Finite Group Databasebeta

← #29576916

Subgroups — #29576916

Order 15616.

536 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 97 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#21455950C211yesmaximal, characteristic
2#21448182C211yesmaximal, characteristic
3#21446763C211yesmaximal, characteristic
4#21444330C211yesmaximal, characteristic
5#21444267C211yesmaximal, characteristic
6#21443676C211yesmaximal, characteristic
7C61×#1300C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9C2×C4×Dic122—21no—
10C2×C4×Dic122C2211yescharacteristic
11#16840093—21no—
12#16840048C412yescomplemented
13#16839973C412yescomplemented
14#16839920—21no—
15#16839870—22no—
16#16839819C2211yescharacteristic
17C4×C4⋊Dic61—21no—
18C4×C4⋊Dic61C2211yescharacteristic
19#16839613—22no—
20#16839613—21no—
21#16839607—21no—
22#16839523—22no—
23#16839498—21no—
24#16839498—21no—
25#16839494—22no—
26C61×#432C2211yescharacteristic
27C4×M16×C61C2211yescharacteristic
28C61×#376C2211yescharacteristic
29C22×Dic122D411yescharacteristic
30C2×Dic244—22no—
31#14519777C2×C411yescharacteristic
32#14519777C2×C411yescharacteristic
33#14519769—21no—
34#14519756—21no—
35#14519756—21no—
36C4×Dic122—28no—
37C4×Dic122—21no—
38C4×Dic122—21no—
39C4×Dic122—21no—
40C4×Dic122—21no—
41C4×Dic122C2×C412yes—
42C4×Dic122D412yes—
43#14519736—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61D411yescharacteristic
46C2×#11758196—22no—
47C2×#11758196—21no—
48C2×C4×Dic61—22no—
49C2×C4×Dic61—21no—
50C2×C4×Dic61—22no—
51C2×C4×Dic61D412yes—
52#14519709—22no—
53C4⋊#92418—21no—
54#14519702—22no—
55#14519702—22no—
56C22⋊#92418—21no—
57#14519688—21no—
58#14519684—22no—
59#14519684—21no—
60C2×M16×C61—21no—
61C2×C42×C61—21no—
62C2×C42×C61C2311yescharacteristic
63C4⋊C8×C61C2×C412yes—
64C22⋊C8×C61—21no—
65#97×C61—21no—
66#97×C61—21no—
67C4×C8×C61—21no—
68#95×C61—21no—
69#95×C61D411yescharacteristic
70#95×C61D411yescharacteristic
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—22no—
79C2×Dic122—21no—
80C2×Dic122—24no—
81C2×Dic122—21no—
82C2×Dic122—21no—
83C2×Dic122C22⋊C412yes—
84M16×C61—22no—
85M16×C61—41no—
86C2×C8×C61—22no—
87C2×C8×C61—41no—
88C42×C61—24no—
89C42×C61C22×C411yescharacteristic
90C42×C61C2×D411yescharacteristic
91C42×C61#4111yescharacteristic
92C42×C61#4111yescharacteristic
93C22⋊Dic61—41no—
94C4⋊Dic61—24no—
95C4⋊Dic61—21no—
96C4⋊Dic61—21no—
97C4⋊Dic61—21no—
98C4⋊Dic61—21no—
99C4⋊Dic61—21no—
100C4⋊Dic61C22⋊C412yes—
101#11758196—28no—
102#11758196—22no—
103#11758196—22no—
104#11758196—41no—
105#11758196—41no—
106C4×Dic61—28no—
107C4×Dic61—22no—
108C4×Dic61—22no—
109C4×Dic61—41no—
110C4×Dic61—42no—
111C4×Dic61—82no—
112C2×#92418—41no—
113Dic244—44no—
114C23×C61#12011yescharacteristic
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—22no—
118C2×C4×C61—22no—
119C2×C4×C61—22no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61C2×C22⋊C411yescharacteristic
126C2×C4×C61C4×D411yescharacteristic
127C2×C4×C61C4×D411yescharacteristic
128C2×C4×C61#12111yescharacteristic
129C2×C4×C61#12111yescharacteristic
130C2×C4×C61#12311yescharacteristic
131C2×Dic61—24no—
132C2×Dic61—22no—
133C2×Dic61—24no—
134C2×Dic61—22no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141Dic122—24no—
142Dic122—22no—
143Dic122—42no—
144Dic122—81no—
145Dic122#10414yes—
146C8×C61—42no—
147C8×C61—42no—
148#92418—81no—
149#9324—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61#38611yescharacteristic
153C22×C61#45511yescharacteristic
154C22×C61#45611yescharacteristic
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—21no—
160C4×C61—42no—
161C4×C61—42no—
162C4×C61C2×#10412yes—
163Dic61—24no—
164Dic61—44no—
165Dic61—42no—
166Dic61—81no—
167Dic61—81no—
168#3113—611no—
169#2276—611no—
170#1986—611no—
171#1506—611no—
172#1491—611no—
173#1300D6111yescharacteristic
174#1285—611no—
175C2×C61—21no—
176C2×C61—21no—
177C2×C61#188211yescharacteristic
178C2×C61#189611yescharacteristic
179C2×C61#190511yescharacteristic
180#533—611no—
181C2×C4×Q8—611no—
182C2×C4×Q8—1221no—
183#483—1221no—
184#475—612no—
185#462—612no—
186#451—1221no—
187#441—1222no—
188#432—611no—
189#432C2×D6111yescharacteristic
190C4×M16C2×D6111yescharacteristic
191C4×C4⋊C4—611no—
192C4×C4⋊C4—1222no—
193C4×C4⋊C4—1221no—
194C4×C4⋊C4—1221no—
195#376—1221no—
196#376C2×D6111yescharacteristic
197#339—1222no—
198#328—1221no—
199#328—1221no—
200Q8⋊C8—1222no—
201C61#932411yescharacteristic, complemented, Sylow
202C22×Q8—611no—
203C2×Q16—1222no—
204C2×M16—21no—
205C4⋊Q8—611no—
206C4⋊Q8—611no—
207#125—1221no—
208C22⋊Q8—1221no—
209C22⋊Q8—1221no—
210C4×Q8—612no—
211C4×Q8—612no—
212C4×Q8—1228no—
213C4×Q8—1221no—
214C4×Q8—1221no—
215C4×Q8—1221no—
216C4×Q8—1221no—
217#117—1221no—
218C2×C4⋊C4—611no—
219C2×C4⋊C4—1222no—
220C2×C4⋊C4—1221no—
221C2×C4⋊C4—1221no—
222C2×C42—21no—
223C2×C42—612no—
224C2×C42—1222no—
225C2×C42—1221no—
226C2×C42—1222no—
227C2×C42C22×D6111yescharacteristic
228#106—1222no—
229C4⋊C8—1221no—
230C4⋊C8C4×D6112yes—
231#103—1222no—
232#103—1222no—
233C22⋊C8—21no—
234C22⋊C8—1221no—
235#97—21no—
236#97—21no—
237#97—1221no—
238C4×C8—21no—
239#95—21no—
240#95—1222no—
241#95—1221no—
242#95D24411yescharacteristic
243#95D24411yescharacteristic
244C2×Q8—612no—
245C2×Q8—1224no—
246C2×Q8—1221no—
247C2×Q8—1221no—
248C2×Q8—1221no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—1221no—
252C22×C4—1222no—
253C22×C4—1221no—
254C22×C4C2×D24411yescharacteristic
255C22×C4D4×D6111yescharacteristic
256C22×C4D4×D6111yescharacteristic
257Q16—2444no—
258M16—22no—
259M16—41no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—2441no—
263C4⋊C4—612no—
264C4⋊C4—1228no—
265C4⋊C4—1224no—
266C4⋊C4—1222no—
267C4⋊C4—1222no—
268C4⋊C4—1221no—
269C4⋊C4—1221no—
270C4⋊C4—1221no—
271C4⋊C4—1221no—
272C4⋊C4—1221no—
273C4⋊C4—2441no—
274C4⋊C4—2441no—
275C22⋊C4—2441no—
276C42—24no—
277C42—1228no—
278C42—1222no—
279C42—1222no—
280C42—4882no—
281C42—2442no—
282C42—2441no—
283C42C2×C4×D6111yescharacteristic
284C42#1175822211yescharacteristic
285C42D4×D6111yescharacteristic
286C42#1175822611yescharacteristic
287C23#1451974711yescharacteristic
288Q8—614no—
289Q8—1224no—
290Q8—1222no—
291Q8—4881no—
292Q8—2442no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—1224no—
304C2×C4—1222no—
305C2×C4—1224no—
306C2×C4—1222no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2442no—
310C2×C4—2441no—
311C2×C4—2442no—
312C2×C4—2441no—
313C2×C4C22⋊C4×D6111yescharacteristic
314C2×C4#1451973911yescharacteristic
315C2×C4C4×D24411yescharacteristic
316C2×C4#1451975011yescharacteristic
317C2×C4#1451975211yescharacteristic
318C2×C4#1451976211yescharacteristic
319C8—42no—
320C8—42no—
321C8—4881no—
322C22—21no—
323C22—21no—
324C22#1683964011yescharacteristic
325C22#1683994111yescharacteristic
326C22#1683994611yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—42no—
333C4—42no—
334C4—1224no—
335C4—4881no—
336C4—4881no—
337C4—2444no—
338C4—2442no—
339C4#104×D6112yes—
340C2—21no—
341C2—21no—
342C2#2144618911yescharacteristic
343C2#2144626111yescharacteristic
344C2#2144631211yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1300C2
socleC22×C61#386
prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#1300D61
hypocenterC61#9324
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
12. #16840048C4semidirect
13. #16839973C4semidirect
201. C61#9324semidirect

Maximal subgroups

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

subgroupindex
#214559502
#214481822
#214467632
#214443302
#214442672
#214436762
C61×#13002
#932461