Finite Group Databasebeta

← #29571752

Subgroups — #29571752

Order 15616.

706 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#16840490C211yesmaximal, characteristic
2#21444492C212yesmaximal
3#21444247C212yesmaximal
4#21444143C211yesmaximal, characteristic
5C61×#1801C211yesmaximal, characteristic
6#16840490—24no—
7#16840490—24no—
8C2×#14519777C2211yescharacteristic
9C2×#14519775—21no—
10C2×#14519769C412yes—
11C2×#14519767—22no—
12C2×C4×Dic122C412yes—
13C2×C4×Dic122C2211yescharacteristic
14C2×#14519742—21no—
15#16839815C2212yes—
16#16839509—24no—
17#16839492—24no—
18#16839488—22no—
19C2×C4×Q8×C61C2211yescharacteristic
20C61×#423C2212yes—
21C2×#101×C61—21no—
22C61×#343—21no—
23C22×Dic122—21no—
24C22×Dic122C2×C411yescharacteristic
25#14519777—22no—
26#14519777D414yes—
27#14519775—28no—
28#14519769—24no—
29#14519769—24no—
30#14519767—216no—
31C4×Dic122—24no—
32C4×Dic122—22no—
33C4×Dic122—24no—
34C4×Dic122D414yes—
35#14519742—28no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61C2×C412yes—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×#11758196—22no—
40C2×#11758196—22no—
41C2×#11758196—22no—
42C2×#11758196—21no—
43C2×C4×Dic61—22no—
44C2×C4×Dic61—21no—
45C4⋊#92418—22no—
46C22⋊#92418—22no—
47C4×#92418—22no—
48C22×Q8×C61D411yescharacteristic
49C2×M16×C61—22no—
50C4×Q8×C61—22no—
51C4×Q8×C61—22no—
52C2×C4⋊C4×C61D412yes—
53C2×C4⋊C4×C61D411yescharacteristic
54C2×C42×C61C2×C412yes—
55C2×C42×C61C2311yescharacteristic
56C4⋊C8×C61—22no—
57C4⋊C8×C61—22no—
58#101×C61—42no—
59C2×Q8×C61—22no—
60C2×Q8×C61—22no—
61C2×Q8×C61—21no—
62C2×Q8×C61—21no—
63C22×C4×C61C22⋊C412yes—
64C22×C4×C61C22⋊C412yes—
65C22×C4×C61C22×C411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—22no—
70C22×Dic61—21no—
71C2×Dic122—21no—
72C2×Dic122—24no—
73C2×Dic122—24no—
74C2×Dic122—42no—
75C2×Dic122—41no—
76C2×Dic122—41no—
77C2×Dic122C22⋊C412yes—
78M16×C61—42no—
79M16×C61—42no—
80C2×C8×C61—42no—
81C4⋊C4×C61—22no—
82C4⋊C4×C61—24no—
83C42×C61—22no—
84C42×C61—22no—
85C42×C61C2×D412yes—
86C42×C61C2×D412yes—
87C4⋊Dic61—21no—
88C4⋊Dic61—22no—
89C4⋊Dic61—24no—
90C4⋊Dic61C22⋊C414yes—
91C4⋊Dic61C22⋊C412yes—
92#11758196—28no—
93#11758196—28no—
94#11758196—28no—
95#11758196—24no—
96C4×Dic61—28no—
97C4×Dic61—24no—
98C2×#92418—42no—
99C23×C61C2×C22⋊C411yescharacteristic
100Q8×C61—42no—
101Q8×C61—42no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—22no—
105C2×C4×C61—22no—
106C2×C4×C61—22no—
107C2×C4×C61—41no—
108C2×C4×C61—42no—
109C2×C4×C61C2×C22⋊C411yescharacteristic
110C2×C4×C61C2×C22⋊C411yescharacteristic
111C2×C4×C61#12012yes—
112C2×C4×C61#12012yes—
113C2×C4×C61#13512yes—
114C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
115C2×C4×C61#13711yescharacteristic
116C2×Dic61—24no—
117C2×Dic61—22no—
118C2×Dic61—22no—
119C2×Dic61—24no—
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123Dic122—24no—
124Dic122—22no—
125Dic122—44no—
126Dic122—44no—
127Dic122#10114yescomplemented
128C8×C61—82no—
129#92418—82no—
130#8390—611nomaximal, Sylow
131C22×C61—21no—
132C22×C61—21no—
133C22×C61#37911yescharacteristic
134C22×C61#41611yescharacteristic
135C22×C61#41711yescharacteristic
136C4×C61—22no—
137C4×C61—21no—
138C4×C61—42no—
139C4×C61—42no—
140C4×C61C2×#10112yes—
141C4×C61#44912yes—
142C4×C61#45212yes—
143Dic61—24no—
144Dic61—42no—
145Dic61—44no—
146C2×#557—611no—
147#1801D6111yescharacteristic
148#1546—612no—
149#1487—612no—
150#1461—611no—
151C2×C61—21no—
152C2×C61—21no—
153C2×C61#178411yescharacteristic
154C2×C61#178811yescharacteristic
155C2×C61#179611yescharacteristic
156#557—1224no—
157#557—1224no—
158C2×C4⋊Q8—611no—
159C2×C4⋊Q8—1221no—
160C2×#125—612no—
161C2×#125—1222no—
162C2×C4×Q8—612no—
163C2×C4×Q8—611no—
164C2×C4×Q8—1221no—
165C2×C4×Q8C2×D6111yescharacteristic
166#431—612no—
167#423C2×D6112yes—
168C2×#101—21no—
169#343—21no—
170#332—1224no—
171Q8⋊C8—1224no—
172#324—1222no—
173C61#839011yescharacteristic, complemented, Sylow
174C22×Q8—611no—
175C22×Q8—1221no—
176C22×Q8C22⋊D6111yescharacteristic
177C2×M16—22no—
178C4⋊Q8—614no—
179C4⋊Q8—1228no—
180C4⋊Q8—1222no—
181#125—1224no—
182#125—1224no—
183#125—12216no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—614no—
187C4×Q8—1224no—
188C4×Q8—1224no—
189C4×Q8—1228no—
190C4×Q8—1222no—
191C2×C4⋊C4—611no—
192C2×C4⋊C4—612no—
193C2×C4⋊C4—1222no—
194C2×C4⋊C4—1222no—
195C2×C4⋊C4—1222no—
196C2×C4⋊C4—1221no—
197C2×C4⋊C4—1221no—
198C2×C4⋊C4D24412yes—
199C2×C4⋊C4C22⋊D6111yescharacteristic
200C2×C42—1222no—
201C2×C42—1221no—
202C2×C42C4×D6112yes—
203C2×C42C22×D6111yescharacteristic
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—1222no—
207#101—42no—
208C22⋊C8—1222no—
209C4×C8—1222no—
210C2×Q8—21no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—21no—
214C2×Q8—612no—
215C2×Q8—1224no—
216C2×Q8—1221no—
217C2×Q8—1224no—
218C2×Q8—2441no—
219C2×Q8—2442no—
220C2×Q8—2441no—
221C22×C4—1221no—
222C22×C4—1221no—
223C22×C4—1222no—
224C22×C4#1175819812yes—
225C22×C4#1175819812yes—
226C22×C4C2×C4×D6111yescharacteristic
227C22×C4C2×D24411yescharacteristic
228C22×C4C2×C22⋊D6111yescharacteristic
229M16—42no—
230M16—42no—
231C2×C8—42no—
232C2×C8—2442no—
233C4⋊C4—22no—
234C4⋊C4—24no—
235C4⋊C4—612no—
236C4⋊C4—614no—
237C4⋊C4—1228no—
238C4⋊C4—1228no—
239C4⋊C4—1228no—
240C4⋊C4—1221no—
241C4⋊C4—1222no—
242C4⋊C4—1224no—
243C4⋊C4—1224no—
244C42—22no—
245C42—22no—
246C42—1228no—
247C42—1224no—
248C42D4×D6112yes—
249C42D4×D6112yes—
250C23C2×#1175819811yescharacteristic
251Q8—42no—
252Q8—42no—
253Q8—614no—
254Q8—1224no—
255Q8—1222no—
256Q8—2444no—
257Q8—2444no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—1224no—
266C2×C4—1222no—
267C2×C4—1222no—
268C2×C4—1224no—
269C2×C4—2442no—
270C2×C4—2442no—
271C2×C4—2442no—
272C2×C4C22⋊C4×D6111yescharacteristic
273C2×C4C22⋊C4×D6111yescharacteristic
274C2×C4#1451974512yes—
275C2×C4#1451974712yes—
276C2×C4#1451979712yes—
277C2×C4#1451980311yescharacteristic
278C2×C4#1451980811yescharacteristic
279C8—82no—
280C8—4882no—
281C22—21no—
282C22—21no—
283C22#1683961711yescharacteristic
284C22#1683975611yescharacteristic
285C22#1683976111yescharacteristic
286C4—21no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—1224no—
291C4—2444no—
292C4—2442no—
293C4#101×D6112yes—
294C4#1683990512yes—
295C4#1683992212yes—
296C2—21no—
297C2—21no—
298C2#2144574111yescharacteristic
299C2#2144575711yescharacteristic
300C2#2144579511yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1801C2
socleC22×C61#379
prime radicalC23×C61C2×C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1801D61
hypocenterC61#8390
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
127. Dic122#101semidirect
173. C61#8390semidirect

Maximal subgroups

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

subgroupindex
C2×#168404902
#214444922
#214442472
#214441432
C61×#18012
#839061