Finite Group Databasebeta

← #29572573

Subgroups — #29572573

Order 15616.

728 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 115 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#21460233C211yesmaximal, characteristic
2#21446796C212yesmaximal, complemented
3#21444556C212yesmaximal, complemented
4#21444554C211yesmaximal, characteristic
5C61×#1962C211yesmaximal, characteristic
6#16840512C414yescomplemented
7#16840498—21no—
8#16840492—21no—
9#16840482—22no—
10D4×Dic122—22no—
11#16840379—24no—
12#16840333—22no—
13#16840319C2212yes—
14#16839786C2212yes—
15#16839522—24no—
16#16839511—21no—
17#16839509—22no—
18#16839507—22no—
19#16839506—21no—
20C61×#532C2211yescharacteristic
21C61×#423C2211yescharacteristic
22C2×#104×C61—22no—
23C43×C61C2211yescharacteristic
24C22×Dic122—22no—
25C2×#11758194—22no—
26#14519777D412yes—
27#14519777D412yes—
28#14519776—24no—
29#14519775—24no—
30#14519770—28no—
31#14519769C2×C414yes—
32#14519764—24no—
33#14519764—24no—
34#14519758—24no—
35#14519758—24no—
36#14519756—24no—
37#14519756—22no—
38#14519756—22no—
39#14519754—28no—
40C4×Dic122—22no—
41C4×Dic122—22no—
42#14519742—24no—
43D4×Dic61—24no—
44#14519736—22no—
45#14519736D414yes—
46C4⋊#92418—22no—
47C4⋊#92418—22no—
48C2×#41×C61D412yes—
49C2×M16×C61—21no—
50C4⋊Q8×C61D411yescharacteristic
51#127×C61D411yescharacteristic
52#124×C61C2×C412yes—
53#121×C61—22no—
54C4×D4×C61—22no—
55C2×C42×C61—22no—
56C2×C42×C61—21no—
57C2×C42×C61C2311yescharacteristic
58C4⋊C8×C61—22no—
59#104×C61—44no—
60#41×C61—42no—
61#41×C61—42no—
62C2×Q8×C61C22⋊C412yes—
63C2×D4×C61—22no—
64C2×D4×C61—22no—
65C2×D4×C61C22⋊C412yes—
66C22×C4×C61—22no—
67C22×C4×C61—41no—
68C22×C4×C61C2×D412yes—
69C22×C4×C61C2×D411yescharacteristic
70C22×Dic61—24no—
71C2×Dic122—22no—
72C2×Dic122—22no—
73C2×Dic122—24no—
74C2×Dic122—82no—
75C2×Dic122—82no—
76M16×C61—42no—
77C2×C8×C61—41no—
78C4⋊C4×C61—22no—
79C22⋊C4×C61—24no—
80C42×C61—22no—
81C42×C61—24no—
82C42×C61—42no—
83C42×C61—41no—
84C42×C61C22×C411yescharacteristic
85C42×C61C2×D411yescharacteristic
86C42×C61C2×D412yes—
87C22⋊Dic61—24no—
88C22⋊Dic61—28no—
89C22⋊Dic61—24no—
90C4⋊Dic61—22no—
91C4⋊Dic61—22no—
92C4⋊Dic61C22⋊C414yes—
93C4⋊Dic61C22⋊C414yes—
94#11758196—28no—
95#11758196—24no—
96#11758196—24no—
97C4×Dic61—24no—
98C4×Dic61—24no—
99#11758194—44no—
100C2×#92418—42no—
101C23×C61—22no—
102C23×C61#12011yescharacteristic
103Q8×C61—42no—
104D4×C61—42no—
105D4×C61—42no—
106D4×C61—42no—
107D4×C61—82no—
108C2×C4×C61—21no—
109C2×C4×C61—22no—
110C2×C4×C61—21no—
111C2×C4×C61—22no—
112C2×C4×C61—22no—
113C2×C4×C61—42no—
114C2×C4×C61—44no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61C2×C22⋊C412yes—
118C2×C4×C61C2×C22⋊C411yescharacteristic
119C2×C4×C61#12011yescharacteristic
120C2×C4×C61#12012yes—
121C2×Dic61—24no—
122C2×Dic61—24no—
123C2×Dic61—84no—
124Dic122—42no—
125Dic122—42no—
126Dic122—42no—
127Dic122—42no—
128Dic122—84no—
129C8×C61—81no—
130#92418—82no—
131#8531—611nomaximal, Sylow
132C22×C61—21no—
133C22×C61—22no—
134C22×C61—42no—
135C22×C61—44no—
136C22×C61#37911yescharacteristic
137C4×C61—21no—
138C4×C61—22no—
139C4×C61—44no—
140C4×C61—42no—
141C4×C61#45311yescharacteristic
142C4×C61#45412yes—
143C4×C61#45511yescharacteristic
144C4×C61#45612yes—
145Dic61—44no—
146Dic61—44no—
147#3550—611no—
148#1994—612no—
149#1962D6111yescharacteristic
150#1561—612no—
151#1560—611no—
152C2×C61—41no—
153C2×C61—82no—
154C2×C61#179211yescharacteristic
155C2×C61#179312yes—
156#563—614no—
157#559—1221no—
158Q82—1221no—
159#556—1222no—
160D4×Q8—1222no—
161#542—1224no—
162#536—1222no—
163#533—612no—
164#532C2×D6111yescharacteristic
165#423—612no—
166#423C2×D6111yescharacteristic
167C2×#104—22no—
168C43C2×D6111yescharacteristic
169#339—1224no—
170#333—1221no—
171#332—1222no—
172#332—1222no—
173#331—1221no—
174C61#853111yescharacteristic, complemented, Sylow
175C2×#41C22⋊D6112yes—
176C22×Q8—1222no—
177C2×M16—21no—
178C2×M16—1222no—
179C4⋊Q8—612no—
180C4⋊Q8—612no—
181C4⋊Q8—1224no—
182C4⋊Q8—1224no—
183C4⋊Q8D24411yescharacteristic
184#127D24411yescharacteristic
185#126—1228no—
186#125—614no—
187#124—1224no—
188#124—1224no—
189#124C4×D6112yes—
190#123—1224no—
191#123—1224no—
192C22⋊Q8—1228no—
193C22⋊Q8—1224no—
194C22⋊Q8—1222no—
195C22⋊Q8—1222no—
196#121—22no—
197C4×Q8—1224no—
198C4×Q8—1222no—
199C4×Q8—1222no—
200C4×D4—22no—
201C4×D4—1224no—
202#117—614no—
203#117—1222no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42C22×D6111yescharacteristic
207C4⋊C8—22no—
208C4⋊C8—1222no—
209C4⋊C8—1222no—
210#104—44no—
211#41—42no—
212#41—42no—
213C2×Q8—1224no—
214C2×Q8—1222no—
215C2×Q8—1222no—
216C2×Q8—4882no—
217C2×Q8—4882no—
218C2×Q8#1175819812yes—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4#1175819812yes—
222C22×C4—22no—
223C22×C4—41no—
224C22×C4—1224no—
225C22×C4D4×D6112yes—
226C22×C4C2×C22⋊D6111yescharacteristic
227M16—42no—
228M16—2444no—
229C2×C8—41no—
230C2×C8—2442no—
231C4⋊C4—22no—
232C4⋊C4—614no—
233C4⋊C4—614no—
234C4⋊C4—1228no—
235C4⋊C4—1224no—
236C4⋊C4—1224no—
237C4⋊C4—1222no—
238C4⋊C4—1222no—
239C22⋊C4—24no—
240C22⋊C4—1224no—
241C22⋊C4—1228no—
242C22⋊C4—1224no—
243C42—22no—
244C42—24no—
245C42—42no—
246C42—41no—
247C42—1224no—
248C42—1224no—
249C42C2×C4×D6111yescharacteristic
250C42C2×D24411yescharacteristic
251C42D4×D6112yes—
252C23—22no—
253C23#1451974511yescharacteristic
254Q8—42no—
255Q8—4884no—
256Q8—2442no—
257Q8—2442no—
258Q8—2442no—
259Q8—2442no—
260D4—42no—
261D4—42no—
262D4—42no—
263D4—82no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—1224no—
274C2×C4—1224no—
275C2×C4—4884no—
276C2×C4C22⋊C4×D6112yes—
277C2×C4C2×#1175819811yescharacteristic
278C2×C4#1451974511yescharacteristic
279C2×C4#1451974712yes—
280C8—81no—
281C8—4882no—
282C22—21no—
283C22—22no—
284C22—42no—
285C22—44no—
286C22#1683961711yescharacteristic
287C4—21no—
288C4—22no—
289C4—44no—
290C4—42no—
291C4—2444no—
292C4—2444no—
293C4#1683992911yescharacteristic
294C4#1683993612yes—
295C4#1683993911yescharacteristic
296C4#1683994612yes—
297C2—41no—
298C2—82no—
299C2#2144577211yescharacteristic
300C2#2144577912yes—
subgroupG/H
centerC22#16839617
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1962C2
socleC22×C61#379
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#1962D61
hypocenterC61#8531
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
2. #21446796C2semidirect
3. #21444556C2semidirect
6. #16840512C4semidirect
174. C61#8531semidirect

Maximal subgroups

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

subgroupindex
#214602332
#214467962
#214445562
#214445542
C61×#19622
#853161