Finite Group Databasebeta

← #29711982

Subgroups — #29711982

Order 15616.

1158 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 191 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×#16840374C211yesmaximal, characteristic
2C2×#16839920C212yesmaximal
3#21448403C211yesmaximal, characteristic
4#21446971C212yesmaximal, complemented
5#21446921C212yesmaximal, complemented
6#21446159C214yesmaximal, complemented
7#21446129C212yesmaximal
8C61×#1818C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic
10C22×Dic244C2214yes—
11#16840374—28no—
12C2×#14519777C2211yescharacteristic
13C2×#14519770—22no—
14C2×#14519769C2211yescharacteristic
15C2×#14519767—22no—
16C2×#14519756C2212yes—
17C2×C4×Dic122C2212yes—
18C2×#14519736C2211yescharacteristic
19#16839920—28no—
20C2×#14519702C2214yes—
21C2×#14519702C2214yes—
22#16839668C2212yes—
23#16839668C2212yes—
24#16839667—22no—
25#16839623—22no—
26#16839525C2214yes—
27C2×#117×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2212yes—
29C61×#336C2212yes—
30C61×#336C2212yes—
31C22×Dic122—21no—
32C22×Dic122—24no—
33C22×Dic122—24no—
34C22×Dic122—42no—
35C22×Dic122C2312yes—
36C22×Dic122C2312yes—
37C23×Dic61—21no—
38C2×Dic244—28no—
39C2×Dic244—28no—
40C2×Dic244—44no—
41#14519777—24no—
42#14519770—216no—
43#14519769—24no—
44#14519767—216no—
45#14519756—24no—
46#14519756D418yes—
47C4×Dic122—24no—
48C4×Dic122—24no—
49#14519736—42no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61C2312yes—
52C2×C4⋊Dic61C2312yes—
53C2×#11758196—22no—
54C2×#11758196—22no—
55C2×#11758196—22no—
56C2×C22⋊Dic61—22no—
57C2×C22⋊Dic61—21no—
58C2×C4×Dic61—22no—
59C2×C4×Dic61—21no—
60#14519702—28no—
61#14519702—44no—
62#14519684—24no—
63#14519684—24no—
64C23×C4×C61C2311yescharacteristic
65C22×C8×C61C2314yes—
66#117×C61—24no—
67C2×C4⋊C4×C61D412yes—
68C2×C22⋊C4×C61D412yes—
69C2×C42×C61C2311yescharacteristic
70C2×C42×C61C2311yescharacteristic
71C22⋊C8×C61—24no—
72C24×C61C2×D411yescharacteristic
73C22×C4×C61—22no—
74C22×C4×C61—22no—
75C22×C4×C61C2×D412yes—
76C22×C4×C61C2×D412yes—
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D414yes—
79C22×C4×C61C2411yescharacteristic
80C22×Dic61—24no—
81C22×Dic61—22no—
82C22×Dic61—22no—
83C22×Dic61—22no—
84C22×Dic61—22no—
85C22×Dic61—42no—
86C22×Dic61—42no—
87C2×Dic122—24no—
88C2×Dic122—24no—
89C2×Dic122—24no—
90C2×Dic122—24no—
91C2×Dic122—24no—
92C2×Dic122—22no—
93C2×Dic122—22no—
94C2×Dic122—416no—
95C2×Dic122—42no—
96C2×Dic122—42no—
97C2×Dic122C2×D414yes—
98C2×Dic122#4114yes—
99C2×C8×C61—24no—
100C2×C8×C61—24no—
101C2×C8×C61—24no—
102C4⋊C4×C61—24no—
103C22⋊C4×C61—24no—
104C42×C61—22no—
105C42×C61—22no—
106C22⋊Dic61—28no—
107C22⋊Dic61—42no—
108C4⋊Dic61—28no—
109C4⋊Dic61—22no—
110C4⋊Dic61—24no—
111C4⋊Dic61C2×D414yes—
112#11758196—28no—
113#11758196—28no—
114#11758196—28no—
115C4×Dic61—28no—
116C4×Dic61—42no—
117Dic244—416no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—24no—
121C23×C61#12012yes—
122C23×C61C22×D411yescharacteristic
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—28no—
126C2×C4×C61—22no—
127C2×C4×C61—22no—
128C2×C4×C61—24no—
129C2×C4×C61—24no—
130C2×C4×C61#12012yes—
131C2×C4×C61C22×D412yes—
132C2×C4×C61C2×#4112yes—
133C2×C4×C61H(5,𝔽2)12yes—
134C2×Dic61—24no—
135C2×Dic61—24no—
136C2×Dic61—24no—
137C2×Dic61—48no—
138C2×Dic61—42no—
139C2×Dic61—42no—
140C2×Dic61—42no—
141C2×Dic61—42no—
142C2×Dic61—44no—
143C2×Dic61—42no—
144C2×Dic61—44no—
145Dic122—28no—
146Dic122—28no—
147Dic122—48no—
148Dic122—44no—
149Dic122—44no—
150C8×C61—44no—
151#23943—611nomaximal, Sylow
152C22×C61—22no—
153C22×C61—24no—
154C22×C61—42no—
155C22×C61—42no—
156C22×C61C2×#12011yescharacteristic
157C22×C61#52511yescharacteristic
158C22×C61#53511yescharacteristic
159C22×C61#57814yes—
160C4×C61—22no—
161C4×C61—42no—
162C4×C61—42no—
163C4×C61#54614yes—
164Dic61—44no—
165Dic61—44no—
166Dic61—44no—
167Dic61—82no—
168C2×#541—611no—
169C2×#451—612no—
170#2310—611no—
171#2026—612no—
172#2016—612no—
173#1878—614no—
174#1873—612no—
175#1818D6111yescharacteristic
176C2×C61—42no—
177C2×C61#240411yescharacteristic
178C2×C61#301112yes—
179C2×C61#312712yes—
180C2×C61#322312yes—
181C23×Q8—611no—
182C22×Q16—614no—
183#541—1228no—
184C2×C4⋊Q8—611no—
185C2×#126—1222no—
186C2×#125—611no—
187C2×#125—1222no—
188C2×C22⋊Q8—612no—
189C2×C4×Q8—612no—
190C2×#117—611no—
191C2×#117C2×D6111yescharacteristic
192#451—1228no—
193C2×#103—614no—
194C2×#103—614no—
195C2×C22⋊C8C2×D6112yes—
196#391—612no—
197#391—612no—
198#391—1222no—
199#381—1222no—
200#340—614no—
201#336C2×D6112yes—
202#336C2×D6112yes—
203C61#2394311yescharacteristic, complemented, Sylow
204C22×Q8—612no—
205C22×Q8—612no—
206C22×Q8—1224no—
207C22×Q8—1221no—
208C22×Q8—1224no—
209C22×Q8—2442no—
210C23×C4—1221no—
211C23×C4C22×D6111yescharacteristic
212C2×Q16—1228no—
213C2×Q16—1228no—
214C2×Q16—2444no—
215C22×C8C22×D6114yes—
216C4⋊Q8—1224no—
217#126—12216no—
218#125—1224no—
219#125—12216no—
220C22⋊Q8—618no—
221C22⋊Q8—1224no—
222C4×Q8—1224no—
223C4×Q8—1224no—
224#117—24no—
225#117—2442no—
226C2×C4⋊C4—612no—
227C2×C4⋊C4—612no—
228C2×C4⋊C4—1222no—
229C2×C4⋊C4—1222no—
230C2×C4⋊C4—1222no—
231C2×C4⋊C4—1222no—
232C2×C4⋊C4D24412yes—
233C2×C22⋊C4—1222no—
234C2×C22⋊C4—1221no—
235C2×C22⋊C4D24412yes—
236C2×C42—1222no—
237C2×C42—1221no—
238C2×C42C22×D6111yescharacteristic
239C2×C42C22×D6111yescharacteristic
240#103—1228no—
241#103—2444no—
242C22⋊C8—24no—
243#95—1224no—
244#95—1224no—
245C24C2×D24411yescharacteristic
246C2×Q8—614no—
247C2×Q8—614no—
248C2×Q8—1224no—
249C2×Q8—1222no—
250C2×Q8—1222no—
251C2×Q8—1224no—
252C2×Q8—1224no—
253C2×Q8—1224no—
254C2×Q8—1224no—
255C2×Q8—24416no—
256C2×Q8—2442no—
257C2×Q8—2442no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—1224no—
261C22×C4—1222no—
262C22×C4—1222no—
263C22×C4—1222no—
264C22×C4—1222no—
265C22×C4—2442no—
266C22×C4—2442no—
267C22×C4C2×D24412yes—
268C22×C4C2×D24412yes—
269C22×C4C2×D24411yescharacteristic
270C22×C4D4×D6114yes—
271C22×C4C23×D6111yescharacteristic
272Q16—24416no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C4⋊C4—24no—
277C4⋊C4—614no—
278C4⋊C4—1228no—
279C4⋊C4—1228no—
280C4⋊C4—1228no—
281C4⋊C4—1222no—
282C4⋊C4—1228no—
283C4⋊C4—1224no—
284C22⋊C4—24no—
285C22⋊C4—1228no—
286C22⋊C4—2442no—
287C42—22no—
288C42—22no—
289C42—1228no—
290C42—2442no—
291C23—21no—
292C23—24no—
293C23—21no—
294C23#1451974712yes—
295C23C22×D24411yescharacteristic
296Q8—1228no—
297Q8—1228no—
298Q8—2448no—
299Q8—2444no—
300Q8—2444no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—28no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—1224no—
309C2×C4—1224no—
310C2×C4—1224no—
311C2×C4—2444no—
312C2×C4—2442no—
313C2×C4—2448no—
314C2×C4—2442no—
315C2×C4—2442no—
316C2×C4—2444no—
317C2×C4—2442no—
318C2×C4—2442no—
319C2×C4#1451974712yes—
320C2×C4C2×D4×D6112yes—
321C2×C4#41×D6112yes—
322C2×C4#1451982312yes—
323C8—44no—
324C22—24no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22C2×#1451974711yescharacteristic
329C22#1684027411yescharacteristic
330C22#1684033111yescharacteristic
331C22#1684057914yes—
332C4—22no—
333C4—42no—
334C4—42no—
335C4—4882no—
336C4—2444no—
337C4—2444no—
338C4—2444no—
339C4#1684040414yes—
340C2—42no—
341C2#2144914011yescharacteristic
342C2#2145511612yes—
343C2#2145612012yes—
344C2#2145690412yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1818C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1818D61
hypocenterC61#23943
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1818

Complements

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

NH ≅ G/Ntype
4. #21446971C2semidirect
5. #21446921C2semidirect
6. #21446159C2semidirect
203. C61#23943semidirect

Maximal subgroups

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

subgroupindex
C2×#168403742
C2×#168399202
#214484032
#214469712
#214469212
#214461592
#214461292
C61×#18182
#2394361