Finite Group Databasebeta

← #29606781

Subgroups — #29606781

Order 15616.

1830 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839673C211yesmaximal, characteristic
2C2×#16839653C213yesmaximal
3#21444090C211yesmaximal, characteristic, complemented
4#21444082C213yesmaximal, complemented
5#21444061C213yesmaximal
6#21444046C213yesmaximal, complemented
7C22×C4×Q8×C61C211yesmaximal, characteristic, complemented
8C22×#11758196C2213yes—
9C22×#11758198C2211yescharacteristic
10C22×#11758198C2213yes—
11C22×#11758198C2213yes—
12#16839673C418yescomplemented
13#16839653—212no—
14C2×#14519684C2213yes—
15C2×#14519684C2213yes—
16C2×#14519684C2213yes—
17C2×#14519684C2216yes—
18C2×#14519684C2213yes—
19C23×Q8×C61C2211yescharacteristic
20C2×C4×Q8×C61—212no—
21C22×C4⋊C4×C61C2213yes—
22C22×C42×C61C2213yes—
23C24×D61C2311yescharacteristic
24C23×Dic61C2311yescharacteristic
25C23×Dic61C2313yes—
26C23×Dic61C2313yes—
27C2×#11758196—26no—
28C2×#11758196—46no—
29C2×#11758198—26no—
30C2×#11758198—26no—
31C2×#11758198—26no—
32C2×#11758198—26no—
33C2×#11758198—46no—
34C2×#11758198C2×C4112yes—
35#14519684—26no—
36#14519684—26no—
37#14519684—26no—
38#14519684—46no—
39#14519684C2×C4112yes—
40C22×Q8×C61—26no—
41C22×Q8×C61—26no—
42C22×Q8×C61C2×C414yes—
43C23×C4×C61C2313yes—
44C23×C4×C61C2313yes—
45C23×C4×C61C2311yescharacteristic
46C4×Q8×C61—232no—
47C2×C4⋊C4×C61—212no—
48C2×C4⋊C4×C61—26no—
49C2×C42×C61—212no—
50C2×C42×C61D4112yes—
51C24×C61C2411yescharacteristic
52C23×D61—23no—
53C23×D61—23no—
54C23×D61—23no—
55C23×D61—21no—
56C23×D61—21no—
57C23×D61—23no—
58C23×D61C22×C412yes—
59C2×Q8×C61—28no—
60C2×Q8×C61—224no—
61C2×Q8×C61—224no—
62C22×C4×C61—26no—
63C22×C4×C61—26no—
64C22×C4×C61—21no—
65C22×C4×C61—23no—
66C22×C4×C61—26no—
67C22×C4×C61—23no—
68C22×C4×C61—26no—
69C22×C4×C61—23no—
70C22×C4×C61C22×C416yes—
71C22×C4×C61C2×D416yes—
72C22×C4×C61C2×D416yes—
73C22×C4×C61C2×D416yes—
74C22×C4×C61#4116yes—
75C22×Dic61—23no—
76C22×Dic61—23no—
77C22×Dic61—26no—
78C22×Dic61—26no—
79C22×Dic61—23no—
80C22×Dic61—23no—
81C22×Dic61—26no—
82C22×Dic61—26no—
83C22×Dic61—23no—
84C22×Dic61—23no—
85C22×Dic61—23no—
86C22×Dic61—21no—
87C22×Dic61C22×C416yes—
88C4⋊C4×C61—224no—
89C42×C61—224no—
90#11758198—224no—
91#11758198—412no—
92#11758198—44no—
93#11758196—412no—
94C23×C61C23×C411yescharacteristic
95C23×C61C22×D413yes—
96C23×C61C2×#4113yes—
97C23×C61H(5,𝔽2)13yes—
98C23×C612−1+411yescharacteristic
99C23×C612−1+411yescharacteristic
100C23×C612−1+413yes—
101C22×D61—26no—
102C22×D61—22no—
103C22×D61—26no—
104C22×D61—41no—
105C22×D61—43no—
106C22×D61—43no—
107C22×D61—43no—
108C22×D61—43no—
109C22×D61—43no—
110C22×D61—43no—
111C22×D61—43no—
112C22×D61—46no—
113Q8×C61—232no—
114C2×C4×C61—212no—
115C2×C4×C61—26no—
116C2×C4×C61—26no—
117C2×C4×C61—212no—
118C2×C4×C61—26no—
119C2×C4×C61—26no—
120C2×C4×C61—26no—
121C2×C4×C61—26no—
122C2×C4×C61—26no—
123C2×C4×C61—46no—
124C2×C4×C61—46no—
125C2×C4×C61C4×D4112yes—
126C2×C4×C61#12014yes—
127C2×Dic61—26no—
128C2×Dic61—212no—
129C2×Dic61—43no—
130C2×Dic61—43no—
131C2×Dic61—46no—
132C2×Dic61—41no—
133C2×Dic61—43no—
134C2×Dic61—43no—
135C2×Dic61—43no—
136C2×Dic61—43no—
137C2×Dic61—43no—
138C2×Dic61—46no—
139C2×Dic61—46no—
140#14301—611nomaximal, Sylow
141C22×C61C2×C4×D413yes—
142C22×C61#51911yescharacteristic
143C22×C61#52013yes—
144C22×C61C2×#12011yescharacteristic
145C22×C61#52513yes—
146C22×C61#53513yes—
147C22×C61#53613yes—
148C22×C61#53613yes—
149C22×C61#53713yes—
150C22×C61#54113yes—
151C22×C61#54213yes—
152C22×C61#54216yes—
153C2×D61—46no—
154C2×D61—46no—
155C2×D61—42no—
156C2×D61—81no—
157C2×D61—81no—
158C2×D61—83no—
159C2×D61—83no—
160C4×C61—24no—
161C4×C61—212no—
162C4×C61—46no—
163Dic61—46no—
164Dic61—81no—
165Dic61—83no—
166C22×C4×Q8D6111yescharacteristic, complemented
167C2×#393—611no—
168C2×#388—613no—
169#1446—611no—
170#1444—613no—
171#1439—613no—
172#1434—613no—
173C2×C61C4×#12011yescharacteristic
174C2×C61#232313yes—
175C2×C61#232613yes—
176C2×C61#261411yescharacteristic
177C2×C61#261511yescharacteristic
178C2×C61#263213yes—
179C2×C61#263713yes—
180D61—82no—
181C23×Q8C2×D6111yescharacteristic
182C2×C4×Q8—212no—
183C22×C4⋊C4—613no—
184C22×C4⋊C4C2×D6113yes—
185C22×C22⋊C4—613no—
186C22×C22⋊C4—611no—
187C22×C22⋊C4—613no—
188C22×C42C2×D6113yes—
189#393—618no—
190#388—12212no—
191C2×#95—613no—
192C2×#95—613no—
193C2×#95—613no—
194C2×#95—613no—
195C2×#95—616no—
196C61#1430111yescharacteristic, complemented, Sylow
197C25—611no—
198C22×Q8—26no—
199C22×Q8—26no—
200C22×Q8C4×D6114yes—
201C23×C4—613no—
202C23×C4—611no—
203C23×C4—613no—
204C23×C4C22×D6113yes—
205C23×C4C22×D6113yes—
206C23×C4C22×D6111yescharacteristic
207C4×Q8—232no—
208C2×C4⋊C4—212no—
209C2×C4⋊C4—26no—
210C2×C4⋊C4—1226no—
211C2×C4⋊C4—2446no—
212C2×C22⋊C4—6112no—
213C2×C22⋊C4—1226no—
214C2×C22⋊C4—1226no—
215C2×C22⋊C4—1226no—
216C2×C22⋊C4—1226no—
217C2×C22⋊C4—2446no—
218C2×C42—212no—
219C2×C42C22⋊D61112yes—
220#95—6112no—
221#95—1226no—
222#95—1226no—
223#95—1226no—
224#95—2446no—
225C24—612no—
226C24—1223no—
227C24—1223no—
228C24—1223no—
229C24—1221no—
230C24—1221no—
231C24—1223no—
232C24C23×D6111yescharacteristic
233C2×Q8—28no—
234C2×Q8—224no—
235C2×Q8—224no—
236C22×C4—26no—
237C22×C4—26no—
238C22×C4—21no—
239C22×C4—23no—
240C22×C4—26no—
241C22×C4—23no—
242C22×C4—26no—
243C22×C4—23no—
244C22×C4—616no—
245C22×C4—1223no—
246C22×C4—1223no—
247C22×C4—1226no—
248C22×C4—1226no—
249C22×C4—1223no—
250C22×C4—1223no—
251C22×C4—1226no—
252C22×C4—1226no—
253C22×C4—1223no—
254C22×C4—1221no—
255C22×C4—1223no—
256C22×C4—1223no—
257C22×C4C2×C4×D6116yes—
258C22×C4#1175822216yes—
259C22×C4C2×C22⋊D6116yes—
260C22×C4C2×C22⋊D6116yes—
261C22×C4C2×C22⋊D6116yes—
262C4⋊C4—224no—
263C4⋊C4—24412no—
264C22⋊C4—12224no—
265C22⋊C4—24412no—
266C22⋊C4—2444no—
267C42—224no—
268C23—1226no—
269C23—1222no—
270C23—1226no—
271C23—2441no—
272C23—2443no—
273C23—2443no—
274C23—2443no—
275C23—2443no—
276C23—2443no—
277C23—2443no—
278C23—2443no—
279C23—2446no—
280C23C22×C4×D6111yescharacteristic
281C23C22×C22⋊D6113yes—
282C23C2×#1175822213yes—
283C23#1451982313yes—
284C23#1451982411yescharacteristic
285C23#1451982511yescharacteristic
286C23#1451982513yes—
287Q8—232no—
288C2×C4—212no—
289C2×C4—26no—
290C2×C4—212no—
291C2×C4—26no—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—26no—
295C2×C4—26no—
296C2×C4—26no—
297C2×C4—46no—
298C2×C4—46no—
299C2×C4—1226no—
300C2×C4—12212no—
301C2×C4—2441no—
302C2×C4—2443no—
303C2×C4—2443no—
304C2×C4—2443no—
305C2×C4—2443no—
306C2×C4—2443no—
307C2×C4—2446no—
308C2×C4—2446no—
309C2×C4—2443no—
310C2×C4—2443no—
311C2×C4—2446no—
312C2×C4C4×C22⋊D61112yes—
313C2×C4#1451974614yes—
314C22—4881no—
315C22—4881no—
316C22—4883no—
317C22—4883no—
318C22—2446no—
319C22—2446no—
320C22—2442no—
321C22C2×C4×C22⋊D6113yes—
322C22#1684023811yescharacteristic
323C22#1684024413yes—
324C22C2×#1451974611yescharacteristic
325C22#1684027313yes—
326C22#1684032613yes—
327C22#1684033613yes—
328C22#1684033613yes—
329C22#1684034013yes—
330C22#1684037013yes—
331C22#1684038413yes—
332C22#1684038416yes—
333C4—24no—
334C4—212no—
335C4—46no—
336C4—4881no—
337C4—4883no—
338C4—2446no—
339C2—4882no—
340C2C4×#1451974611yescharacteristic
341C2#2144849513yes—
342C2#2144852613yes—
343C2#2145153811yescharacteristic
344C2#2145154311yescharacteristic
345C2#2145167913yes—
346C2#2145172513yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC22×C4×Q8×C61C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenterC22×C4×Q8D61
hypocenterC61#14301
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #21444090C2semidirect
4. #21444082C2semidirect
6. #21444046C2semidirect
7. C22×C4×Q8×C61C2semidirect
12. #16839673C4semidirect
166. C22×C4×Q8D61semidirect
196. C61#14301semidirect

Maximal subgroups

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

subgroupindex
C2×#168396732
C2×#168396532
#214440902
#214440822
#214440612
#214440462
C22×C4×Q8×C612
#1430161