4⋊C4—1224no—
220C2×C4⋊C4—1224no—
221C2×C4⋊C4—12216no—
222C2×C4⋊C4—12216no—
223C2×C4⋊C4—1224no—
224C2×C4⋊C4—1222no—
225C2×C4⋊C4—1224no—
226C2×C4⋊C4—1222no—
227C22⋊C8—24no—
228C22⋊C8—44no—
229C22⋊C8—1224no—
230C22⋊C8—2444no—
231C24—24no—
232C24—24no—
233C24—22no—
234C24—22no—
235C24C8×D6112yes—
236C24#1175818912yes—
237C24#1175819812yes—
238C24C2×C4×D6111yescharacteristic
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—416no—
249C22×C4—614no—
250C22×C4—614no—
251C22×C4—614no—
252C22×C4—614no—
253C22×C4—12216no—
254C22×C4—1228no—
255C22×C4—12216no—
256C22×C4—1228no—
257C22×C4—1228no—
258C22×C4—1222no—
259C22×C4—1222no—
260C22×C4—1222no—
261C22×C4—1222no—
262C22×C4—1228no—
263C22×C4—1222no—
264C22×C4—1222no—
265C22×C4—24416no—
266C22×C4—2444no—
267C22×C4—2444no—
268C22×C4—2444no—
269C22×C4#1175819812yes—
270C22×C4C2×D24411yescharacteristic
271C22×C4D4×D6114yes—
272C22×C4C2×C22⋊D6111yescharacteristic
273C2×C8—44no—
274C2×C8—42no—
275C2×C8—2444no—
276C2×C8—2442no—
277C4⋊C4—12216no—
278C4⋊C4—1228no—
279C4⋊C4—1228no—
280C23—28no—
281C23—216no—
282C23—28no—
283C23—24no—
284C23—24no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—28no—
291C23—28no—
292C23—22no—
293C23#1451968914yes—
294C23#1451969114yes—
295C23#1451969514yes—
296C23C22⋊C4×D6112yes—
297C23C2×#1175819811yescharacteristic
298C23C2×C8×D6111yescharacteristic
299C23M16×D6112yes—
300C23C2×#1175818911yescharacteristic
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—44no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—44no—
310C2×C4—618no—
311C2×C4—12232no—
312C2×C4—1224no—
313C2×C4—1224no—
314C2×C4—1224no—
315C2×C4—1224no—
316C2×C4—1224no—
317C2×C4—1224no—
318C2×C4—1224no—
319C2×C4—24416no—
320C2×C4—2444no—
321C2×C4—2444no—
322C2×C4—2444no—
323C2×C4#1451974512yes—
324C2×C4#1451974712yes—
325C8—82no—
326C8—4882no—
327C22—216no—
328C22—216no—
329C22—24no—
330C22—22no—
331C22—22no—
332C22—24no—
333C22—24no—
334C22—24no—
335C22—24no—
336C22—24no—
337C22—22no—
338C22—24no—
339C22—22no—
340C22#1683948718yescomplemented
341C22#1683961711yescharacteristic
342C22C22⋊C8×D6112yes—
343C22C2×#1451968911yescharacteristic
344C22#1683972412yes—
345C22#1683972511yescharacteristic
346C22C2×#1451969112yes—
347C22C2×#1451969512yes—
348C4—22no—
349C4—22no—
350C4—42no—
351C4—1228no—
352C4—2444no—
353C2—22no—
354C2—22no—
355C2—28no—
356C2C2×#1683948712yes—
357C2#2144414812yes—
358C2#2144572811yescharacteristic
359C2#2144573311yescharacteristic
360C2#2144574211yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC23×C61C22×C8
FrattiniC23×C4C22×D61
FittingC61×#1779C2
socleC23×C61C2×C22⋊C4
prime radicalC25×C61C2×C4
cosocleC23×C4×C61C23
hypercenter#1779D61
hypocenterC61#8299
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
3. C2×#16839487C2semidirect
25. C23×Dic61C8semidirect
143. C2×Dic61C22⋊C8semidirect
160. C22×C61#324semidirect
198. C61#8299semidirect
340. C22#16839487semidirect

Maximal subgroups

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

subgroupindex
C23×#117581962
#214457292
C2×#168394872
C61×#17792
#829961