Finite Group Databasebeta

← #29571769

Subgroups — #29571769

Order 15616.

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

normal subgroups only

This group has 738 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×#16840485C211yesmaximal, characteristic
2#21445815C211yesmaximal, characteristic, complemented
3#21444499C212yesmaximal, complemented
4#21444245C212yesmaximal
5C61×#1461C211yesmaximal, characteristic, complemented
6#16840485—24no—
7#16840485—24no—
8C2×#14519777C2211yescharacteristic
9C2×#14519772—22no—
10C2×#14519765C412yes—
11C2×#14519753—21no—
12C2×C4×Dic122C412yes—
13C2×C4×D244C2211yescharacteristic
14C2×#14519735—21no—
15#16839786C2212yes—
16C2×#14519696—21no—
17#16839531—21no—
18#16839510—24no—
19#16839491—24no—
20C2×C4×Q8×C61C2211yescharacteristic
21C61×#431C2212yes—
22C61×#324—22no—
23C22×Dic122C2×C412yes—
24C22×D244C2×C411yescharacteristic
25C22×C4×D61—21no—
26C2×#11758194—22no—
27#14519777—22no—
28#14519777D414yes—
29#14519772—216no—
30#14519765—24no—
31#14519765—24no—
32#14519753—28no—
33C4×Dic122—24no—
34C4×Dic122—24no—
35C4×D244—22no—
36C4×D244D414yes—
37#14519735—28no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×#11758196—22no—
42C2×#11758198—22no—
43C2×#11758198—22no—
44C2×#11758198—21no—
45C2×C4×Dic61—22no—
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—21no—
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—22no—
71C2×D244—21no—
72C2×D244—42no—
73C2×D244C22⋊C412yes—
74C2×C4×D61—24no—
75C2×C4×D61—42no—
76C2×C4×D61—41no—
77C2×C4×D61—41no—
78C2×Dic122—22no—
79C2×Dic122—44no—
80C2×Dic122C22⋊C414yes—
81C2×C8×C61—42no—
82C4⋊C4×C61—22no—
83C4⋊C4×C61—24no—
84C42×C61—22no—
85C42×C61—22no—
86C42×C61C2×D412yes—
87C42×C61C2×D412yes—
88#11758198—28no—
89#11758198—28no—
90#11758198—24no—
91C4⋊Dic61—21no—
92C4⋊Dic61—24no—
93C4⋊Dic61—24no—
94C4⋊Dic61C22⋊C412yes—
95#11758196—28no—
96C4×Dic61—28no—
97#11758194—42no—
98#11758194—42no—
99C2×#92418—42no—
100C23×C61C2×C22⋊C411yescharacteristic
101C22×D61—22no—
102C22×D61—42no—
103C22×D61—42no—
104C22×D61—41no—
105C22×D61—41no—
106Q8×C61—42no—
107Q8×C61—42no—
108C2×C4×C61—22no—
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—22no—
112C2×C4×C61—22no—
113C2×C4×C61—41no—
114C2×C4×C61—42no—
115C2×C4×C61C2×C22⋊C411yescharacteristic
116C2×C4×C61C2×C22⋊C411yescharacteristic
117C2×C4×C61#12012yes—
118C2×C4×C61#12012yes—
119C2×C4×C61#13512yes—
120C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C61#13711yescharacteristic
122C2×Dic61—22no—
123C2×Dic61—24no—
124C2×Dic61—44no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127D244—24no—
128D244—42no—
129C4×D61—44no—
130C4×D61—44no—
131Dic122—44no—
132Dic122—44no—
133C8×C61—82no—
134#92418—82no—
135#8394—611nomaximal, Sylow
136C22×C61—21no—
137C22×C61—21no—
138C22×C61#37911yescharacteristic
139C22×C61#41611yescharacteristic
140C22×C61#41711yescharacteristic
141C2×D61—42no—
142C2×D61—42no—
143C2×D61—42no—
144C2×D61—82no—
145C2×D61—82no—
146C4×C61—22no—
147C4×C61—21no—
148C4×C61—42no—
149C4×C61—42no—
150C4×C61#41312yes—
151C4×C61#44912yes—
152C4×C61#45212yes—
153Dic61—44no—
154Dic61—42no—
155C2×#556—611no—
156#1801—611no—
157#1548—612no—
158#1486—612no—
159#1461D6111yescharacteristic, complemented
160C2×C61—21no—
161C2×C61—21no—
162C2×C61#178511yescharacteristic
163C2×C61#178811yescharacteristic
164C2×C61#179611yescharacteristic
165D61—82no—
166#556—1224no—
167#556—1224no—
168C2×C4⋊Q8—611no—
169C2×#126—1222no—
170C2×#124—612no—
171C2×C22⋊Q8—1221no—
172C2×C4×Q8—612no—
173C2×C4×Q8C2×D6111yescharacteristic
174C2×C4×D4—611no—
175C2×#117—1221no—
176#431C2×D6112yes—
177#423—612no—
178C2×#101—1221no—
179#343—1221no—
180#333—1224no—
181D4⋊C8—1224no—
182#324—22no—
183C61#839411yescharacteristic, complemented, Sylow
184C22×Q8—612no—
185C22×Q8D24411yescharacteristic
186C22×D4—611no—
187C23×C4—1221no—
188C2×M16—1222no—
189C4⋊Q8—614no—
190C4⋊Q8—1222no—
191#126—12216no—
192#124—1224no—
193#124—1224no—
194C22⋊Q8—1228no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—1224no—
198C4×Q8—1224no—
199C4×D4—614no—
200C4×D4—1222no—
201#117—1228no—
202C2×C4⋊C4—611no—
203C2×C4⋊C4—1222no—
204C2×C4⋊C4—1221no—
205C2×C4⋊C4—1221no—
206C2×C4⋊C4D24411yescharacteristic
207C2×C4⋊C4C22⋊D6112yes—
208C2×C22⋊C4—1222no—
209C2×C22⋊C4—1222no—
210C2×C22⋊C4—1221no—
211C2×C42—1222no—
212C2×C42C4×D6112yes—
213C2×C42C22×D6111yescharacteristic
214C4⋊C8—22no—
215C4⋊C8—1222no—
216C4⋊C8—1222no—
217#101—2442no—
218C22⋊C8—22no—
219C4×C8—22no—
220C24—1221no—
221C2×Q8—21no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—21no—
225C2×Q8—614no—
226C2×Q8—1222no—
227C2×Q8—2444no—
228C2×D4—612no—
229C2×D4—1221no—
230C2×D4—2442no—
231C22×C4—1221no—
232C22×C4—1222no—
233C22×C4—1224no—
234C22×C4—2441no—
235C22×C4—2442no—
236C22×C4—2441no—
237C22×C4#1175819812yes—
238C22×C4#1175819812yes—
239C22×C4C2×C4×D6111yescharacteristic
240C22×C4C2×D24411yescharacteristic
241C22×C4C2×C22⋊D6111yescharacteristic
242M16—2442no—
243M16—2442no—
244C2×C8—42no—
245C2×C8—2442no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—612no—
249C4⋊C4—1228no—
250C4⋊C4—1221no—
251C4⋊C4—1224no—
252C4⋊C4—1224no—
253C22⋊C4—1228no—
254C22⋊C4—1228no—
255C22⋊C4—1224no—
256C42—22no—
257C42—22no—
258C42—1228no—
259C42D4×D6112yes—
260C42D4×D6112yes—
261C23—1222no—
262C23—2442no—
263C23—2442no—
264C23—2441no—
265C23—2441no—
266C23C2×#1175819811yescharacteristic
267Q8—42no—
268Q8—42no—
269Q8—2444no—
270Q8—2444no—
271D4—1224no—
272D4—2442no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—1222no—
281C2×C4—1224no—
282C2×C4—2444no—
283C2×C4—2441no—
284C2×C4—2441no—
285C2×C4—2444no—
286C2×C4—2444no—
287C2×C4C22⋊C4×D6111yescharacteristic
288C2×C4C22⋊C4×D6111yescharacteristic
289C2×C4#1451974512yes—
290C2×C4#1451974712yes—
291C2×C4#1451979912yes—
292C2×C4#1451980511yescharacteristic
293C2×C4#1451981011yescharacteristic
294C8—82no—
295C8—4882no—
296C22—21no—
297C22—21no—
298C22—4882no—
299C22—4882no—
300C22—2442no—
301C22—2442no—
302C22—2442no—
303C22#1683961711yescharacteristic
304C22#1683975911yescharacteristic
305C22#1683976411yescharacteristic
306C4—21no—
307C4—22no—
308C4—42no—
309C4—42no—
310C4—2444no—
311C4—2442no—
312C4#1683974512yes—
313C4#1683990912yes—
314C4#1683992612yes—
315C2—21no—
316C2—21no—
317C2—4882no—
318C2#2144574611yescharacteristic
319C2#2144576011yescharacteristic
320C2#2144579211yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1461C2
socleC22×C61#379
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1461D61
hypocenterC61#8394
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
3. #21444499C2semidirect
5. C61×#1461C2semidirect
159. #1461D61semidirect
183. C61#8394semidirect

Maximal subgroups

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

subgroupindex
C2×#168404852
#214458152
#214444992
#214442452
C61×#14612
#839461