Finite Group Databasebeta

← #29606785

Subgroups — #29606785

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
1C22×Q8×Dic61C211yesmaximal, characteristic, complemented
2C2×#16839655C213yesmaximal
3#21444090C211yesmaximal, characteristic, complemented
4#21444084C213yesmaximal, complemented
5#21444064C213yesmaximal
6#21444044C213yesmaximal, complemented
7C2×C61×#393C211yesmaximal, characteristic
8C2×Q8×Dic61—212no—
9C22×C4⋊Dic61C2213yes—
10C22×C4⋊Dic61C2213yes—
11C22×C22⋊Dic61C2211yescharacteristic
12C22×C22⋊Dic61C2213yes—
13C22×C4×Dic61C2213yes—
14#16839655—212no—
15C2×#14519684C2213yes—
16C2×#14519684C2213yes—
17C2×#14519684C2213yes—
18C2×#14519684C2216yes—
19C23×Q8×C61C2211yescharacteristic
20C22×C22⋊C4×C61C2213yes—
21C61×#393C418yescomplemented
22C2×#95×C61C2213yes—
23C23×Dic61C2311yescharacteristic
24C23×Dic61C2313yes—
25C23×Dic61C2313yes—
26C23×Dic61C2311yescharacteristic
27Q8×Dic61—232no—
28C2×C4⋊Dic61—26no—
29C2×C4⋊Dic61—212no—
30C2×C4⋊Dic61—26no—
31C2×C4⋊Dic61—46no—
32C2×C22⋊Dic61—26no—
33C2×C22⋊Dic61—26no—
34C2×C22⋊Dic61—46no—
35C2×C4×Dic61—212no—
36C2×C4×Dic61D4112yes—
37#14519684—26no—
38#14519684—26no—
39#14519684—26no—
40#14519684—46no—
41C25×C61C2311yescharacteristic
42C22×Q8×C61—26no—
43C22×Q8×C61—26no—
44C22×Q8×C61C2×C414yes—
45C23×C4×C61C2313yes—
46C23×C4×C61C2313yes—
47C2×C22⋊C4×C61—26no—
48C2×C22⋊C4×C61—26no—
49C2×C22⋊C4×C61C2×C4112yes—
50#95×C61C2×C4112yes—
51C24×C61—23no—
52C24×C61—23no—
53C24×C61—23no—
54C24×C61—21no—
55C24×C61—21no—
56C24×C61—23no—
57C24×C61C22×C412yes—
58C24×C61C2411yescharacteristic
59C2×Q8×C61—224no—
60C2×Q8×C61—28no—
61C2×Q8×C61—224no—
62C22×C4×C61—26no—
63C22×C4×C61—26no—
64C22×C4×C61—23no—
65C22×C4×C61—23no—
66C22×C4×C61—26no—
67C22×C4×C61—26no—
68C22×C4×C61C22×C416yes—
69C22×C4×C61C22×C416yes—
70C22×C4×C61C2×D416yes—
71C22×C4×C61#4116yes—
72C22×Dic61—21no—
73C22×Dic61—23no—
74C22×Dic61—26no—
75C22×Dic61—23no—
76C22×Dic61—26no—
77C22×Dic61—23no—
78C22×Dic61—23no—
79C22×Dic61—23no—
80C22×Dic61—26no—
81C22×Dic61—26no—
82C22×Dic61—23no—
83C22×Dic61—23no—
84C22×Dic61—23no—
85C22×Dic61—21no—
86C22×Dic61C2×D416yes—
87C22×Dic61C2×D416yes—
88C22⋊C4×C61—224no—
89C22⋊Dic61—412no—
90C22⋊Dic61—44no—
91C4⋊Dic61—224no—
92C4⋊Dic61—412no—
93C4×Dic61—224no—
94C23×C61—26no—
95C23×C61—22no—
96C23×C61—26no—
97C23×C61—41no—
98C23×C61—43no—
99C23×C61—43no—
100C23×C61—43no—
101C23×C61—43no—
102C23×C61—43no—
103C23×C61—43no—
104C23×C61—43no—
105C23×C61—46no—
106C23×C61C23×C411yescharacteristic
107C23×C61C22×D413yes—
108C23×C61C2×#4113yes—
109C23×C61H(5,𝔽2)13yes—
110C23×C612−1+411yescharacteristic
111C23×C612−1+411yescharacteristic
112C23×C612−1+413yes—
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—212no—
120C2×C4×C61—43no—
121C2×C4×C61—43no—
122C2×C4×C61—46no—
123C2×C4×C61C4×D4112yes—
124C2×Dic61—26no—
125C2×Dic61—26no—
126C2×Dic61—26no—
127C2×Dic61—26no—
128C2×Dic61—26no—
129C2×Dic61—46no—
130C2×Dic61—46no—
131C2×Dic61—41no—
132C2×Dic61—43no—
133C2×Dic61—43no—
134C2×Dic61—43no—
135C2×Dic61—43no—
136C2×Dic61—43no—
137C2×Dic61—46no—
138C2×Dic61—46no—
139C2×Dic61#12014yes—
140#14301—611nomaximal, Sylow
141C22×C61—46no—
142C22×C61—46no—
143C22×C61—42no—
144C22×C61—81no—
145C22×C61—81no—
146C22×C61—83no—
147C22×C61—83no—
148C22×C61C2×C4×D413yes—
149C22×C61#51911yescharacteristic
150C22×C61#52013yes—
151C22×C61C2×#12011yescharacteristic
152C22×C61#52513yes—
153C22×C61#53513yes—
154C22×C61#53613yes—
155C22×C61#53613yes—
156C22×C61#53713yes—
157C22×C61#54113yes—
158C22×C61#54216yes—
159C22×C61#54213yes—
160C4×C61—212no—
161C4×C61—46no—
162Dic61—24no—
163Dic61—46no—
164Dic61—81no—
165Dic61—83no—
166C22×C4×Q8—611no—
167C2×#393D6111yescharacteristic
168C2×#388—613no—
169#1446—611no—
170#1444—613no—
171#1439—613no—
172#1434—613no—
173C2×C61—82no—
174C2×C61C4×#12011yescharacteristic
175C2×C61#232313yes—
176C2×C61#232613yes—
177C2×C61#261411yescharacteristic
178C2×C61#261511yescharacteristic
179C2×C61#263213yes—
180C2×C61#263713yes—
181C23×Q8C2×D6111yescharacteristic
182C2×C4×Q8—12212no—
183C22×C4⋊C4—613no—
184C22×C4⋊C4—613no—
185C22×C22⋊C4—611no—
186C22×C22⋊C4—613no—
187C22×C22⋊C4C2×D6113yes—
188C22×C42—613no—
189#393Dic6118yescomplemented
190#388—12212no—
191C2×#95—613no—
192C2×#95—613no—
193C2×#95—613no—
194C2×#95—616no—
195C2×#95C2×D6113yes—
196C61#1430111yescharacteristic, complemented, Sylow
197C25C22×D6111yescharacteristic
198C22×Q8—26no—
199C22×Q8—26no—
200C22×Q8C2×Dic6114yes—
201C23×C4—611no—
202C23×C4—613no—
203C23×C4—611no—
204C23×C4—613no—
205C23×C4C22×D6113yes—
206C23×C4C22×D6113yes—
207C4×Q8—12232no—
208C2×C4⋊C4—12212no—
209C2×C4⋊C4—1226no—
210C2×C4⋊C4—1226no—
211C2×C4⋊C4—2446no—
212C2×C22⋊C4—26no—
213C2×C22⋊C4—26no—
214C2×C22⋊C4—1226no—
215C2×C22⋊C4—1226no—
216C2×C22⋊C4—2446no—
217C2×C22⋊C4C2×Dic61112yes—
218C2×C42—6112no—
219C2×C42—12212no—
220#95—1226no—
221#95—1226no—
222#95—1226no—
223#95—2446no—
224#95C2×Dic61112yes—
225C24—23no—
226C24—23no—
227C24—23no—
228C24—23no—
229C24—21no—
230C24—21no—
231C24C22×Dic6112yes—
232C24C23×D6111yescharacteristic
233C2×Q8—28no—
234C2×Q8—224no—
235C2×Q8—224no—
236C22×C4—26no—
237C22×C4—26no—
238C22×C4—23no—
239C22×C4—23no—
240C22×C4—26no—
241C22×C4—26no—
242C22×C4—616no—
243C22×C4—616no—
244C22×C4—1223no—
245C22×C4—1221no—
246C22×C4—1226no—
247C22×C4—1223no—
248C22×C4—1226no—
249C22×C4—1223no—
250C22×C4—1223no—
251C22×C4—1223no—
252C22×C4—1226no—
253C22×C4—1226no—
254C22×C4—1223no—
255C22×C4—1223no—
256C22×C4—1223no—
257C22×C4—1221no—
258C22×C4D4×D6116yes—
259C22×C4#1175822416yes—
260C22×C4C22×Dic6116yes—
261C22×C4C22×Dic6116yes—
262C4⋊C4—12224no—
263C4⋊C4—24412no—
264C22⋊C4—224no—
265C22⋊C4—2444no—
266C22⋊C4—24412no—
267C42—12224no—
268C23—26no—
269C23—22no—
270C23—26no—
271C23—41no—
272C23—43no—
273C23—43no—
274C23—43no—
275C23—43no—
276C23—43no—
277C23—43no—
278C23—46no—
279C23—43no—
280C23C23×Dic6111yescharacteristic
281C23C2×D4×D6113yes—
282C23C2×#1175822413yes—
283C23#1451982313yes—
284C23#1451982413yes—
285C23#1451982511yescharacteristic
286C23#1451982511yescharacteristic
287Q8—232no—
288C2×C4—212no—
289C2×C4—26no—
290C2×C4—26no—
291C2×C4—212no—
292C2×C4—26no—
293C2×C4—212no—
294C2×C4—43no—
295C2×C4—43no—
296C2×C4—46no—
297C2×C4—614no—
298C2×C4—1226no—
299C2×C4—1226no—
300C2×C4—1226no—
301C2×C4—1226no—
302C2×C4—1226no—
303C2×C4—2441no—
304C2×C4—2443no—
305C2×C4—2443no—
306C2×C4—2443no—
307C2×C4—2443no—
308C2×C4—2443no—
309C2×C4—2446no—
310C2×C4—2446no—
311C2×C4—2446no—
312C2×C4—2446no—
313C2×C4D4×Dic61112yes—
314C22—46no—
315C22—46no—
316C22—42no—
317C22—81no—
318C22—81no—
319C22—83no—
320C22—83no—
321C22C2×D4×Dic6113yes—
322C22#1684024011yescharacteristic
323C22#1684024513yes—
324C22#120×D6111yescharacteristic
325C22#1684026813yes—
326C22#1684033013yes—
327C22#1684033213yes—
328C22#1684033513yes—
329C22#1684034213yes—
330C22#1684037213yes—
331C22#1684038116yes—
332C22#1684038513yes—
333C4—212no—
334C4—46no—
335C4—1224no—
336C4—4881no—
337C4—4883no—
338C4—2446no—
339C2—82no—
340C2#120×Dic6111yescharacteristic
341C2#2144850313yes—
342C2#2144853213yes—
343C2#2145153711yescharacteristic
344C2#2145154611yescharacteristic
345C2#2145167813yes—
346C2#2145172413yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#393C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×#393D61
hypocenterC61#14301
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic61C2semidirect
3. #21444090C2semidirect
4. #21444084C2semidirect
6. #21444044C2semidirect
21. C61×#393C4semidirect
189. #393Dic61semidirect
196. C61#14301semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic612
C2×#168396552
#214440902
#214440842
#214440642
#214440442
C2×C61×#3932
#1430161