Finite Group Databasebeta

← #29653145

Subgroups — #29653145

Order 15616.

890 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 187 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#21459629C211yesmaximal, characteristic
2#21455858C212yesmaximal
3#21444860C214yesmaximal
4#21444792C214yesmaximal
5#21444382C212yesmaximal
6#21444143C211yesmaximal, characteristic
7C61×#2581C211yesmaximal, characteristic
8Q8×Dic122—22no—
9#16840491—22no—
10#16840488—24no—
11#16840393—21no—
12#16840393C2214yes—
13#16840319—22no—
14#16840319C2212yes—
15#16840240—21no—
16#16840097C2214yes—
17#16840057—22no—
18#16839972—22no—
19#16839918C2214yes—
20C4×#11758202—24no—
21#16839815C2212yes—
22#16839498C2218yes—
23#16839488C2214yes—
24C2×C4×Q8×C61C2211yescharacteristic
25C2×C4×Q8×C61C2211yescharacteristic
26C61×#400C2212yes—
27C61×#395C2212yes—
28C61×#391—21no—
29C61×#389—21no—
30C61×#387—22no—
31C61×#382C2211yescharacteristic
32C2×#11758202—24no—
33#14519777—22no—
34#14519777—22no—
35#14519777D412yes—
36#14519777D412yes—
37#14519776—24no—
38#14519776—28no—
39#14519769—24no—
40#14519769—22no—
41#14519768—24no—
42#14519768—28no—
43#14519756—22no—
44#14519756—24no—
45#14519756C2314yes—
46Q8×Dic61—24no—
47Q8×Dic61—24no—
48C4×Dic122—24no—
49C4×Dic122—24no—
50C4×Dic122—22no—
51C4×Dic122D414yes—
52#14519736—24no—
53#14519736—22no—
54#14519710—24no—
55C4⋊#92418—22no—
56#14519701—24no—
57#14519700—24no—
58C22⋊#92418C2314yes—
59C4×#92418—22no—
60C22×Q8×C61D411yescharacteristic
61C22×Q8×C61C2311yescharacteristic
62C4×Q8×C61—22no—
63C4×Q8×C61—22no—
64C4×Q8×C61—22no—
65C4×Q8×C61—22no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—22no—
69C2×C4⋊C4×C61D411yescharacteristic
70C2×C4⋊C4×C61C2311yescharacteristic
71C2×C4⋊C4×C61C2312yes—
72C2×C42×C61—21no—
73C2×C42×C61D412yes—
74C2×C42×C61C2311yescharacteristic
75#95×C61—21no—
76#95×C61—22no—
77#95×C61—22no—
78#95×C61—22no—
79#95×C61C2312yes—
80C2×Q8×C61—22no—
81C2×Q8×C61—22no—
82C2×Q8×C61—22no—
83C2×Q8×C61—42no—
84C2×Q8×C61Q1614yescomplemented
85C22×C4×C61—21no—
86C22×C4×C61—22no—
87C22×C4×C61—21no—
88C22×C4×C61C2×D412yes—
89C22×C4×C61C2×D412yes—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2411yescharacteristic
93C2×Dic122—22no—
94C2×Dic122—24no—
95C2×Dic122C2×D414yes—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—24no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61—44no—
100C4⋊C4×C61—41no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61—42no—
103C42×C61—22no—
104C42×C61—22no—
105C42×C61—42no—
106C42×C61C2×D412yes—
107C42×C61C2×D412yes—
108C22⋊Dic61—24no—
109#11758202—48no—
110C4⋊Dic61—24no—
111C4⋊Dic61—24no—
112C4⋊Dic61—24no—
113C4⋊Dic61—22no—
114C4⋊Dic61—22no—
115C4⋊Dic61—22no—
116C4⋊Dic61C2×D414yes—
117#11758196—28no—
118#11758196—24no—
119#11758196—24no—
120C4×Dic61—28no—
121C4×Dic61—24no—
122C2×#92418—24no—
123C23×C61C22×D411yescharacteristic
124Q8×C61—28no—
125Q8×C61—24no—
126Q8×C61—42no—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61—41no—
140C2×C4×C61—42no—
141C2×C4×C61#12011yescharacteristic
142C2×C4×C61#12011yescharacteristic
143C2×C4×C61#12012yes—
144C2×C4×C61C2×Q1614yes—
145C2×C4×C61C2×Q1612yes—
146C2×C4×C61#13712yes—
147C2×C4×C61C22×D412yes—
148C2×Dic61—24no—
149C2×Dic61—24no—
150Dic122—44no—
151Dic122—44no—
152Dic122—84no—
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—42no—
165C4×C61—42no—
166C4×C61—82no—
167C4×C61—81no—
168C4×C61#45114yes—
169Dic61—44no—
170Dic61—44no—
171#3502—611no—
172#3104—612no—
173#2581D6111yescharacteristic
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—1222no—
184#557—1224no—
185#557—1222no—
186#544—614no—
187#544—1221no—
188#533—612no—
189#533—1222no—
190#519—1221no—
191C2×C4×Q8C2×D6111yescharacteristic
192C2×C4×Q8C2×D6111yescharacteristic
193#484—614no—
194#477—1222no—
195#462—1222no—
196#451—614no—
197C4×Q16—1224no—
198#431—612no—
199#400C2×D6112yes—
200#395C2×D6112yes—
201#391—21no—
202#389—21no—
203#387—22no—
204#382C2×D6111yescharacteristic
205#328—618no—
206#324—614no—
207C61#1822111yescharacteristic, complemented, Sylow
208C22×Q8C22⋊D6111yescharacteristic
209C22×Q8C22×D6111yescharacteristic
210C2×Q16—1224no—
211C4⋊Q8—612no—
212C4⋊Q8—612no—
213C4⋊Q8—1222no—
214C4⋊Q8—1222no—
215C4⋊Q8—1228no—
216C4⋊Q8—1224no—
217#125—1224no—
218#125—1228no—
219#125—1224no—
220#125—1222no—
221C22⋊Q8—614no—
222C22⋊Q8—1222no—
223C22⋊Q8—1224no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—614no—
229C4×Q8—1224no—
230C4×Q8—1224no—
231C4×Q8—1224no—
232C4×Q8—1224no—
233C4×Q8—1222no—
234#117—1222no—
235#117—1224no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4C22⋊D6111yescharacteristic
240C2×C4⋊C4C22×D6111yescharacteristic
241C2×C4⋊C4C22×D6112yes—
242C2×C42—21no—
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—21no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95C22×D6112yes—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—42no—
260C2×Q8—614no—
261C2×Q8—1224no—
262C2×Q8—1222no—
263C2×Q8#1175820214yescomplemented
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
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—22no—
276C4⋊C4—22no—
277C4⋊C4—44no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—614no—
282C4⋊C4—1228no—
283C4⋊C4—1224no—
284C4⋊C4—1224no—
285C4⋊C4—1224no—
286C4⋊C4—1224no—
287C4⋊C4—1224no—
288C4⋊C4—1222no—
289C4⋊C4—1222no—
290C4⋊C4—1222no—
291C22⋊C4—1224no—
292C42—22no—
293C42—22no—
294C42—42no—
295C42—1228no—
296C42—1224no—
297C42D4×D6112yes—
298C42D4×D6112yes—
299C23C22×C22⋊D6111yescharacteristic
300Q8—28no—
301Q8—24no—
302Q8—42no—
303Q8—4884no—
304Q8—2444no—
305Q8—2444no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—1224no—
321C2×C4—1224no—
322C2×C4#1451974511yescharacteristic
323C2×C4#1451974511yescharacteristic
324C2×C4#1451974512yes—
325C2×C4C2×#1175820214yes—
326C2×C4C2×#1175820212yes—
327C2×C4#1451980712yes—
328C2×C4C2×D4×D6112yes—
329C8—2444no—
330C22—21no—
331C22—21no—
332C22C2×#1451974511yescharacteristic
333C22C22×#1175820211yescharacteristic
334C22C2×#1451980711yescharacteristic
335C4—21no—
336C4—21no—
337C4—22no—
338C4—24no—
339C4—42no—
340C4—42no—
341C4—82no—
342C4—81no—
343C4—2444no—
344C4—2444no—
345C4#1683991714yes—
346C2—21no—
347C2—21no—
348C2C2×#1683991711yescharacteristic
349C2#2145518211yescharacteristic
350C2#2145521011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2581C2
socleC22×C61C2×#120
prime radicalC23×C61C22×D4
hypercenter#2581D61
hypocenterC61#18221
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2581

Complements

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

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

Maximal subgroups

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

subgroupindex
#214596292
#214558582
#214448602
#214447922
#214443822
#214441432
C61×#25812
#1822161