Finite Group Databasebeta

← #29598243

Subgroups — #29598243

Order 15616.

590 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840137C211yesmaximal, characteristic
2C2×#16839564C211yesmaximal, characteristic, complemented
3C2×#16839563C211yesmaximal, characteristic, complemented
4C2×#16839562C211yesmaximal, characteristic, complemented
5C2×#16839537C211yesmaximal, characteristic, complemented
6#21445982C211yesmaximal, characteristic, complemented
7C2×C61×#357C211yesmaximal, characteristic, complemented
8C22×D488C2211yescharacteristic
9C22×#11758199—21no—
10C2×#14519773—21no—
11C22×#11758185C2211yescharacteristic
12#16840137—22no—
13#16840137C414yescomplemented
14C2×#14519711C2211yescharacteristic
15C2×#14519708—21no—
16C2×C8×Dic61C2211yescharacteristic
17#16839627—21no—
18#16839564—22no—
19#16839563—22no—
20#16839562—22no—
21#16839537—22no—
22C22×D8×C61C2211yescharacteristic
23C22×C16×C61C2211yescharacteristic
24C2×#107×C61C2211yescharacteristic
25C61×#357—22no—
26C22×D244—21no—
27C22×C22⋊D61—42no—
28C2×D488—21no—
29C2×D488—21no—
30C2×D488—21no—
31C2×D488—21no—
32C2×D488—21no—
33C2×D488C2×C412yes—
34C2×#11758199—24no—
35C2×#11758199—42no—
36C2×#11758199—42no—
37C22×#92418D411yescharacteristic
38#14519773—24no—
39#14519773—42no—
40C2×C4⋊Dic61—21no—
41C2×C4×Dic61D411yescharacteristic
42C2×#11758185—21no—
43C2×#11758185—21no—
44C2×#11758185D412yescomplemented
45#14519711—21no—
46#14519711—21no—
47#14519708—42no—
48C8×Dic61C2×C412yes—
49C8×Dic61D412yes—
50#14519684—42no—
51C22×D4×C61—21no—
52C2×D8×C61—21no—
53C2×D8×C61—21no—
54C2×D8×C61—21no—
55C2×D8×C61—21no—
56C2×D8×C61—21no—
57C2×D8×C61C2×C412yes—
58C22×C8×C61C2311yescharacteristic
59C2×C4⋊C4×C61—21no—
60C2×C16×C61—21no—
61C2×C16×C61—21no—
62C2×C16×C61D412yescomplemented
63#107×C61—21no—
64#107×C61—21no—
65C24×C61—41no—
66C23×D61—41no—
67C2×D4×C61—22no—
68C2×D4×C61—41no—
69C2×D4×C61—41no—
70C2×D4×C61—41no—
71C2×D4×C61—41no—
72C2×D4×C61—41no—
73C22×C4×C61—41no—
74C22×C4×C61C2×D411yescharacteristic
75C2×C22⋊D61—48no—
76C2×C22⋊D61—84no—
77C2×C22⋊D61—84no—
78C22×Dic61—41no—
79C22×Dic61D812yes—
80C2×D244—22no—
81C2×D244—41no—
82C2×D244—41no—
83C2×D244—41no—
84C2×D244—41no—
85C2×D244—41no—
86D8×C61—22no—
87D8×C61—22no—
88D8×C61—41no—
89D8×C61—41no—
90C2×C8×C61C22×C411yescharacteristic
91C2×C8×C61C2×D411yescharacteristic
92C2×C8×C61C2×D411yescharacteristic
93C2×C8×C61C2×D411yescharacteristic
94C2×C8×C61#4111yescharacteristic
95C2×C8×C61#4111yescharacteristic
96C4⋊C4×C61—41no—
97C4⋊C4×C61—41no—
98#11758199—48no—
99C4⋊Dic61—41no—
100C4⋊Dic61—41no—
101C4×Dic61C22⋊C412yes—
102C4×Dic61SD1612yes—
103C2×#92418C22⋊C412yes—
104C2×#92418D812yes—
105C2×#92418SD1612yes—
106D488—22no—
107D488—22no—
108D488—41no—
109D488—41no—
110C16×C61—22no—
111#11758185—22no—
112C23×C61—42no—
113C23×C61—81no—
114C23×C61—81no—
115C23×C61—82no—
116C23×C61—82no—
117C23×C61C2×D811yescharacteristic
118C22×D61—42no—
119C22×D61—81no—
120C22×D61—81no—
121C22×D61—82no—
122C22×D61—82no—
123D4×C61—42no—
124D4×C61—42no—
125D4×C61—81no—
126D4×C61—81no—
127C2×C4×C61—81no—
128C2×C4×C61—82no—
129C2×C4×C61C2×C22⋊C411yescharacteristic
130C2×C4×C61#12411yescharacteristic
131C2×C4×C61#12711yescharacteristic
132C2×C4×C61C2×D811yescharacteristic
133C2×C4×C61C2×SD1611yescharacteristic
134C2×C4×C61C2×SD1611yescharacteristic
135C22⋊D61—816no—
136C2×Dic61—22no—
137C2×Dic61—22no—
138C2×Dic61—81no—
139C2×Dic61—82no—
140C2×Dic61#10214yes—
141D244—42no—
142D244—42no—
143D244—81no—
144D244—81no—
145C8×C61C4×D411yescharacteristic
146C8×C61C4×D411yescharacteristic
147C8×C61#12111yescharacteristic
148C8×C61#12111yescharacteristic
149#92418#10214yes—
150#13206—611nomaximal, Sylow
151C22×C61—84no—
152C22×C61—82no—
153C22×C61—162no—
154C22×C61—161no—
155C22×C61—161no—
156C22×C61C2×#10211yescharacteristic
157C22×C61#48611yescharacteristic
158C22×C61#49311yescharacteristic
159C22×C61#50912yes—
160C22×C61#50912yes—
161C2×D61—84no—
162C2×D61—82no—
163C2×D61—162no—
164C2×D61—161no—
165C2×D61—161no—
166C4×C61—161no—
167C4×C61#39011yescharacteristic
168C4×C61C2×#10211yescharacteristic
169C4×C61#48611yescharacteristic
170C4×C61#49211yescharacteristic
171Dic61—24no—
172Dic61—161no—
173C2×#493—611no—
174C2×#357—611no—
175C2×#357—611no—
176C2×#357—611no—
177C2×#357D6111yescharacteristic, complemented
178C2×#346—611no—
179#1846—611no—
180C2×C61—162no—
181C2×C61#195711yescharacteristic
182C2×C61#214012yes—
183C2×C61#224412yes—
184C2×C61#225011yescharacteristic
185C2×C61#225011yescharacteristic
186D61—162no—
187C22×D8—611no—
188C22×D8—1221no—
189C22×D8C2×D6111yescharacteristic
190C2×#127—1221no—
191C22×C16—611no—
192C22×C16C2×D6111yescharacteristic
193#493—614no—
194#493—1222no—
195C2×#107—611no—
196C2×#107C2×D6111yescharacteristic
197C2×#106—1221no—
198C2×C4×C8—611no—
199#382—1221no—
200#357—22no—
201#357—1222no—
202#357—1222no—
203#357—1222no—
204#346—1222no—
205C61#1320611yescharacteristic, complemented, Sylow
206C22×D4—21no—
207C22×D4—1221no—
208C22×D4—2442no—
209C2×D8—21no—
210C2×D8—21no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8—21no—
214C2×D8—612no—
215C2×D8—1224no—
216C2×D8—1221no—
217C2×D8—1221no—
218C2×D8—1221no—
219C2×D8—1221no—
220C2×D8—1221no—
221C2×D8—2442no—
222C2×D8—2442no—
223C2×D8C4×D6112yes—
224C22×C8—611no—
225C22×C8C22×D6111yescharacteristic
226#127—1224no—
227#127—2442no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—1221no—
230C2×C42—611no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—612no—
234C2×C16—1221no—
235C2×C16—1221no—
236C2×C16C22⋊D6112yescomplemented
237#107—21no—
238#107—21no—
239#107—1221no—
240#107—1221no—
241#106—2442no—
242C4×C8—612no—
243C4×C8—612no—
244#95—2442no—
245C24—41no—
246C24—2441no—
247C2×D4—22no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—1222no—
254C2×D4—4884no—
255C2×D4—4884no—
256C2×D4—2448no—
257C2×D4—2441no—
258C2×D4—2441no—
259C2×D4—2441no—
260C2×D4—2441no—
261C2×D4—2441no—
262C22×C4—41no—
263C22×C4—612no—
264C22×C4—2441no—
265C22×C4D4×D6111yescharacteristic
266D8—22no—
267D8—22no—
268D8—41no—
269D8—41no—
270D8—1222no—
271D8—1222no—
272D8—2448no—
273D8—2441no—
274D8—2441no—
275C2×C8—612no—
276C2×C8—612no—
277C2×C8—612no—
278C2×C8C2×C4×D6111yescharacteristic
279C2×C8#1175822211yescharacteristic
280C2×C8D4×D6111yescharacteristic
281C2×C8D4×D6111yescharacteristic
282C2×C8#1175822411yescharacteristic
283C2×C8C2×C22⋊D6111yescharacteristic
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—2441no—
287C4⋊C4—2441no—
288C42—612no—
289C42—612no—
290C16—22no—
291C16—1222no—
292C23—42no—
293C23—82no—
294C23—82no—
295C23—81no—
296C23—81no—
297C23—4882no—
298C23—4882no—
299C23—4881no—
300C23—4881no—
301C23—2442no—
302C23D8×D6111yescharacteristic
303D4—42no—
304D4—42no—
305D4—81no—
306D4—81no—
307D4—48816no—
308D4—4881no—
309D4—4881no—
310D4—2442no—
311D4—2442no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—614no—
315C2×C4—1222no—
316C2×C4—1222no—
317C2×C4—4882no—
318C2×C4—4881no—
319C2×C4C22⋊C4×D6111yescharacteristic
320C2×C4#1451976311yescharacteristic
321C2×C4#1451977311yescharacteristic
322C2×C4D8×D6111yescharacteristic
323C2×C4SD16×D6111yescharacteristic
324C2×C4SD16×D6111yescharacteristic
325C8—614no—
326C8#1451973711yescharacteristic
327C8C4×C22⋊D6111yescharacteristic
328C8#1451974911yescharacteristic
329C8#1451975011yescharacteristic
330C22—84no—
331C22—82no—
332C22—162no—
333C22—161no—
334C22—161no—
335C22—9762no—
336C22—9761no—
337C22—9761no—
338C22—4884no—
339C22—4882no—
340C22#102×D6111yescharacteristic
341C22#1684010711yescharacteristic
342C22#1684013711yescharacteristic
343C22#1684019612yes—
344C22#1684019912yes—
345C4—161no—
346C4—1224no—
347C4—9761no—
348C4#1683966211yescharacteristic
349C4#102×D6111yescharacteristic
350C4#1684010711yescharacteristic
351C4#1684013311yescharacteristic
352C2—162no—
353C2—9762no—
354C2#2144662311yescharacteristic
355C2#2144753312yes—
356C2#2144805912yes—
357C2#2144808211yescharacteristic
358C2#2144808411yescharacteristic
subgroupG/H
centerC23D8×D61
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC2×C61×#357C2
socleC23×C61C2×D8
prime radicalC2×#16840137C2
cosocleC22×C8×C61C23
hypercenterC2×#357D61
hypocenterC61#13206
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#16839564C2semidirect
3. C2×#16839563C2semidirect
4. C2×#16839562C2semidirect
5. C2×#16839537C2semidirect
6. #21445982C2semidirect
7. C2×C61×#357C2semidirect
13. #16840137C4semidirect
44. C2×#11758185D4semidirect
62. C2×C16×C61D4semidirect
177. C2×#357D61semidirect
205. C61#13206semidirect
236. C2×C16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401372
C2×#168395642
C2×#168395632
C2×#168395622
C2×#168395372
#214459822
C2×C61×#3572
#1320661