Finite Group Databasebeta

← #29571747

Subgroups — #29571747

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×#16840482C211yesmaximal, characteristic
2#21444492C212yesmaximal, complemented
3#21444247C212yesmaximal, complemented
4#21444140C211yesmaximal, characteristic
5C61×#1800C211yesmaximal, characteristic
6#16840482—24no—
7#16840482—24no—
8C2×#14519777C2211yescharacteristic
9C2×#14519770—22no—
10C2×#14519764—21no—
11C2×#14519756C412yes—
12C2×C4×Dic122C2211yescharacteristic
13C2×#14519742—21no—
14C2×#14519736C412yes—
15#16839815C2212yes—
16#16839509—24no—
17#16839492—24no—
18#16839486—22no—
19C2×C4×D4×C61C2211yescharacteristic
20C61×#423C2212yes—
21C2×#100×C61—21no—
22C61×#343—21no—
23C22×Dic122—21no—
24C22×Dic122C2×C411yescharacteristic
25#14519777—22no—
26#14519777D414yes—
27#14519770—216no—
28#14519764—28no—
29#14519756—24no—
30#14519756—24no—
31C4×Dic122—22no—
32C4×Dic122D414yes—
33#14519742—28no—
34#14519736—24no—
35#14519736—24no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×C4⋊Dic61C2×C412yes—
39C2×#11758196—22no—
40C2×#11758196—21no—
41C2×C22⋊Dic61—22no—
42C2×C22⋊Dic61—22no—
43C2×C4×Dic61—22no—
44C2×C4×Dic61—21no—
45C4⋊#92418—22no—
46C22⋊#92418—22no—
47C4×#92418—22no—
48C22×D4×C61D411yescharacteristic
49C23×C4×C61C2×C412yes—
50C2×M16×C61—22no—
51C4×D4×C61—22no—
52C4×D4×C61—22no—
53C2×C4⋊C4×C61D411yescharacteristic
54C2×C22⋊C4×C61D412yes—
55C2×C42×C61C2311yescharacteristic
56C4⋊C8×C61—22no—
57C4⋊C8×C61—22no—
58#100×C61—42no—
59C24×C61C22⋊C412yes—
60C2×D4×C61—22no—
61C2×D4×C61—22no—
62C2×D4×C61—21no—
63C2×D4×C61—21no—
64C22×C4×C61—22no—
65C22×C4×C61—22no—
66C22×C4×C61—44no—
67C22×C4×C61C22⋊C412yes—
68C22×C4×C61C22×C411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×Dic61—22no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C2×Dic122—21no—
75C2×Dic122—24no—
76C2×Dic122—42no—
77C2×Dic122—42no—
78C2×Dic122—41no—
79C2×Dic122—41no—
80C2×Dic122C22⋊C412yes—
81M16×C61—42no—
82M16×C61—42no—
83C2×C8×C61—42no—
84C4⋊C4×C61—22no—
85C22⋊C4×C61—24no—
86C42×C61C2×D412yes—
87C42×C61C2×D412yes—
88C22⋊Dic61—28no—
89C22⋊Dic61—28no—
90C4⋊Dic61—21no—
91C4⋊Dic61—22no—
92C4⋊Dic61—24no—
93C4⋊Dic61C22⋊C412yes—
94C4⋊Dic61C22⋊C414yes—
95#11758196—28no—
96#11758196—24no—
97C4×Dic61—28no—
98C4×Dic61—24no—
99C2×#92418—42no—
100C23×C61—22no—
101C23×C61—22no—
102C23×C61—22no—
103C23×C61—42no—
104C23×C61—42no—
105C23×C61C2×C22⋊C411yescharacteristic
106D4×C61—42no—
107D4×C61—42no—
108C2×C4×C61—22no—
109C2×C4×C61—22no—
110C2×C4×C61—41no—
111C2×C4×C61—44no—
112C2×C4×C61—44no—
113C2×C4×C61—42no—
114C2×C4×C61C2×C22⋊C411yescharacteristic
115C2×C4×C61C2×C22⋊C411yescharacteristic
116C2×C4×C61#12012yes—
117C2×C4×C61#12012yes—
118C2×C4×C61#13512yes—
119C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C61#13711yescharacteristic
121C2×Dic61—22no—
122C2×Dic61—22no—
123C2×Dic61—24no—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128Dic122—24no—
129Dic122—42no—
130Dic122—44no—
131Dic122—44no—
132C8×C61—82no—
133#92418—82no—
134#8389—611nomaximal, Sylow
135C22×C61—21no—
136C22×C61—21no—
137C22×C61—42no—
138C22×C61—42no—
139C22×C61—42no—
140C22×C61—82no—
141C22×C61—82no—
142C22×C61#37911yescharacteristic
143C22×C61#41611yescharacteristic
144C22×C61#41711yescharacteristic
145C4×C61—22no—
146C4×C61—21no—
147C4×C61—42no—
148C4×C61#41312yes—
149C4×C61#44912yes—
150C4×C61#45212yes—
151Dic61—42no—
152Dic61—42no—
153Dic61—44no—
154C2×#556—611no—
155#1800D6111yescharacteristic
156#1546—612no—
157#1487—612no—
158#1460—611no—
159C2×C61—21no—
160C2×C61—21no—
161C2×C61—82no—
162C2×C61#178511yescharacteristic
163C2×C61#178811yescharacteristic
164C2×C61#179611yescharacteristic
165#556—1224no—
166#556—1224no—
167C2×C4⋊Q8—611no—
168C2×#126—1222no—
169C2×#124—1221no—
170C2×C22⋊Q8—612no—
171C2×C4×Q8—611no—
172C2×C4×Q8—1221no—
173C2×C4×D4C2×D6111yescharacteristic
174C2×#117—612no—
175#431—612no—
176#423C2×D6112yes—
177C2×#100—21no—
178#343—21no—
179#332—1224no—
180Q8⋊C8—1224no—
181#323—1222no—
182C61#838911yescharacteristic, complemented, Sylow
183C22×Q8—611no—
184C22×Q8—1221no—
185C22×D4C22⋊D6111yescharacteristic
186C23×C4C4×D6112yes—
187C2×M16—22no—
188C4⋊Q8—614no—
189C4⋊Q8—1222no—
190#126—12216no—
191#124—1228no—
192C22⋊Q8—1224no—
193C22⋊Q8—1224no—
194C4×Q8—614no—
195C4×Q8—1228no—
196C4×Q8—1222no—
197C4×D4—22no—
198C4×D4—22no—
199#117—1224no—
200#117—1224no—
201C2×C4⋊C4—611no—
202C2×C4⋊C4—612no—
203C2×C4⋊C4—1222no—
204C2×C4⋊C4—1221no—
205C2×C4⋊C4—1221no—
206C2×C4⋊C4C22⋊D6111yescharacteristic
207C2×C22⋊C4—1222no—
208C2×C22⋊C4—1222no—
209C2×C22⋊C4D24412yes—
210C2×C42—1222no—
211C2×C42—1221no—
212C2×C42C22×D6111yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—1222no—
216#100—42no—
217C22⋊C8—1222no—
218C4×C8—1222no—
219C24#1175819812yes—
220C2×Q8—612no—
221C2×Q8—1221no—
222C2×Q8—1224no—
223C2×Q8—2442no—
224C2×Q8—2441no—
225C2×Q8—2442no—
226C2×Q8—2441no—
227C2×D4—21no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—44no—
234C22×C4—1221no—
235C22×C4—1222no—
236C22×C4—1221no—
237C22×C4#1175819812yes—
238C22×C4C2×C4×D6111yescharacteristic
239C22×C4C2×D24411yescharacteristic
240C22×C4C2×C22⋊D6111yescharacteristic
241M16—42no—
242M16—42no—
243C2×C8—42no—
244C2×C8—2442no—
245C4⋊C4—22no—
246C4⋊C4—612no—
247C4⋊C4—614no—
248C4⋊C4—1228no—
249C4⋊C4—1221no—
250C4⋊C4—1222no—
251C4⋊C4—1224no—
252C4⋊C4—1224no—
253C22⋊C4—24no—
254C22⋊C4—1228no—
255C22⋊C4—1228no—
256C42—1228no—
257C42—1224no—
258C42D4×D6112yes—
259C42D4×D6112yes—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23C2×#1175819811yescharacteristic
266Q8—1224no—
267Q8—2442no—
268Q8—2444no—
269Q8—2444no—
270D4—42no—
271D4—42no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—1222no—
279C2×C4—1224no—
280C2×C4—1222no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4—2442no—
284C2×C4—2442no—
285C2×C4C22⋊C4×D6111yescharacteristic
286C2×C4C22⋊C4×D6111yescharacteristic
287C2×C4#1451974512yes—
288C2×C4#1451974712yes—
289C2×C4#1451979712yes—
290C2×C4#1451980311yescharacteristic
291C2×C4#1451980811yescharacteristic
292C8—82no—
293C8—4882no—
294C22—21no—
295C22—21no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—82no—
300C22—82no—
301C22#1683961711yescharacteristic
302C22#1683975611yescharacteristic
303C22#1683976111yescharacteristic
304C4—21no—
305C4—22no—
306C4—42no—
307C4—2442no—
308C4—2444no—
309C4—2442no—
310C4#1683974312yes—
311C4#1683990512yes—
312C4#1683992212yes—
313C2—21no—
314C2—21no—
315C2—82no—
316C2#2144574411yescharacteristic
317C2#2144575711yescharacteristic
318C2#2144579511yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1800C2
socleC22×C61#379
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenter#1800D61
hypocenterC61#8389
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #21444492C2semidirect
3. #21444247C2semidirect
182. C61#8389semidirect

Maximal subgroups

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

subgroupindex
C2×#168404822
#214444922
#214442472
#214441402
C61×#18002
#838961