Finite Group Databasebeta

← #29571757

Subgroups — #29571757

Order 15616.

966 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 966 conjugacy classes of subgroups — too many to draw. Only its 147 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×D4×Dic122C211yesmaximal, characteristic
2#21444501C212yesmaximal, complemented
3#21444247C212yesmaximal, complemented
4#21444140C211yesmaximal, characteristic
5C61×#1800C211yesmaximal, characteristic
6C23×Dic122C412yes—
7D4×Dic122—24no—
8D4×Dic122—24no—
9C2×#14519777C2211yescharacteristic
10C2×#14519776—21no—
11C2×#14519756C412yes—
12C2×#14519754—22no—
13C2×C4×Dic122C2211yescharacteristic
14C2×D4×Dic61—21no—
15#16839815C2212yes—
16#16839510—24no—
17#16839492—24no—
18#16839486—22no—
19C2×C4×D4×C61C2211yescharacteristic
20C61×#423C2212yes—
21C2×#100×C61—21no—
22C61×#343—21no—
23C22×Dic122—22no—
24C22×Dic122—24no—
25C22×Dic122—24no—
26C22×Dic122—48no—
27C22×Dic122C2×C411yescharacteristic
28C22×Dic122C2×C412yes—
29C23×Dic61—22no—
30#14519777—22no—
31#14519777D414yes—
32#14519776—28no—
33#14519756—24no—
34#14519756—24no—
35#14519754—216no—
36C4×Dic122—22no—
37C4×Dic122D414yes—
38D4×Dic61—28no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×#11758196—22no—
43C2×#11758196—21no—
44C2×C22⋊Dic61—22no—
45C2×C4×Dic61—21no—
46C4⋊#92418—22no—
47C22⋊#92418—22no—
48C4×#92418—22no—
49C22×D4×C61D411yescharacteristic
50C23×C4×C61C2×C412yes—
51C2×M16×C61—22no—
52C4×D4×C61—22no—
53C4×D4×C61—22no—
54C2×C4⋊C4×C61D411yescharacteristic
55C2×C22⋊C4×C61D412yes—
56C2×C42×C61C2311yescharacteristic
57C4⋊C8×C61—22no—
58C4⋊C8×C61—22no—
59#100×C61—42no—
60C24×C61C22⋊C412yes—
61C2×D4×C61—22no—
62C2×D4×C61—22no—
63C2×D4×C61SD1612yes—
64C2×D4×C61Q1612yes—
65C22×C4×C61—22no—
66C22×C4×C61—22no—
67C22×C4×C61—44no—
68C22×C4×C61C22⋊C412yes—
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C22×Dic61—21no—
73C22×Dic61—28no—
74C22×Dic61—22no—
75C22×Dic61—21no—
76C22×Dic61—48no—
77C2×Dic122—21no—
78C2×Dic122—24no—
79C2×Dic122—28no—
80C2×Dic122—22no—
81C2×Dic122—416no—
82C2×Dic122—48no—
83C2×Dic122—48no—
84C2×Dic122—48no—
85C2×Dic122—44no—
86C2×Dic122C22⋊C412yes—
87C2×Dic122C22⋊C414yes—
88M16×C61—42no—
89M16×C61—42no—
90C2×C8×C61—42no—
91C4⋊C4×C61—22no—
92C22⋊C4×C61—24no—
93C42×C61C2×D412yes—
94C42×C61C2×D412yes—
95C22⋊Dic61—28no—
96C4⋊Dic61—21no—
97C4⋊Dic61—24no—
98C4⋊Dic61—24no—
99C4⋊Dic61C22⋊C412yes—
100#11758196—28no—
101#11758196—24no—
102C4×Dic61—24no—
103C2×#92418—42no—
104C23×C61—22no—
105C23×C61—22no—
106C23×C61—42no—
107C23×C61—42no—
108C23×C61#10314yes—
109C23×C61C2×C22⋊C411yescharacteristic
110D4×C61—24no—
111D4×C61—24no—
112C2×C4×C61—22no—
113C2×C4×C61—22no—
114C2×C4×C61—41no—
115C2×C4×C61—44no—
116C2×C4×C61—44no—
117C2×C4×C61—42no—
118C2×C4×C61C2×C22⋊C411yescharacteristic
119C2×C4×C61C2×C22⋊C411yescharacteristic
120C2×C4×C61#12012yes—
121C2×C4×C61#12012yes—
122C2×C4×C61C2×SD1611yescharacteristic
123C2×C4×C61C2×Q1611yescharacteristic
124C2×C4×C61#13712yes—
125C2×Dic61—24no—
126C2×Dic61—22no—
127C2×Dic61—22no—
128C2×Dic61—24no—
129C2×Dic61—416no—
130C2×Dic61—44no—
131C2×Dic61—42no—
132Dic122—24no—
133Dic122—22no—
134Dic122—416no—
135Dic122—44no—
136Dic122—44no—
137Dic122#10014yescomplemented
138C8×C61—82no—
139#92418—82no—
140#8391—611nomaximal, Sylow
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—24no—
144C22×C61—44no—
145C22×C61—44no—
146C22×C61—42no—
147C22×C61—42no—
148C22×C61#37911yescharacteristic
149C22×C61C2×#10311yescharacteristic
150C22×C61#41911yescharacteristic
151C4×C61—22no—
152C4×C61—21no—
153C4×C61—42no—
154C4×C61C2×#10012yes—
155C4×C61#44812yes—
156C4×C61#45112yes—
157Dic61—24no—
158Dic61—42no—
159Dic61—44no—
160C2×D4×Q8—611no—
161#1800D6111yescharacteristic
162#1548—612no—
163#1487—612no—
164#1460—611no—
165C2×C61—21no—
166C2×C61—21no—
167C2×C61—44no—
168C2×C61#178311yescharacteristic
169C2×C61#178711yescharacteristic
170C2×C61#179611yescharacteristic
171C23×Q8—612no—
172D4×Q8—1224no—
173D4×Q8—1224no—
174C2×C4⋊Q8—611no—
175C2×C4⋊Q8—1221no—
176C2×C22⋊Q8—612no—
177C2×C22⋊Q8—1222no—
178C2×C4×Q8—611no—
179C2×C4×D4—1221no—
180C2×C4×D4C2×D6111yescharacteristic
181#431—612no—
182#423C2×D6112yes—
183C2×#100—21no—
184#343—21no—
185#333—1224no—
186Q8⋊C8—1224no—
187#323—1222no—
188C61#839111yescharacteristic, complemented, Sylow
189C22×Q8—611no—
190C22×Q8—612no—
191C22×Q8—1224no—
192C22×Q8—1224no—
193C22×Q8—1222no—
194C22×Q8—2448no—
195C22×D4C22⋊D6111yescharacteristic
196C23×C4—1222no—
197C23×C4C4×D6112yes—
198C2×M16—22no—
199C4⋊Q8—614no—
200C4⋊Q8—1228no—
201C4⋊Q8—1222no—
202C22⋊Q8—12216no—
203C22⋊Q8—1224no—
204C22⋊Q8—1224no—
205C4×Q8—614no—
206C4×Q8—1222no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—1228no—
210C2×C4⋊C4—611no—
211C2×C4⋊C4—1222no—
212C2×C4⋊C4—1221no—
213C2×C4⋊C4—1221no—
214C2×C4⋊C4—1221no—
215C2×C4⋊C4C22⋊D6111yescharacteristic
216C2×C22⋊C4—1222no—
217C2×C22⋊C4D24412yes—
218C2×C42—1221no—
219C2×C42C22×D6111yescharacteristic
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—1222no—
223#100—42no—
224C22⋊C8—1222no—
225C4×C8—1222no—
226C24#1175819812yes—
227C2×Q8—612no—
228C2×Q8—614no—
229C2×Q8—1224no—
230C2×Q8—1228no—
231C2×Q8—1221no—
232C2×Q8—1222no—
233C2×Q8—2444no—
234C2×Q8—2448no—
235C2×Q8—2448no—
236C2×Q8—24416no—
237C2×Q8—2448no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4#1175820012yes—
241C2×D4#1175820212yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—44no—
245C22×C4—1221no—
246C22×C4—1228no—
247C22×C4—1221no—
248C22×C4—1222no—
249C22×C4—2448no—
250C22×C4#1175819812yes—
251C22×C4C2×C4×D6111yescharacteristic
252C22×C4C2×D24411yescharacteristic
253C22×C4C2×C22⋊D6111yescharacteristic
254M16—42no—
255M16—42no—
256C2×C8—42no—
257C2×C8—2442no—
258C4⋊C4—22no—
259C4⋊C4—612no—
260C4⋊C4—1228no—
261C4⋊C4—1221no—
262C4⋊C4—1224no—
263C4⋊C4—1224no—
264C4⋊C4—1224no—
265C22⋊C4—24no—
266C22⋊C4—1228no—
267C42—1224no—
268C42D4×D6112yes—
269C42D4×D6112yes—
270C23—22no—
271C23—22no—
272C23—42no—
273C23—42no—
274C23#1451970014yes—
275C23C2×#1175819811yescharacteristic
276Q8—614no—
277Q8—1224no—
278Q8—1222no—
279Q8—2444no—
280Q8—24416no—
281Q8—2444no—
282D4—24no—
283D4—24no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—1224no—
291C2×C4—1222no—
292C2×C4—1222no—
293C2×C4—1224no—
294C2×C4—24416no—
295C2×C4—2442no—
296C2×C4—2444no—
297C2×C4C22⋊C4×D6111yescharacteristic
298C2×C4C22⋊C4×D6111yescharacteristic
299C2×C4#1451974512yes—
300C2×C4#1451974712yes—
301C2×C4C2×#1175820011yescharacteristic
302C2×C4C2×#1175820211yescharacteristic
303C2×C4#1451980712yes—
304C8—82no—
305C8—4882no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—44no—
310C22—44no—
311C22—42no—
312C22—42no—
313C22#1683961711yescharacteristic
314C22C2×#1451970011yescharacteristic
315C22#1683977011yescharacteristic
316C4—21no—
317C4—22no—
318C4—42no—
319C4—1224no—
320C4—2444no—
321C4—2442no—
322C4#100×D6112yes—
323C4#1683990012yes—
324C4#1683991712yes—
325C2—21no—
326C2—21no—
327C2—44no—
328C2#2144573811yescharacteristic
329C2#2144575311yescharacteristic
330C2#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#8391
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. #21444501C2semidirect
3. #21444247C2semidirect
137. Dic122#100semidirect
188. C61#8391semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1222
#214445012
#214442472
#214441402
C61×#18002
#839161