Finite Group Databasebeta

← #29607271

Subgroups — #29607271

Order 15616.

2070 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839621C213yesmaximal
2C2×#16839619C213yesmaximal
3C2×#95×D61C211yesmaximal, characteristic
4#21444084C211yesmaximal, characteristic, complemented
5#21444083C213yesmaximal, complemented
6#21444082C213yesmaximal, complemented
7C61×#1444C211yesmaximal, characteristic, complemented
8C23×C4×D61C2213yes—
9C22×C4⋊Dic61C2213yes—
10C22×#11758196C2213yes—
11C22×#11758196C2213yes—
12C22×#11758198C2213yes—
13C22×#11758198C2211yescharacteristic
14#16839621—212no—
15#16839619—212no—
16C2×#14519684C2216yes—
17C2×#14519684C2213yes—
18C2×#14519684C2213yes—
19#95×D61C418yescomplemented
20C22×C4⋊C4×C61C2213yes—
21C2×#95×C61C2213yes—
22C2×#95×C61C2211yescharacteristic
23C24×D61C2311yescharacteristic
24C23×Dic61C2311yescharacteristic
25C23×Dic61C2313yes—
26C23×Dic61C2313yes—
27C22×C4×D61—46no—
28C22×C4×D61—46no—
29C22×C4×D61C2×C4112yes—
30C22×C4×D61D4112yes—
31C22×C4×D61Q8112yes—
32C2×C4⋊Dic61—26no—
33C2×C4⋊Dic61—46no—
34C2×#11758196—26no—
35C2×#11758196—26no—
36C2×#11758196—46no—
37C2×#11758196—46no—
38C2×#11758198—212no—
39C2×#11758198—26no—
40C2×#11758198—26no—
41#14519684—26no—
42#14519684—212no—
43#14519684C2×C4112yes—
44C23×C4×C61C2311yescharacteristic
45C23×C4×C61C2313yes—
46C23×C4×C61C2313yes—
47C2×C4⋊C4×C61—26no—
48C2×C4⋊C4×C61—46no—
49#95×C61—26no—
50#95×C61C2×C414yes—
51C24×C61C2411yescharacteristic
52C23×D61—23no—
53C23×D61—23no—
54C23×D61—21no—
55C23×D61—21no—
56C23×D61C22×C412yes—
57C23×D61C2×D416yes—
58C23×D61C2×Q816yes—
59C22×C4×C61—26no—
60C22×C4×C61—26no—
61C22×C4×C61—23no—
62C22×C4×C61—23no—
63C22×C4×C61—26no—
64C22×C4×C61—23no—
65C22×C4×C61—26no—
66C22×C4×C61—23no—
67C22×C4×C61—23no—
68C22×C4×C61—21no—
69C22×C4×C61C22×C416yes—
70C22×C4×C61C2×D416yes—
71C22×C4×C61C2×Q816yes—
72C22×Dic61—26no—
73C22×Dic61—26no—
74C22×Dic61—23no—
75C22×Dic61—23no—
76C22×Dic61—26no—
77C22×Dic61—23no—
78C22×Dic61—26no—
79C22×Dic61—23no—
80C22×Dic61—21no—
81C22×Dic61—23no—
82C22×Dic61C22×C416yes—
83C22×Dic61C2×D416yes—
84C22×Dic61C2×Q816yes—
85C2×C4×D61—424no—
86C2×C4×D61—412no—
87C2×C4×D61—412no—
88C2×C4×D61—424no—
89C2×C4×D61C4⋊C4148yescomplemented
90C4⋊C4×C61—412no—
91#11758198—412no—
92#11758198—44no—
93C4⋊Dic61—412no—
94#11758196—412no—
95#11758196—412no—
96C23×C61C23×C411yescharacteristic
97C23×C61C22×D413yes—
98C23×C61C22×Q813yes—
99C23×C61H(5,𝔽2)13yes—
100C23×C61H(5,𝔽2)13yes—
101C23×C612−1+411yescharacteristic
102C23×C612−1+411yescharacteristic
103C22×D61—26no—
104C22×D61—22no—
105C22×D61—26no—
106C22×D61—26no—
107C22×D61—26no—
108C22×D61—26no—
109C22×D61—212no—
110C22×D61—26no—
111C22×D61—26no—
112C22×D61C2×C4⋊C4112yes—
113C22×D61#12714yes—
114C22×D61C4⋊Q8112yes—
115C2×C4×C61—212no—
116C2×C4×C61—26no—
117C2×C4×C61—26no—
118C2×C4×C61—212no—
119C2×C4×C61—41no—
120C2×C4×C61—43no—
121C2×C4×C61—43no—
122C2×C4×C61—43no—
123C2×C4×C61—46no—
124C2×C4×C61—43no—
125C2×C4×C61—46no—
126C2×C4×C61—43no—
127C2×C4×C61C2×C4⋊C4112yes—
128C2×Dic61—212no—
129C2×Dic61—26no—
130C2×Dic61—26no—
131C2×Dic61—212no—
132C2×Dic61—41no—
133C2×Dic61—43no—
134C2×Dic61—43no—
135C2×Dic61—43no—
136C2×Dic61—46no—
137C2×Dic61—43no—
138C2×Dic61—43no—
139C2×Dic61—46no—
140C2×Dic61C2×C4⋊C4112yes—
141C4×D61—448no—
142#14353—611nomaximal, Sylow
143C22×C61C22×C4⋊C413yes—
144C22×C61#51813yes—
145C22×C61#51911yescharacteristic
146C22×C61C2×#12711yescharacteristic
147C22×C61C2×C4⋊Q813yes—
148C22×C61#53413yes—
149C22×C61#53413yes—
150C22×C61#53713yes—
151C22×C61#53713yes—
152C22×C61#54316yes—
153C22×C61#54413yes—
154C22×C61#54413yes—
155C2×D61—212no—
156C2×D61—212no—
157C2×D61—24no—
158C2×D61—24no—
159C2×D61—212no—
160C2×D61—212no—
161C2×D61#38418yes—
162C4×C61—212no—
163C4×C61—83no—
164C4×C61—81no—
165Dic61—212no—
166Dic61—81no—
167Dic61—83no—
168C2×#380—613no—
169C2×#380—613no—
170C22×#95—611no—
171#1444—613no—
172#1444—611no—
173#1444—613no—
174#1444D6111yescharacteristic, complemented
175C2×C61C2×#38411yescharacteristic
176C2×C61#231513yes—
177C2×C61#231613yes—
178C2×C61#265611yescharacteristic
179C2×C61#265611yescharacteristic
180C2×C61#265913yes—
181C2×C61#265913yes—
182D61—28no—
183C24×C4—613no—
184C22×C4⋊C4—613no—
185C22×C4⋊C4—613no—
186C22×C4⋊C4—613no—
187C22×C4⋊C4C2×D6113yes—
188C22×C22⋊C4—611no—
189C22×C22⋊C4—613no—
190#380—12212no—
191#380—12212no—
192C2×#95—613no—
193C2×#95—618no—
194C2×#95—616no—
195C2×#95—613no—
196C2×#95C2×D6113yes—
197C2×#95C2×D6111yescharacteristic
198C61#1435311yescharacteristic, complemented, Sylow
199C25—611no—
200C23×C4—613no—
201C23×C4—6112no—
202C23×C4—6112no—
203C23×C4—6112no—
204C23×C4—611no—
205C23×C4—613no—
206C23×C4—2446no—
207C23×C4—2446no—
208C23×C4C22×D6111yescharacteristic
209C23×C4C22×D6113yes—
210C23×C4C22×D6113yes—
211C2×C4⋊C4—26no—
212C2×C4⋊C4—46no—
213C2×C4⋊C4—1226no—
214C2×C4⋊C4—1226no—
215C2×C4⋊C4—1226no—
216C2×C4⋊C4—2446no—
217C2×C4⋊C4—2446no—
218C2×C4⋊C4—2446no—
219C2×C22⋊C4—1226no—
220C2×C22⋊C4—1226no—
221C2×C22⋊C4—12212no—
222#95—26no—
223#95—6112no—
224#95—12212no—
225#95—1226no—
226#95C4×D6114yes—
227C24—612no—
228C24—616no—
229C24—616no—
230C24—1223no—
231C24—1223no—
232C24—1221no—
233C24—1221no—
234C24C23×D6111yescharacteristic
235C22×C4—26no—
236C22×C4—26no—
237C22×C4—23no—
238C22×C4—23no—
239C22×C4—26no—
240C22×C4—23no—
241C22×C4—26no—
242C22×C4—23no—
243C22×C4—23no—
244C22×C4—21no—
245C22×C4—616no—
246C22×C4—616no—
247C22×C4—616no—
248C22×C4—6148no—
249C22×C4—1226no—
250C22×C4—1226no—
251C22×C4—1223no—
252C22×C4—1226no—
253C22×C4—1223no—
254C22×C4—1223no—
255C22×C4—1226no—
256C22×C4—1223no—
257C22×C4—1221no—
258C22×C4—1223no—
259C22×C4—24424no—
260C22×C4—24412no—
261C22×C4—24412no—
262C22×C4—24424no—
263C22×C4C2×C4×D6116yes—
264C22×C4D4×D6116yes—
265C22×C4Q8×D6116yes—
266C4⋊C4—412no—
267C4⋊C4—24412no—
268C4⋊C4—24412no—
269C4⋊C4—24412no—
270C22⋊C4—24412no—
271C22⋊C4—2444no—
272C23—6112no—
273C23—614no—
274C23—6112no—
275C23—1226no—
276C23—1222no—
277C23—1226no—
278C23—1226no—
279C23—1226no—
280C23—1226no—
281C23—12212no—
282C23—1226no—
283C23—1226no—
284C23C22×C4×D6111yescharacteristic
285C23C2×D4×D6113yes—
286C23C2×Q8×D6113yes—
287C23#1451982213yes—
288C23#1451982313yes—
289C23#1451982411yescharacteristic
290C23#1451982511yescharacteristic
291C2×C4—212no—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—212no—
295C2×C4—43no—
296C2×C4—41no—
297C2×C4—46no—
298C2×C4—43no—
299C2×C4—43no—
300C2×C4—43no—
301C2×C4—46no—
302C2×C4—43no—
303C2×C4—6112no—
304C2×C4—12212no—
305C2×C4—1226no—
306C2×C4—1226no—
307C2×C4—12212no—
308C2×C4—24448no—
309C2×C4—2441no—
310C2×C4—2443no—
311C2×C4—2443no—
312C2×C4—2443no—
313C2×C4—2446no—
314C2×C4—2443no—
315C2×C4—2443no—
316C2×C4—2446no—
317C2×C4C4⋊C4×D61112yes—
318C22—618no—
319C22—1224no—
320C22—1224no—
321C22—12212no—
322C22—12212no—
323C22—12212no—
324C22—12212no—
325C22C2×C4⋊C4×D6113yes—
326C22#1684023413yes—
327C22#1684023811yescharacteristic
328C22#127×D6111yescharacteristic
329C22C4⋊Q8×D6113yes—
330C22#1684032213yes—
331C22#1684032313yes—
332C22#1684034113yes—
333C22#1684034213yes—
334C22#1684038716yes—
335C22#1684038913yes—
336C22#1684039213yes—
337C4—212no—
338C4—83no—
339C4—81no—
340C4—12212no—
341C4—4881no—
342C4—4883no—
343C2—1228no—
344C2#384×D6111yescharacteristic
345C2#2144843013yes—
346C2#2144843413yes—
347C2#2145192011yescharacteristic
348C2#2145192211yescharacteristic
349C2#2145193213yes—
350C2#2145193413yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1444C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenter#1444D61
hypocenterC61#14353
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
4. #21444084C2semidirect
5. #21444083C2semidirect
6. #21444082C2semidirect
7. C61×#1444C2semidirect
19. #95×D61C4semidirect
89. C2×C4×D61C4⋊C4semidirect
174. #1444D61semidirect
198. C61#14353semidirect

Maximal subgroups

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

subgroupindex
C2×#168396212
C2×#168396192
C2×#95×D612
#214440842
#214440832
#214440822
C61×#14442
#1435361