Finite Group Databasebeta

← #29653146

Subgroups — #29653146

Order 15616.

890 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 203 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#21455854C212yesmaximal
2#21451181C211yesmaximal, characteristic
3#21444861C214yesmaximal
4#21444793C214yesmaximal
5#21444383C212yesmaximal
6#21444143C211yesmaximal, characteristic
7C61×#3502C211yesmaximal, characteristic
8C2×C4×Dic122C2211yescharacteristic
9#16840095C2214yes—
10#16840054—22no—
11#16839969—22no—
12#16839917C2214yes—
13C4×#11758202—24no—
14#16839815C2212yes—
15#16839701C2212yes—
16#16839682C2212yes—
17#16839669—21no—
18#16839658—21no—
19#16839646—22no—
20#16839628C2211yescharacteristic
21#16839499C2218yes—
22#16839488C2214yes—
23Q82×C61—22no—
24C61×#557—22no—
25C61×#557—24no—
26C61×#544—21no—
27C61×#544C2214yes—
28C61×#533—22no—
29C61×#533C2212yes—
30C61×#519—21no—
31C2×C4×Q8×C61C2211yescharacteristic
32C22×Dic122C2311yescharacteristic
33C2×#11758202—24no—
34C4×Dic122—22no—
35C4×Dic122—22no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61C2311yescharacteristic
38C2×#11758196—22no—
39C2×#11758196—21no—
40C2×C4×Dic61—21no—
41#14519710—24no—
42C4⋊#92418—22no—
43#14519701—24no—
44#14519700—24no—
45C22⋊#92418C2314yes—
46C4×#92418—22no—
47#14519684—22no—
48#14519684—22no—
49#14519684—22no—
50#14519684—21no—
51#14519684C2312yes—
52C22×Q8×C61D411yescharacteristic
53C4⋊Q8×C61—22no—
54C4⋊Q8×C61—22no—
55C4⋊Q8×C61—24no—
56C4⋊Q8×C61—28no—
57C4⋊Q8×C61D412yes—
58C4⋊Q8×C61D412yes—
59#125×C61—24no—
60#125×C61—24no—
61#125×C61—28no—
62#125×C61—22no—
63C22⋊Q8×C61—22no—
64C22⋊Q8×C61—24no—
65C22⋊Q8×C61C2314yes—
66C4×Q8×C61—24no—
67C4×Q8×C61—24no—
68C4×Q8×C61—24no—
69C4×Q8×C61—24no—
70C4×Q8×C61—22no—
71C4×Q8×C61—22no—
72C4×Q8×C61—22no—
73C4×Q8×C61D414yes—
74#117×C61—24no—
75#117×C61—22no—
76C2×C4⋊C4×C61D411yescharacteristic
77C2×C4⋊C4×C61C2312yes—
78C2×C42×C61D412yes—
79C2×C42×C61C2311yescharacteristic
80C2×Q8×C61—24no—
81C2×Q8×C61—22no—
82C2×Q8×C61—22no—
83C2×Q8×C61—22no—
84C2×Q8×C61Q1614yescomplemented
85C2×Q8×C61C2×D414yes—
86C22×C4×C61C2×D412yes—
87C22×C4×C61C2×D412yes—
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2411yescharacteristic
91C22×Dic61—21no—
92C22×Dic61—22no—
93C22×Dic61—21no—
94C2×Dic122—22no—
95C2×Dic122—42no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61—24no—
98C4⋊C4×C61—24no—
99C4⋊C4×C61—28no—
100C4⋊C4×C61—22no—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—24no—
105C4⋊C4×C61—24no—
106C4⋊C4×C61—24no—
107C4⋊C4×C61C2×D414yes—
108C22⋊C4×C61—24no—
109C42×C61—28no—
110C42×C61—22no—
111C42×C61—22no—
112C42×C61—24no—
113C42×C61C2×D412yes—
114C42×C61C2×D412yes—
115#11758202—48no—
116C4⋊Dic61—22no—
117C4⋊Dic61—42no—
118#11758196—44no—
119#11758196—41no—
120#11758196—41no—
121C4×Dic61—42no—
122C2×#92418—24no—
123C23×C61C22×D411yescharacteristic
124Q8×C61—28no—
125Q8×C61—44no—
126Q8×C61—44no—
127Q8×C61—84no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—24no—
131C2×C4×C61—24no—
132C2×C4×C61—22no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—42no—
136C2×C4×C61#12011yescharacteristic
137C2×C4×C61#12011yescharacteristic
138C2×C4×C61#12012yes—
139C2×C4×C61C2×Q1614yes—
140C2×C4×C61C2×Q1612yes—
141C2×C4×C61#13712yes—
142C2×C4×C61C22×D412yes—
143C2×Dic61—22no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—42no—
147C2×Dic61—42no—
148C2×Dic61—42no—
149C2×Dic61—41no—
150C2×Dic61—42no—
151Dic122—24no—
152Dic122—42no—
153#92418—44no—
154#18221—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61—21no—
157C22×C61C2×#12011yescharacteristic
158C22×C61C22×Q1611yescharacteristic
159C22×C61C2×#13711yescharacteristic
160C4×C61—24no—
161C4×C61—21no—
162C4×C61—21no—
163C4×C61—22no—
164C4×C61—44no—
165C4×C61—42no—
166C4×C61—44no—
167C4×C61#45114yes—
168Dic61—42no—
169Dic61—82no—
170Dic61—81no—
171#3502D6111yescharacteristic
172#3104—612no—
173#2581—611no—
174#1617—614no—
175#1605—614no—
176#1518—612no—
177#1461—611no—
178C2×C61—21no—
179C2×C61—21no—
180C2×C61C2×#45111yescharacteristic
181C2×C61#302011yescharacteristic
182C2×C61#302511yescharacteristic
183Q82—22no—
184#557—22no—
185#557—24no—
186#544—21no—
187#544C2×D6114yes—
188#533—22no—
189#533C2×D6112yes—
190#519—21no—
191C2×C4×Q8—611no—
192C2×C4×Q8C2×D6111yescharacteristic
193#484—614no—
194#477—1222no—
195#462—1222no—
196#451—614no—
197C4×Q16—1224no—
198#431—612no—
199#400—612no—
200#395—612no—
201#391—1221no—
202#389—1221no—
203#387—1222no—
204#382—611no—
205#328—618no—
206#324—614no—
207C61#1822111yescharacteristic, complemented, Sylow
208C22×Q8—611no—
209C22×Q8C22⋊D6111yescharacteristic
210C2×Q16—1224no—
211C4⋊Q8—22no—
212C4⋊Q8—22no—
213C4⋊Q8—24no—
214C4⋊Q8—28no—
215C4⋊Q8C22⋊D6112yes—
216C4⋊Q8C22⋊D6112yes—
217#125—24no—
218#125—24no—
219#125—28no—
220#125—22no—
221C22⋊Q8—22no—
222C22⋊Q8—24no—
223C22⋊Q8C22×D6114yes—
224C4×Q8—24no—
225C4×Q8—24no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—1222no—
232C4×Q8—1222no—
233C4×Q8C22⋊D6114yes—
234#117—24no—
235#117—22no—
236C2×C4⋊C4—611no—
237C2×C4⋊C4—1221no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4—1222no—
240C2×C4⋊C4C22⋊D6111yescharacteristic
241C2×C4⋊C4C22×D6112yes—
242C2×C42—1221no—
243C2×C42C22⋊D6112yes—
244C2×C42C22×D6111yescharacteristic
245#107—1224no—
246C4⋊C8—1222no—
247#103—1224no—
248#103—1224no—
249C22⋊C8—614no—
250C4×C8—1222no—
251#95—612no—
252#95—1221no—
253#95—1222no—
254#95—1222no—
255#95—1222no—
256C2×Q8—24no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—1222no—
261C2×Q8—2442no—
262C2×Q8#1175820214yescomplemented
263C2×Q8C2×C22⋊D6114yes—
264C22×C4—1221no—
265C22×C4—1222no—
266C22×C4—1221no—
267C22×C4C2×C22⋊D6112yes—
268C22×C4C2×C22⋊D6112yes—
269C22×C4C2×C22⋊D6111yescharacteristic
270C22×C4C2×C22⋊D6111yescharacteristic
271C22×C4C23×D6111yescharacteristic
272Q16—2448no—
273C2×C8—1224no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—28no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—24no—
285C4⋊C4—1222no—
286C4⋊C4—2444no—
287C4⋊C4—2442no—
288C4⋊C4—2441no—
289C4⋊C4—2441no—
290C4⋊C4C2×C22⋊D6114yes—
291C22⋊C4—24no—
292C42—28no—
293C42—22no—
294C42—22no—
295C42—24no—
296C42—2442no—
297C42C2×C22⋊D6112yes—
298C42C2×C22⋊D6112yes—
299C23C22×C22⋊D6111yescharacteristic
300Q8—28no—
301Q8—44no—
302Q8—44no—
303Q8—84no—
304Q8—1224no—
305Q8—2442no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—1222no—
315C2×C4—2442no—
316C2×C4—2442no—
317C2×C4—2442no—
318C2×C4—2441no—
319C2×C4—2442no—
320C2×C4—2441no—
321C2×C4—2441no—
322C2×C4#1451974611yescharacteristic
323C2×C4#1451974611yescharacteristic
324C2×C4#1451974612yes—
325C2×C4C2×#1175820214yes—
326C2×C4C2×#1175820212yes—
327C2×C4#1451980812yes—
328C2×C4C22×C22⋊D6112yes—
329C8—2444no—
330C22—21no—
331C22—21no—
332C22C2×#1451974611yescharacteristic
333C22C22×#1175820211yescharacteristic
334C22C2×#1451980811yescharacteristic
335C4—21no—
336C4—21no—
337C4—22no—
338C4—24no—
339C4—44no—
340C4—42no—
341C4—44no—
342C4—4882no—
343C4—4881no—
344C4—2442no—
345C4#1683991814yes—
346C2—21no—
347C2—21no—
348C2C2×#1683991811yescharacteristic
349C2#2145518311yescharacteristic
350C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3502C2
socleC22×C61C2×#120
prime radicalC23×C61C22×D4
hypercenter#3502D61
hypocenterC61#18221
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
84. C2×Q8×C61Q16semidirect
207. C61#18221semidirect
262. C2×Q8#11758202semidirect

Maximal subgroups

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

subgroupindex
#214558542
#214511812
#214448612
#214447932
#214443832
#214441432
C61×#35022
#1822161