Finite Group Databasebeta

← #29576942

Subgroups — #29576942

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#21455950C211yesmaximal, characteristic
2#21448465C211yesmaximal, characteristic
3#21446762C211yesmaximal, characteristic
4#21444332C211yesmaximal, characteristic
5#21444265C211yesmaximal, characteristic
6#21443711C211yesmaximal, characteristic
7C61×#1285C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9C2×C4×Dic122C2211yescharacteristic
10#16840093—21no—
11#16840048C412yescomplemented
12#16839973C412yescomplemented
13#16839920—21no—
14#16839870—22no—
15C4×#11758194C2211yescharacteristic
16#16839667—21no—
17#16839658—21no—
18#16839646—22no—
19#16839629—21no—
20#16839626—21no—
21#16839608C2211yescharacteristic
22#16839522—22no—
23#16839500—21no—
24#16839500—21no—
25#16839492—22no—
26C61×#432C2211yescharacteristic
27C61×#432C2211yescharacteristic
28C4×C4⋊C4×C61C2211yescharacteristic
29C22×Dic122D411yescharacteristic
30C2×Dic244—22no—
31C2×#11758194—21no—
32#14519777C2×C411yescharacteristic
33#14519777C2×C411yescharacteristic
34#14519769—21no—
35#14519756—21no—
36#14519756—21no—
37C4×Dic122—21no—
38C4×Dic122—21no—
39C4×Dic122—21no—
40C4×Dic122—21no—
41C4×Dic122C2×C412yes—
42C4×Dic122D412yes—
43#14519736—21no—
44C2×C4⋊Dic61D411yescharacteristic
45C2×#11758196—21no—
46C2×#11758196—22no—
47C2×C4×Dic61—21no—
48C2×C4×Dic61—21no—
49#14519709—22no—
50#14519702—22no—
51#14519702—22no—
52#14519688—21no—
53C4×#92418—21no—
54#14519684—22no—
55#14519684—22no—
56#14519684—22no—
57#14519684—21no—
58#14519684D411yescharacteristic
59#14519684D411yescharacteristic
60C2×C4⋊C4×C61—21no—
61C2×C42×C61D412yes—
62C2×C42×C61C2311yescharacteristic
63C4⋊C8×C61—21no—
64C4⋊C8×C61C2×C412yes—
65C22⋊C8×C61—21no—
66C22⋊C8×C61—21no—
67#97×C61—21no—
68#97×C61—21no—
69#95×C61—21no—
70C22×C4×C61—22no—
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×Dic61—21no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C2×Dic122—21no—
79C2×Dic122—21no—
80C2×Dic122—21no—
81C2×Dic122—42no—
82C2×Dic122C22⋊C412yes—
83C2×C8×C61—22no—
84C2×C8×C61—41no—
85C4⋊C4×C61—42no—
86C42×C61—42no—
87C42×C61C22×C411yescharacteristic
88C42×C61C2×D411yescharacteristic
89C42×C61#4111yescharacteristic
90C42×C61#4111yescharacteristic
91C22⋊Dic61—41no—
92C4⋊Dic61—21no—
93C4⋊Dic61—21no—
94C4⋊Dic61—21no—
95C4⋊Dic61—21no—
96C4⋊Dic61—21no—
97C4⋊Dic61C22⋊C412yes—
98#11758196—22no—
99#11758196—22no—
100#11758196—41no—
101#11758196—41no—
102#11758196—82no—
103C4×Dic61—22no—
104C4×Dic61—22no—
105C4×Dic61—42no—
106C4×Dic61—41no—
107#11758194—22no—
108#11758194—41no—
109C2×#92418—41no—
110Dic244—44no—
111C23×C61#12011yescharacteristic
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—22no—
115C2×C4×C61—41no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61C2×C22⋊C411yescharacteristic
120C2×C4×C61C4×D411yescharacteristic
121C2×C4×C61C4×D411yescharacteristic
122C2×C4×C61#12111yescharacteristic
123C2×C4×C61#12111yescharacteristic
124C2×C4×C61#12311yescharacteristic
125C2×Dic61—22no—
126C2×Dic61—22no—
127C2×Dic61—42no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138Dic122—22no—
139Dic122—22no—
140Dic122—42no—
141Dic122—42no—
142Dic122—81no—
143C8×C61—42no—
144C8×C61—81no—
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—82no—
158C4×C61#42112yes—
159Dic61—42no—
160Dic61—42no—
161Dic61—81no—
162Dic61—81no—
163Dic61—81no—
164Dic61—81no—
165#3113—611no—
166#2320—611no—
167#1986—611no—
168#1506—611no—
169#1491—611no—
170#1300—611no—
171#1285D6111yescharacteristic
172C2×C61—21no—
173C2×C61—21no—
174C2×C61#188211yescharacteristic
175C2×C61#189711yescharacteristic
176C2×C61#190511yescharacteristic
177#533—611no—
178C2×C4×Q8—611no—
179#483—1221no—
180#475—612no—
181#462—612no—
182#451—1221no—
183#441—1222no—
184#432C2×D6111yescharacteristic
185#432C2×D6111yescharacteristic
186C4×M16—611no—
187#391—1221no—
188#389—1221no—
189#387—1222no—
190#383—1221no—
191#382—1221no—
192C4×C4⋊C4C2×D6111yescharacteristic
193#376—611no—
194#339—1222no—
195#328—1221no—
196#328—1221no—
197Q8⋊C8—1222no—
198C61#932811yescharacteristic, complemented, Sylow
199C22×Q8—611no—
200C2×Q16—1222no—
201C2×M16—1221no—
202C4⋊Q8—611no—
203C4⋊Q8—611no—
204#125—1221no—
205C22⋊Q8—1221no—
206C22⋊Q8—1221no—
207C4×Q8—612no—
208C4×Q8—612no—
209C4×Q8—1221no—
210C4×Q8—1221no—
211C4×Q8—1221no—
212C4×Q8—1221no—
213#117—1221no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—611no—
216C2×C4⋊C4—1221no—
217C2×C4⋊C4—1222no—
218C2×C42—1221no—
219C2×C42—1221no—
220C2×C42D24412yes—
221C2×C42C22×D6111yescharacteristic
222#106—1222no—
223C4⋊C8—21no—
224C4⋊C8C4×D6112yes—
225#103—1222no—
226#103—1222no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229#97—21no—
230#97—21no—
231#97—1221no—
232C4×C8—1221no—
233#95—21no—
234#95—611no—
235#95—611no—
236#95—1222no—
237#95—1222no—
238#95—1222no—
239#95—1221no—
240C2×Q8—612no—
241C2×Q8—1221no—
242C2×Q8—1221no—
243C2×Q8—1221no—
244C2×Q8—2442no—
245C22×C4—22no—
246C22×C4—1221no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4C2×D24411yescharacteristic
251C22×C4D4×D6111yescharacteristic
252C22×C4D4×D6111yescharacteristic
253Q16—2444no—
254M16—1222no—
255M16—2441no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—2441no—
259C4⋊C4—42no—
260C4⋊C4—612no—
261C4⋊C4—1222no—
262C4⋊C4—1222no—
263C4⋊C4—1221no—
264C4⋊C4—1221no—
265C4⋊C4—1221no—
266C4⋊C4—1221no—
267C4⋊C4—1221no—
268C4⋊C4—4882no—
269C4⋊C4—2441no—
270C4⋊C4—2441no—
271C22⋊C4—2441no—
272C42—42no—
273C42—1222no—
274C42—1222no—
275C42—2442no—
276C42—2441no—
277C42C2×C4×D6111yescharacteristic
278C42#1175822211yescharacteristic
279C42D4×D6111yescharacteristic
280C42#1175822611yescharacteristic
281C23#1451974711yescharacteristic
282Q8—1222no—
283Q8—1222no—
284Q8—4881no—
285Q8—2442no—
286Q8—2442no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—1222no—
295C2×C4—1222no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2442no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4C22⋊C4×D6111yescharacteristic
308C2×C4#1451973911yescharacteristic
309C2×C4C4×D24411yescharacteristic
310C2×C4#1451975011yescharacteristic
311C2×C4#1451975211yescharacteristic
312C2×C4#1451976211yescharacteristic
313C8—42no—
314C8—81no—
315C8—2442no—
316C22—21no—
317C22—21no—
318C22#1683964011yescharacteristic
319C22#1683994211yescharacteristic
320C22#1683994711yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—82no—
327C4—4881no—
328C4—4881no—
329C4—4881no—
330C4—4881no—
331C4—2442no—
332C4—2442no—
333C4#1683977712yes—
334C2—21no—
335C2—21no—
336C2#2144618711yescharacteristic
337C2#2144626611yescharacteristic
338C2#2144631411yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1285C2
socleC22×C61#386
prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#1285D61
hypocenterC61#9328
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
11. #16840048C4semidirect
12. #16839973C4semidirect
198. C61#9328semidirect

Maximal subgroups

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

subgroupindex
#214559502
#214484652
#214467622
#214443322
#214442652
#214437112
C61×#12852
#932861