Finite Group Databasebeta

← #29689464

Subgroups — #29689464

Order 15616.

1130 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840388C211yesmaximal, characteristic
2#21448417C211yesmaximal, characteristic
3#21446804C214yesmaximal, complemented
4#21446803C214yesmaximal, complemented
5#21445891C214yesmaximal
6C61×#2312C211yesmaximal, characteristic
7#16840388—24no—
8#16840388—24no—
9C2×#14519756C2212yes—
10C2×#14519756C2212yes—
11C2×#14519754C414yescomplemented
12C2×#14519754C414yescomplemented
13C22×C22⋊Dic61C2212yes—
14#16839655—24no—
15#16839622C2212yes—
16#16839618C2212yes—
17C4×C22⋊Dic61C2212yes—
18#16839528C22116yes—
19C22×C22⋊C4×C61C2211yescharacteristic
20C61×#388—24no—
21C61×#380C2212yes—
22C61×#379C2212yes—
23C4×C22⋊C4×C61C2212yes—
24C22×Dic122C2×C412yes—
25C22×Dic122C2×C412yes—
26C23×Dic61C2×C412yes—
27C23×Dic61C2×C412yes—
28#14519756—24no—
29#14519756—24no—
30#14519756—24no—
31#14519756—24no—
32#14519754—216no—
33#14519754—216no—
34C2×C4⋊Dic61—22no—
35C2×C4⋊Dic61C2×C412yes—
36C2×C4⋊Dic61C2×C412yes—
37C2×#11758196C2×C414yes—
38C2×#11758196C2×C414yes—
39C2×C22⋊Dic61—24no—
40C2×C22⋊Dic61—24no—
41C2×C22⋊Dic61—22no—
42C2×C22⋊Dic61—22no—
43C2×C22⋊Dic61C2×C414yes—
44C2×C22⋊Dic61C2314yes—
45C2×C22⋊Dic61C2314yes—
46C2×C4×Dic61—22no—
47#14519684—22no—
48#14519684—22no—
49C25×C61C2311yescharacteristic
50C23×C4×C61C2311yescharacteristic
51C23×C4×C61C2311yescharacteristic
52C2×C4⋊C4×C61—22no—
53C2×C22⋊C4×C61—22no—
54C2×C22⋊C4×C61—22no—
55C2×C22⋊C4×C61C2×C414yes—
56C2×C22⋊C4×C61D414yes—
57C2×C22⋊C4×C61D414yes—
58C2×C22⋊C4×C61C2314yes—
59C2×C42×C61—22no—
60#95×C61—22no—
61#95×C61—22no—
62C24×C61—24no—
63C24×C61—81no—
64C24×C61—81no—
65C24×C61C22×C412yes—
66C24×C61C2×D412yes—
67C24×C61C2×D412yes—
68C24×C61C2411yescharacteristic
69C22×C4×C61—24no—
70C22×C4×C61—81no—
71C22×C4×C61—81no—
72C22×C4×C61C22×C412yes—
73C22×C4×C61C22×C412yes—
74C22×C4×C61C2×D412yes—
75C22×C4×C61C2×D412yes—
76C22×C4×C61C2×D412yes—
77C22×C4×C61C2×D412yes—
78C22×Dic61—24no—
79C22×Dic61—24no—
80C22×Dic61—24no—
81C22×Dic61—82no—
82C22×Dic61—82no—
83C22×Dic61C22×C414yes—
84C22×Dic61C22×C414yes—
85C2×Dic122—24no—
86C2×Dic122—24no—
87C2×Dic122—82no—
88C2×Dic122—82no—
89C4⋊C4×C61—82no—
90C22⋊C4×C61—28no—
91C22⋊C4×C61—82no—
92C22⋊C4×C61—82no—
93C22⋊C4×C61C22⋊C4116yes—
94C42×C61—82no—
95C22⋊Dic61—28no—
96C22⋊Dic61—24no—
97C22⋊Dic61—24no—
98C22⋊Dic61—28no—
99C22⋊Dic61—82no—
100C22⋊Dic61—82no—
101C4⋊Dic61—24no—
102C4⋊Dic61—24no—
103C4⋊Dic61—82no—
104#11758196—28no—
105#11758196—28no—
106C4×Dic61—82no—
107C23×C61—24no—
108C23×C61—24no—
109C23×C61—24no—
110C23×C61—42no—
111C23×C61—42no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—42no—
115C23×C61—42no—
116C23×C61—42no—
117C23×C61—42no—
118C23×C61—88no—
119C23×C61C2×C22⋊C414yes—
120C23×C61C23×C411yescharacteristic
121C23×C61C22×D411yescharacteristic
122C23×C61C22×D411yescharacteristic
123C23×C612−1+414yes—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—42no—
129C2×C4×C61—42no—
130C2×C4×C61—44no—
131C2×C4×C61—44no—
132C2×C4×C61—44no—
133C2×C4×C61C2×C22⋊C414yes—
134C2×C4×C61C2×C22⋊C414yes—
135C2×Dic61—24no—
136C2×Dic61—24no—
137C2×Dic61—44no—
138C2×Dic61—44no—
139C2×Dic61—44no—
140C2×Dic61—44no—
141C2×Dic61—44no—
142C2×Dic61—84no—
143C2×Dic61—84no—
144Dic122—84no—
145Dic122—84no—
146#21827—611nomaximal, Sylow
147C22×C61—21no—
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—24no—
154C22×C61—24no—
155C22×C61—24no—
156C22×C61—44no—
157C22×C61—48no—
158C22×C61—84no—
159C22×C61—84no—
160C22×C61C22×C22⋊C411yescharacteristic
161C22×C61#51912yes—
162C22×C61#53612yes—
163C22×C61#53612yes—
164C4×C61—24no—
165C4×C61—24no—
166C4×C61—84no—
167Dic61—44no—
168Dic61—44no—
169Dic61—84no—
170C2×#543—611no—
171#2312—611no—
172#2312D6111yescharacteristic
173#1996—614no—
174#1996—614no—
175#1819—614no—
176C2×C61—22no—
177C2×C61—22no—
178C2×C61—22no—
179C2×C61—44no—
180C2×C61#231011yescharacteristic
181C2×C61#288311yescharacteristic
182C2×C61#288311yescharacteristic
183#543—1224no—
184#543—1224no—
185C2×C22⋊Q8—612no—
186C2×C22⋊Q8—612no—
187C2×C22⋊Q8—614no—
188C2×C22⋊Q8—614no—
189C22×C22⋊C4—612no—
190C22×C22⋊C4C2×D6111yescharacteristic
191#388—24no—
192#388—1224no—
193#380—612no—
194#380C2×D6112yes—
195#379—612no—
196#379C2×D6112yes—
197C4×C22⋊C4—612no—
198C4×C22⋊C4C2×D6112yes—
199#342—6116no—
200C61#2182711yescharacteristic, complemented, Sylow
201C25C22×D6111yescharacteristic
202C22×Q8—612no—
203C22×Q8—612no—
204C23×C4—612no—
205C23×C4—612no—
206C23×C4C22×D6111yescharacteristic
207C23×C4C22×D6111yescharacteristic
208C22⋊Q8—1224no—
209C22⋊Q8—1224no—
210C22⋊Q8—1224no—
211C22⋊Q8—1224no—
212C22⋊Q8—12216no—
213C22⋊Q8—12216no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—614no—
216C2×C4⋊C4—614no—
217C2×C4⋊C4—612no—
218C2×C4⋊C4—612no—
219C2×C4⋊C4—1222no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—614no—
223C2×C22⋊C4—614no—
224C2×C22⋊C4—614no—
225C2×C22⋊C4—1224no—
226C2×C22⋊C4—1224no—
227C2×C22⋊C4—1222no—
228C2×C22⋊C4—1222no—
229C2×C22⋊C4C4×D6114yes—
230C2×C22⋊C4D24414yes—
231C2×C22⋊C4C22⋊D6114yes—
232C2×C22⋊C4C22×D6114yes—
233C2×C42—22no—
234C2×C42—1222no—
235#95—22no—
236#95—22no—
237#95—1222no—
238#95—1222no—
239C24—24no—
240C24—81no—
241C24—81no—
242C24C2×C4×D6112yes—
243C24C2×D24412yes—
244C24C2×C22⋊D6112yes—
245C24C23×D6111yescharacteristic
246C2×Q8—1224no—
247C2×Q8—1224no—
248C2×Q8—4882no—
249C2×Q8—4882no—
250C22×C4—24no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—614no—
254C22×C4—614no—
255C22×C4—1224no—
256C22×C4—1224no—
257C22×C4—1224no—
258C22×C4—4882no—
259C22×C4—4882no—
260C22×C4C2×C4×D6112yes—
261C22×C4C2×C4×D6112yes—
262C22×C4C2×D24412yes—
263C22×C4C2×D24412yes—
264C22×C4C2×C22⋊D6112yes—
265C22×C4C2×C22⋊D6112yes—
266C4⋊C4—82no—
267C4⋊C4—1228no—
268C4⋊C4—1228no—
269C4⋊C4—1224no—
270C4⋊C4—1224no—
271C4⋊C4—4882no—
272C22⋊C4—28no—
273C22⋊C4—82no—
274C22⋊C4—82no—
275C22⋊C4—1224no—
276C22⋊C4—1224no—
277C22⋊C4—1228no—
278C22⋊C4—1228no—
279C22⋊C4—4882no—
280C22⋊C4—4882no—
281C22⋊C4#11758198116yes—
282C42—82no—
283C42—4882no—
284C23—24no—
285C23—24no—
286C23—24no—
287C23—41no—
288C23—41no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—88no—
296C23C2×#1175819814yes—
297C23C22×C4×D6111yescharacteristic
298C23C22×C22⋊D6111yescharacteristic
299C23C22×D24411yescharacteristic
300C23#1451982414yes—
301Q8—4884no—
302Q8—4884no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—1224no—
313C2×C4—1224no—
314C2×C4—4884no—
315C2×C4—4884no—
316C2×C4—2444no—
317C2×C4—2444no—
318C2×C4—2444no—
319C2×C4—2444no—
320C2×C4—2444no—
321C2×C4C2×#1175819814yes—
322C2×C4C2×#1175819814yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—24no—
330C22—24no—
331C22—24no—
332C22—44no—
333C22—48no—
334C22—84no—
335C22—84no—
336C22C22×#1175819811yescharacteristic
337C22#1684023912yes—
338C22#1684033312yes—
339C22#1684033812yes—
340C4—24no—
341C4—24no—
342C4—84no—
343C4—4884no—
344C4—2444no—
345C4—2444no—
346C2—22no—
347C2—22no—
348C2—22no—
349C2—44no—
350C2#2144840311yescharacteristic
351C2#2145427911yescharacteristic
352C2#2145427911yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#2312C2
socleC22×C61C22×C22⋊C4
prime radicalC25×C61C23
cosocleC24×C61C24
hypercenter#2312D61
hypocenterC61#21827
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C22⋊C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
3. #21446804C2semidirect
4. #21446803C2semidirect
11. C2×#14519754C4semidirect
12. C2×#14519754C4semidirect
200. C61#21827semidirect

Maximal subgroups

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

subgroupindex
C2×#168403882
#214484172
#214468042
#214468032
#214458912
C61×#23122
#2182761