Finite Group Databasebeta

← #29654862

Subgroups — #29654862

Order 15616.

998 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 207 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#21456757C212yesmaximal
2#21450818C211yesmaximal, characteristic
3#21444759C218yesmaximal, complemented
4#21444318C212yesmaximal, complemented
5#21444143C211yesmaximal, characteristic, complemented
6C61×#3502C211yesmaximal, characteristic, complemented
7C2×#14519774C2211yescharacteristic
8C2×#14519766—21no—
9#16840136—22no—
10#16840110—22no—
11#16839962—24no—
12#16839912C2218yes—
13#16839815C2212yes—
14#16839679C2214yes—
15#16839664—22no—
16#16839663—21no—
17C4×C4⋊Dic61C2211yescharacteristic
18#16839499C2218yes—
19#16839488C2214yes—
20Q82×C61—22no—
21C61×#557—22no—
22C61×#557—24no—
23C61×#544—21no—
24C61×#544C2214yes—
25C61×#533—22no—
26C61×#533C2212yes—
27C61×#519—21no—
28C2×C4×Q8×C61C2211yescharacteristic
29C22×D244—21no—
30C22×D244—21no—
31C22×D244C2312yes—
32C2×#11758201—28no—
33#14519774—22no—
34#14519774—22no—
35#14519766—44no—
36C2×C4⋊Dic61—21no—
37C2×#11758198—24no—
38C2×#11758198—24no—
39C2×C4×Dic61—21no—
40C4⋊#92418—22no—
41#14519697—28no—
42C22⋊#92418C2314yes—
43C4×#92418—22no—
44#14519684C2312yes—
45C22×Q8×C61D411yescharacteristic
46C4⋊Q8×C61—22no—
47C4⋊Q8×C61—22no—
48C4⋊Q8×C61—24no—
49C4⋊Q8×C61—28no—
50C4⋊Q8×C61D412yes—
51C4⋊Q8×C61D412yes—
52#125×C61—24no—
53#125×C61—24no—
54#125×C61—28no—
55#125×C61—22no—
56C22⋊Q8×C61—22no—
57C22⋊Q8×C61—24no—
58C22⋊Q8×C61C2314yes—
59C4×Q8×C61—24no—
60C4×Q8×C61—24no—
61C4×Q8×C61—24no—
62C4×Q8×C61—24no—
63C4×Q8×C61—22no—
64C4×Q8×C61—22no—
65C4×Q8×C61—22no—
66C4×Q8×C61D414yes—
67#117×C61—24no—
68#117×C61—22no—
69C2×C4⋊C4×C61D411yescharacteristic
70C2×C4⋊C4×C61C2312yes—
71C2×C42×C61D412yes—
72C2×C42×C61C2311yescharacteristic
73C23×D61—22no—
74C2×Q8×C61—24no—
75C2×Q8×C61—22no—
76C2×Q8×C61—22no—
77C2×Q8×C61—22no—
78C2×Q8×C61SD1614yescomplemented
79C2×Q8×C61C2×D414yes—
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D412yes—
82C22×C4×C61C2×D412yes—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2411yescharacteristic
85C22×Dic61—22no—
86C2×D244—24no—
87C2×D244—44no—
88C2×D244—41no—
89C2×D244—42no—
90C2×D244—41no—
91C2×D244—42no—
92C2×D244—42no—
93C2×D244—82no—
94C4⋊C4×C61—24no—
95C4⋊C4×C61—24no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61—28no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61—22no—
100C4⋊C4×C61—22no—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—24no—
103C4⋊C4×C61—24no—
104C4⋊C4×C61—24no—
105C4⋊C4×C61C2×D414yes—
106C22⋊C4×C61—24no—
107C42×C61—28no—
108C42×C61—22no—
109C42×C61—22no—
110C42×C61—24no—
111C42×C61C2×D412yes—
112C42×C61C2×D412yes—
113#11758201—88no—
114#11758198—48no—
115#11758198—84no—
116C4⋊Dic61—42no—
117C4×Dic61—42no—
118C2×#92418—24no—
119C23×C61C22×D411yescharacteristic
120C22×D61—44no—
121C22×D61—44no—
122C22×D61—42no—
123C22×D61—42no—
124C22×D61—42no—
125Q8×C61—28no—
126Q8×C61—44no—
127Q8×C61—44no—
128Q8×C61—84no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—24no—
133C2×C4×C61—24no—
134C2×C4×C61—22no—
135C2×C4×C61—41no—
136C2×C4×C61—42no—
137C2×C4×C61#12011yescharacteristic
138C2×C4×C61#12012yes—
139C2×C4×C61#12011yescharacteristic
140C2×C4×C61C2×SD1614yes—
141C2×C4×C61C2×SD1612yes—
142C2×C4×C61#13712yes—
143C2×C4×C61C22×D412yes—
144C2×Dic61—42no—
145C2×Dic61—42no—
146C2×Dic61—42no—
147D244—48no—
148D244—82no—
149D244—84no—
150D244—82no—
151#92418—44no—
152#18403—611nomaximal, Sylow
153C22×C61—21no—
154C22×C61—21no—
155C22×C61C2×#12011yescharacteristic
156C22×C61C22×SD1611yescharacteristic
157C22×C61C2×#13711yescharacteristic
158C2×D61—84no—
159C2×D61—84no—
160C2×D61—82no—
161C2×D61—82no—
162C2×D61—82no—
163C4×C61—24no—
164C4×C61—21no—
165C4×C61—22no—
166C4×C61—44no—
167C4×C61—42no—
168C4×C61—44no—
169C4×C61#44814yes—
170C4×C61#45412yes—
171Dic61—82no—
172#3502D6111yescharacteristic, complemented
173#3208—612no—
174#2548—611no—
175#1599—618no—
176#1503—612no—
177#1461—611no—
178C2×C61—21no—
179C2×C61—21no—
180C2×C61C2×#44811yescharacteristic
181C2×C61C2×#45411yescharacteristic
182C2×C61#302711yescharacteristic
183D61—162no—
184Q82—22no—
185#557—22no—
186#557—24no—
187#544—21no—
188#544C2×D6114yes—
189#533—22no—
190#533C2×D6112yes—
191C2×#127—611no—
192C2×#124—1221no—
193#519—21no—
194C2×C4×Q8C2×D6111yescharacteristic
195#492—1222no—
196#486—1222no—
197#460—1224no—
198#450—618no—
199#431—612no—
200#394—614no—
201#390—1221no—
202#390—1222no—
203C4×C4⋊C4—611no—
204#328—618no—
205#324—614no—
206C61#1840311yescharacteristic, complemented, Sylow
207C22×Q8C22⋊D6111yescharacteristic
208C22×D4—612no—
209C22×D4—1221no—
210C22×D4—1221no—
211C2×SD16—1228no—
212C4⋊Q8—22no—
213C4⋊Q8—22no—
214C4⋊Q8—24no—
215C4⋊Q8—28no—
216C4⋊Q8C22⋊D6112yes—
217C4⋊Q8C22⋊D6112yes—
218#127—1222no—
219#127—1222no—
220#125—24no—
221#125—24no—
222#125—28no—
223#125—22no—
224#124—2444no—
225C22⋊Q8—22no—
226C22⋊Q8—24no—
227C22⋊Q8C22×D6114yes—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8C22⋊D6114yes—
236#117—24no—
237#117—22no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4C22⋊D6111yescharacteristic
240C2×C4⋊C4C22×D6112yes—
241C2×C22⋊C4—1224no—
242C2×C22⋊C4—1224no—
243C2×C42—1221no—
244C2×C42C22⋊D6112yes—
245C2×C42C22×D6111yescharacteristic
246C4⋊C8—1222no—
247#102—1228no—
248C22⋊C8—614no—
249C4×C8—1222no—
250#95—612no—
251C24—1222no—
252C2×Q8—24no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8#1175820114yescomplemented
257C2×Q8C2×C22⋊D6114yes—
258C2×D4—1224no—
259C2×D4—4882no—
260C2×D4—2444no—
261C2×D4—2442no—
262C2×D4—2441no—
263C2×D4—2442no—
264C2×D4—2441no—
265C2×D4—2442no—
266C22×C4—1222no—
267C22×C4C2×C22⋊D6111yescharacteristic
268C22×C4C2×C22⋊D6112yes—
269C22×C4C2×C22⋊D6112yes—
270C22×C4C2×C22⋊D6111yescharacteristic
271C22×C4C23×D6111yescharacteristic
272SD16—4888no—
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—2442no—
286C4⋊C4C2×C22⋊D6114yes—
287C22⋊C4—24no—
288C22⋊C4—4884no—
289C22⋊C4—2448no—
290C42—28no—
291C42—22no—
292C42—22no—
293C42—24no—
294C42—2442no—
295C42C2×C22⋊D6112yes—
296C42C2×C22⋊D6112yes—
297C23—2444no—
298C23—2444no—
299C23—2442no—
300C23—2442no—
301C23—2442no—
302C23C22×C22⋊D6111yescharacteristic
303Q8—28no—
304Q8—44no—
305Q8—44no—
306Q8—84no—
307D4—4882no—
308D4—4884no—
309D4—4882no—
310D4—2448no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—2442no—
320C2×C4—2442no—
321C2×C4—2442no—
322C2×C4#1451974611yescharacteristic
323C2×C4#1451974612yes—
324C2×C4#1451974611yescharacteristic
325C2×C4C2×#1175820114yes—
326C2×C4C2×#1175820112yes—
327C2×C4#1451980812yes—
328C2×C4C22×C22⋊D6112yes—
329C8—2444no—
330C22—21no—
331C22—21no—
332C22—4884no—
333C22—4884no—
334C22—4882no—
335C22—4882no—
336C22—4882no—
337C22C2×#1451974611yescharacteristic
338C22C22×#1175820111yescharacteristic
339C22C2×#1451980811yescharacteristic
340C4—21no—
341C4—22no—
342C4—24no—
343C4—44no—
344C4—42no—
345C4—44no—
346C4—4882no—
347C4#1683990114yes—
348C4#1683993512yes—
349C2—21no—
350C2—21no—
351C2—9762no—
352C2C2×#1683990111yescharacteristic
353C2C2×#1683993511yescharacteristic
354C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3502C2
socleC22×C61C2×#120
prime radicalC2×#14519774C22
hypercenter#3502D61
hypocenterC61#18403
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
3. #21444759C2semidirect
4. #21444318C2semidirect
5. #21444143C2semidirect
6. C61×#3502C2semidirect
78. C2×Q8×C61SD16semidirect
172. #3502D61semidirect
206. C61#18403semidirect
256. C2×Q8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214567572
#214508182
#214447592
#214443182
#214441432
C61×#35022
#1840361