Finite Group Databasebeta

← #29576940

Subgroups — #29576940

Order 15616.

444 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 93 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#21455945C211yesmaximal, characteristic
2#21448471C211yesmaximal, characteristic
3#21446764C211yesmaximal, characteristic
4#21444265C211yesmaximal, characteristic
5#21443712C211yesmaximal, characteristic
6#21443674C211yesmaximal, characteristic
7C61×#1506C211yesmaximal, characteristic
8C2×C4×Dic122C2211yescharacteristic
9#16840090—21no—
10#16840044C412yescomplemented
11#16839969C412yescomplemented
12#16839917—21no—
13#16839867—22no—
14#16839819C2211yescharacteristic
15C4×#11758194C2211yescharacteristic
16#16839668—21no—
17#16839660—21no—
18#16839644—22no—
19#16839629—21no—
20#16839626—21no—
21C4×C4⋊Dic61C2211yescharacteristic
22#16839521—22no—
23#16839492—22no—
24C61×#533C2211yescharacteristic
25C61×#432C2211yescharacteristic
26C61×#376C2211yescharacteristic
27C61×#328—21no—
28C61×#328—21no—
29C22×Dic122D411yescharacteristic
30C2×#11758202—22no—
31C2×#11758194—21no—
32C4×Dic122—21no—
33C4×Dic122—21no—
34C4×Dic122C2×C412yes—
35C4×Dic122D412yes—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61D411yescharacteristic
38C2×#11758196—21no—
39C2×#11758196—22no—
40C2×C4×Dic61—21no—
41C2×C4×Dic61D412yes—
42#14519708—22no—
43C4⋊#92418C2×C412yes—
44#14519701—22no—
45#14519700—22no—
46C22⋊#92418—21no—
47#14519688—21no—
48#14519688—21no—
49C4×#92418—21no—
50#14519684—22no—
51#14519684—22no—
52#14519684—22no—
53#14519684—21no—
54C4⋊Q8×C61C2×C411yescharacteristic
55C4⋊Q8×C61C2×C411yescharacteristic
56#125×C61—21no—
57C22⋊Q8×C61—21no—
58C22⋊Q8×C61—21no—
59C4×Q8×C61—21no—
60C4×Q8×C61—21no—
61#117×C61—21no—
62C2×C42×C61—21no—
63C2×C42×C61C2311yescharacteristic
64C4⋊C8×C61—21no—
65C22⋊C8×C61—21no—
66#97×C61—21no—
67#95×C61—21no—
68#95×C61D411yescharacteristic
69#95×C61D411yescharacteristic
70C2×Q8×C61—21no—
71C2×Q8×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—22no—
80C2×Dic122—21no—
81C2×Dic122—42no—
82C2×Dic122C22⋊C412yes—
83C2×C8×C61—41no—
84C4⋊C4×C61—21no—
85C4⋊C4×C61—21no—
86C4⋊C4×C61—21no—
87C4⋊C4×C61—21no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C22⋊C4×C61—41no—
91C42×C61—42no—
92C42×C61—41no—
93C42×C61C22×C411yescharacteristic
94C42×C61C2×D411yescharacteristic
95C42×C61#4111yescharacteristic
96C42×C61#4111yescharacteristic
97#11758202—44no—
98C4⋊Dic61—21no—
99C4⋊Dic61—42no—
100C4⋊Dic61C22⋊C412yes—
101#11758196—22no—
102#11758196—22no—
103#11758196—82no—
104C4×Dic61—22no—
105C4×Dic61—22no—
106C4×Dic61—42no—
107#11758194—22no—
108#11758194—41no—
109C2×#92418—22no—
110C2×#92418—41no—
111C23×C61#12011yescharacteristic
112Q8×C61—42no—
113Q8×C61—81no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—22no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61C2×C22⋊C411yescharacteristic
127C2×C4×C61C4×D411yescharacteristic
128C2×C4×C61C4×D411yescharacteristic
129C2×C4×C61#12111yescharacteristic
130C2×C4×C61#12111yescharacteristic
131C2×C4×C61#12311yescharacteristic
132C2×Dic61—22no—
133C2×Dic61—22no—
134C2×Dic61—42no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—42no—
138C2×Dic61—42no—
139C2×Dic61—42no—
140Dic122—22no—
141Dic122—22no—
142Dic122—42no—
143C8×C61—81no—
144#92418—42no—
145#92418—42no—
146#9328—611nomaximal, Sylow
147C22×C61—21no—
148C22×C61—21no—
149C22×C61#38611yescharacteristic
150C22×C61#45511yescharacteristic
151C22×C61#45611yescharacteristic
152C4×C61—21no—
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—81no—
158C4×C61—81no—
159C4×C61—81no—
160C4×C61—81no—
161C4×C61#42112yes—
162Dic61—42no—
163Dic61—42no—
164Dic61—82no—
165#3113—611no—
166#2320—611no—
167#1986—611no—
168#1506D6111yescharacteristic
169#1491—611no—
170#1300—611no—
171#1285—611no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61#188211yescharacteristic
175C2×C61#189711yescharacteristic
176C2×C61#190511yescharacteristic
177#533C2×D6111yescharacteristic
178C2×C4×Q8—611no—
179#483—1221no—
180#475—612no—
181#462—612no—
182#451—1221no—
183#441—1222no—
184#432—611no—
185#432C2×D6111yescharacteristic
186C4×M16—611no—
187#391—1221no—
188#389—1221no—
189#387—1222no—
190#383—1221no—
191#382—1221no—
192C4×C4⋊C4—611no—
193#376C2×D6111yescharacteristic
194#339—1222no—
195#328—21no—
196#328—21no—
197Q8⋊C8—1222no—
198C61#932811yescharacteristic, complemented, Sylow
199C22×Q8—611no—
200C2×Q16—1222no—
201C2×M16—1221no—
202C4⋊Q8C4×D6111yescharacteristic
203C4⋊Q8C4×D6111yescharacteristic
204#125—21no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—612no—
210C4×Q8—612no—
211C4×Q8—1221no—
212C4×Q8—1221no—
213#117—21no—
214C2×C4⋊C4—611no—
215C2×C4⋊C4—1221no—
216C2×C4⋊C4—1222no—
217C2×C4⋊C4—1221no—
218C2×C42—21no—
219C2×C42—612no—
220C2×C42—1221no—
221C2×C42C22×D6111yescharacteristic
222#106—1222no—
223C4⋊C8—21no—
224C4⋊C8—612no—
225#103—1222no—
226#103—1222no—
227C22⋊C8—21no—
228C22⋊C8—1221no—
229#97—21no—
230#97—1221no—
231#97—1221no—
232C4×C8—1221no—
233#95—21no—
234#95—1222no—
235#95—1222no—
236#95—1222no—
237#95—1221no—
238#95C22⋊D6111yescharacteristic
239#95C22⋊D6111yescharacteristic
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—612no—
243C2×Q8—1221no—
244C2×Q8—2442no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4—1222no—
250C22×C4D4×D6111yescharacteristic
251C22×C4D4×D6111yescharacteristic
252C22×C4C2×C22⋊D6111yescharacteristic
253Q16—2444no—
254M16—1222no—
255M16—2441no—
256C2×C8—41no—
257C2×C8—1222no—
258C2×C8—2441no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—612no—
266C4⋊C4—1222no—
267C4⋊C4—1222no—
268C4⋊C4—1221no—
269C4⋊C4—4882no—
270C4⋊C4—2442no—
271C22⋊C4—41no—
272C42—42no—
273C42—41no—
274C42—1222no—
275C42—1222no—
276C42—2442no—
277C42C2×C4×D6111yescharacteristic
278C42#1175822211yescharacteristic
279C42D4×D6111yescharacteristic
280C42#1175822411yescharacteristic
281C23#1451974511yescharacteristic
282Q8—42no—
283Q8—81no—
284Q8—1222no—
285Q8—1222no—
286Q8—2442no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—1222no—
300C2×C4—1222no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4C22⋊C4×D6111yescharacteristic
308C2×C4#1451973711yescharacteristic
309C2×C4C4×C22⋊D6111yescharacteristic
310C2×C4#1451974811yescharacteristic
311C2×C4#1451975011yescharacteristic
312C2×C4#1451975911yescharacteristic
313C8—81no—
314C8—2442no—
315C8—2442no—
316C22—21no—
317C22—21no—
318C22#1683963811yescharacteristic
319C22#1683993811yescharacteristic
320C22#1683994311yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4—4882no—
331C4—2442no—
332C4—2442no—
333C4#1683977712yes—
334C2—21no—
335C2—21no—
336C2#2144618511yescharacteristic
337C2#2144626411yescharacteristic
338C2#2144631611yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1506C2
socleC22×C61#386
prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#1506D61
hypocenterC61#9328
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
10. #16840044C4semidirect
11. #16839969C4semidirect
198. C61#9328semidirect

Maximal subgroups

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

subgroupindex
#214559452
#214484712
#214467642
#214442652
#214437122
#214436742
C61×#15062
#932861