Finite Group Databasebeta

← #29571729

Subgroups — #29571729

Order 15616.

802 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16840497C211yesmaximal, characteristic
2#21445815C211yesmaximal, characteristic, complemented
3#21444486C212yesmaximal
4#21444249C212yesmaximal, complemented
5C61×#1461C211yesmaximal, characteristic, complemented
6#16840497—24no—
7#16840497—24no—
8C2×#14519774C2211yescharacteristic
9C2×#14519766—21no—
10C2×#14519765C412yes—
11C2×#14519762—22no—
12C2×C4×Dic122C2211yescharacteristic
13C2×C4×D244C412yes—
14C2×#14519739—21no—
15#16839786C2212yes—
16C2×#14519696—21no—
17#16839531—21no—
18#16839505—24no—
19#16839494—24no—
20C2×C4×Q8×C61C2211yescharacteristic
21C61×#431C2212yes—
22C61×#324—22no—
23C22×Dic122C2×C411yescharacteristic
24C22×D244—21no—
25C22×D244—21no—
26C22×D244C2×C412yes—
27C22×C4×D61—22no—
28C2×#11758194—22no—
29#14519774—22no—
30#14519774D414yes—
31#14519766—28no—
32#14519765—24no—
33#14519765—24no—
34#14519762—216no—
35C4×Dic122—22no—
36C4×Dic122D414yes—
37C4×D244—24no—
38C4×D244—24no—
39#14519739—28no—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×#11758196—21no—
42C2×#11758198—22no—
43C2×#11758198—22no—
44C2×#11758198—22no—
45C2×C4×Dic61—21no—
46C4⋊#92418—22no—
47C4⋊#92418—22no—
48#14519696—42no—
49C22×Q8×C61D411yescharacteristic
50C4×Q8×C61—22no—
51C4×Q8×C61—22no—
52C2×C4⋊C4×C61D411yescharacteristic
53C2×C4⋊C4×C61D412yes—
54C2×C42×C61C2×C412yes—
55C2×C42×C61C2311yescharacteristic
56C4⋊C8×C61—22no—
57C22⋊C8×C61—22no—
58C4×C8×C61—22no—
59C23×D61—22no—
60C2×Q8×C61—22no—
61C2×Q8×C61—22no—
62C2×Q8×C61—21no—
63C2×Q8×C61—21no—
64C22×C4×C61C22⋊C412yes—
65C22×C4×C61C22⋊C412yes—
66C22×C4×C61C22×C411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×Dic61—21no—
70C22×Dic61—21no—
71C2×D244—22no—
72C2×D244—24no—
73C2×D244—24no—
74C2×D244—42no—
75C2×D244—41no—
76C2×D244—41no—
77C2×D244—44no—
78C2×D244—82no—
79C2×D244C22⋊C414yes—
80C2×C4×D61—28no—
81C2×C4×D61—84no—
82C2×Dic122—21no—
83C2×Dic122—42no—
84C2×Dic122C22⋊C412yes—
85C2×C8×C61—42no—
86C4⋊C4×C61—22no—
87C4⋊C4×C61—24no—
88C42×C61—22no—
89C42×C61—22no—
90C42×C61C2×D412yes—
91C42×C61C2×D412yes—
92#11758198—28no—
93#11758198—28no—
94#11758198—28no—
95C4⋊Dic61—21no—
96C4⋊Dic61C22⋊C412yes—
97#11758196—24no—
98C4×Dic61—24no—
99#11758194—42no—
100#11758194—42no—
101C2×#92418—42no—
102C23×C61C2×C22⋊C411yescharacteristic
103C22×D61—24no—
104C22×D61—44no—
105C22×D61—44no—
106C22×D61—42no—
107C22×D61—42no—
108Q8×C61—42no—
109Q8×C61—42no—
110C2×C4×C61—22no—
111C2×C4×C61—22no—
112C2×C4×C61—22no—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61—41no—
116C2×C4×C61—42no—
117C2×C4×C61C2×C22⋊C411yescharacteristic
118C2×C4×C61C2×C22⋊C411yescharacteristic
119C2×C4×C61#12012yes—
120C2×C4×C61#12012yes—
121C2×C4×C61#13512yes—
122C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C61#13711yescharacteristic
124C2×Dic61—22no—
125C2×Dic61—22no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128D244—44no—
129D244—44no—
130D244—44no—
131D244—44no—
132D244—84no—
133C4×D61—88no—
134Dic122—24no—
135Dic122—42no—
136C8×C61—82no—
137#92418—82no—
138#8386—611nomaximal, Sylow
139C22×C61—21no—
140C22×C61—21no—
141C22×C61#37911yescharacteristic
142C22×C61#41611yescharacteristic
143C22×C61#41711yescharacteristic
144C2×D61—44no—
145C2×D61—44no—
146C2×D61—44no—
147C2×D61—84no—
148C2×D61—84no—
149C4×C61—22no—
150C4×C61—21no—
151C4×C61—42no—
152C4×C61—42no—
153C4×C61#41312yes—
154C4×C61#44912yes—
155C4×C61#45212yes—
156Dic61—42no—
157Dic61—42no—
158C2×#559—611no—
159#1801—611no—
160#1545—612no—
161#1487—612no—
162#1461D6111yescharacteristic, complemented
163C2×C61—21no—
164C2×C61—21no—
165C2×C61#178511yescharacteristic
166C2×C61#178811yescharacteristic
167C2×C61#179511yescharacteristic
168D61—84no—
169#559—1224no—
170#559—1224no—
171C2×#127—611no—
172C2×#124—612no—
173C2×#124—1221no—
174C2×#123—1222no—
175C2×C4×Q8—611no—
176C2×C4×Q8C2×D6111yescharacteristic
177C2×C4×D4—612no—
178C2×C4×D4—1221no—
179#431C2×D6112yes—
180#423—612no—
181C2×#101—1221no—
182#343—1221no—
183#331—1224no—
184Q8⋊C8—1224no—
185#324—22no—
186C61#838611yescharacteristic, complemented, Sylow
187C22×Q8—611no—
188C22×Q8D24411yescharacteristic
189C22×D4—612no—
190C22×D4—1221no—
191C22×D4—1221no—
192C23×C4—1222no—
193C2×M16—1222no—
194#127—614no—
195#127—1222no—
196#124—1224no—
197#124—1224no—
198#124—1228no—
199#123—12216no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—614no—
203C4×Q8—1222no—
204C4×D4—1224no—
205C4×D4—1224no—
206C4×D4—1228no—
207C2×C4⋊C4—611no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4D24411yescharacteristic
210C2×C4⋊C4C22⋊D6112yes—
211C2×C22⋊C4—1222no—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4—1222no—
214C2×C42—1221no—
215C2×C42C4×D6112yes—
216C2×C42C22×D6111yescharacteristic
217C4⋊C8—22no—
218C4⋊C8—1222no—
219C4⋊C8—1222no—
220#101—2442no—
221C22⋊C8—22no—
222C4×C8—22no—
223C24—1222no—
224C2×Q8—21no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—21no—
228C2×Q8—612no—
229C2×Q8—1221no—
230C2×Q8—2442no—
231C2×D4—614no—
232C2×D4—1222no—
233C2×D4—1224no—
234C2×D4—1224no—
235C2×D4—4882no—
236C2×D4—2444no—
237C2×D4—2441no—
238C2×D4—2442no—
239C2×D4—2441no—
240C22×C4—1228no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4—4884no—
244C22×C4#1175819812yes—
245C22×C4#1175819812yes—
246C22×C4C2×C4×D6111yescharacteristic
247C22×C4C2×D24411yescharacteristic
248C22×C4C2×C22⋊D6111yescharacteristic
249M16—2442no—
250M16—2442no—
251C2×C8—42no—
252C2×C8—2442no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—612no—
256C4⋊C4—1221no—
257C4⋊C4—1224no—
258C22⋊C4—1228no—
259C22⋊C4—1228no—
260C22⋊C4—1228no—
261C42—22no—
262C42—22no—
263C42—1224no—
264C42D4×D6112yes—
265C42D4×D6112yes—
266C23—1224no—
267C23—2444no—
268C23—2444no—
269C23—2442no—
270C23—2442no—
271C23C2×#1175819811yescharacteristic
272Q8—42no—
273Q8—42no—
274Q8—1224no—
275Q8—2442no—
276D4—4884no—
277D4—2444no—
278D4—2444no—
279D4—2444no—
280D4—2444no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—1222no—
289C2×C4—1222no—
290C2×C4—4888no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4C22⋊C4×D6111yescharacteristic
294C2×C4C22⋊C4×D6111yescharacteristic
295C2×C4#1451974512yes—
296C2×C4#1451974712yes—
297C2×C4#1451979912yes—
298C2×C4#1451980511yescharacteristic
299C2×C4#1451981011yescharacteristic
300C8—82no—
301C8—4882no—
302C22—21no—
303C22—21no—
304C22—4884no—
305C22—4884no—
306C22—2444no—
307C22—2444no—
308C22—2444no—
309C22#1683961711yescharacteristic
310C22#1683975911yescharacteristic
311C22#1683976411yescharacteristic
312C4—21no—
313C4—22no—
314C4—42no—
315C4—42no—
316C4—2442no—
317C4—2442no—
318C4#1683974512yes—
319C4#1683990912yes—
320C4#1683992612yes—
321C2—21no—
322C2—21no—
323C2—4884no—
324C2#2144574611yescharacteristic
325C2#2144576011yescharacteristic
326C2#2144578711yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1461C2
socleC22×C61#379
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenter#1461D61
hypocenterC61#8386
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #21445815C2semidirect
4. #21444249C2semidirect
5. C61×#1461C2semidirect
162. #1461D61semidirect
186. C61#8386semidirect

Maximal subgroups

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

subgroupindex
C2×#168404972
#214458152
#214444862
#214442492
C61×#14612
#838661