Finite Group Databasebeta

← #29583673

Subgroups — #29583673

Order 15616.

408 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 95 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×#16840053C211yesmaximal, characteristic
2#21445808C211yesmaximal, characteristic
3#21444593C211yesmaximal, characteristic, complemented
4#21444503C211yesmaximal, characteristic, complemented
5#21444501C211yesmaximal, characteristic, complemented
6#21444211C211yesmaximal, characteristic
7C61×#1301C211yesmaximal, characteristic, complemented
8C2×#14519777C2211yescharacteristic
9C2×C4×D244C2211yescharacteristic
10#16840053—22no—
11#16840053—22no—
12#16839815C2211yescharacteristic
13C2×#14519709—21no—
14#16839786C2211yescharacteristic
15C2×#14519699—21no—
16C2×#14519693—21no—
17#16839530—21no—
18#16839514—22no—
19#16839510—22no—
20#16839509—22no—
21#16839491—22no—
22C61×#431C2211yescharacteristic
23C61×#423C2211yescharacteristic
24C2×C4⋊C8×C61C2211yescharacteristic
25C22×Dic122—21no—
26C22×D244D411yescharacteristic
27C22×C4×D61—21no—
28C2×#11758194—21no—
29#14519777—21no—
30#14519777—21no—
31#14519777—21no—
32#14519777D412yes—
33C4×D244—21no—
34C4×D244—21no—
35C4×D244—21no—
36C4×D244—21no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61D411yescharacteristic
40C2×#11758198—21no—
41#14519709—24no—
42C4⋊#92418—21no—
43C4⋊#92418—21no—
44C4⋊#92418D412yes—
45#14519699—24no—
46#14519693—42no—
47C22⋊#92418—21no—
48C4×#92418—21no—
49C2×M16×C61—21no—
50C22×C8×C61—21no—
51C2×C42×C61C2311yescharacteristic
52C4⋊C8×C61—21no—
53C4⋊C8×C61—21no—
54C4⋊C8×C61—21no—
55C4⋊C8×C61—21no—
56C4⋊C8×C61D412yes—
57C22⋊C8×C61—21no—
58C4×C8×C61—21no—
59C23×D61—21no—
60C22×C4×C61C2×D411yescharacteristic
61C22×C4×C61#4111yescharacteristic
62C22×C4×C61#4111yescharacteristic
63C22×Dic61—21no—
64C22×Dic61—41no—
65C2×D244—21no—
66C2×D244—21no—
67C2×D244—21no—
68C2×D244—21no—
69C2×D244—21no—
70C2×D244Q1612yes—
71C2×C4×D61—41no—
72C2×C4×D61—41no—
73C2×C4×D61—82no—
74C2×Dic122—22no—
75C2×Dic122—41no—
76C2×Dic122—82no—
77M16×C61—41no—
78M16×C61—41no—
79C2×C8×C61—22no—
80C2×C8×C61—41no—
81C2×C8×C61—41no—
82C2×C8×C61—41no—
83C2×C8×C61—41no—
84C42×C61C2×D411yescharacteristic
85C42×C61C2×D411yescharacteristic
86C42×C61C2×D411yescharacteristic
87C42×C61#4111yescharacteristic
88#11758198—41no—
89#11758198—41no—
90C4⋊Dic61—22no—
91C4⋊Dic61—21no—
92C4⋊Dic61—21no—
93C4⋊Dic61—22no—
94C4⋊Dic61—41no—
95C4⋊Dic61—41no—
96#11758194—41no—
97#11758194—41no—
98C2×#92418—22no—
99C2×#92418—41no—
100C23×C61#12311yescharacteristic
101C22×D61—22no—
102C22×D61—41no—
103C22×D61—41no—
104C22×D61—81no—
105C22×D61—81no—
106C2×C4×C61—22no—
107C2×C4×C61—41no—
108C2×C4×C61#12011yescharacteristic
109C2×C4×C61#12111yescharacteristic
110C2×C4×C61#12111yescharacteristic
111C2×C4×C61#12111yescharacteristic
112C2×C4×C61#12311yescharacteristic
113C2×C4×C61#12411yescharacteristic
114C2×C4×C61C2×Q1611yescharacteristic
115C2×C4×C61#13511yescharacteristic
116C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
117C2×C4×C61#13711yescharacteristic
118C2×Dic61—42no—
119C2×Dic61—41no—
120C2×Dic61—82no—
121C2×Dic61—81no—
122D244—22no—
123D244—22no—
124D244—41no—
125D244—41no—
126C4×D61—82no—
127C4×D61—82no—
128Dic122—82no—
129Dic122—82no—
130C8×C61—42no—
131C8×C61—81no—
132C8×C61—81no—
133#92418—42no—
134#92418—81no—
135#10453—611nomaximal, Sylow
136C22×C61—21no—
137C22×C61—21no—
138C22×C61#39411yescharacteristic
139C22×C61#48411yescharacteristic
140C22×C61#48511yescharacteristic
141C2×D61—42no—
142C2×D61—82no—
143C2×D61—81no—
144C2×D61—81no—
145C2×D61—82no—
146C4×C61—21no—
147C4×C61—21no—
148C4×C61—21no—
149C4×C61#44911yescharacteristic
150C4×C61#45111yescharacteristic
151C4×C61#46711yescharacteristic
152C4×C61#46811yescharacteristic
153C4×C61#47112yes—
154Dic61—82no—
155Dic61—81no—
156C2×#476—611no—
157#1800—611no—
158#1571—611no—
159#1549—611no—
160#1548—611no—
161#1478—611no—
162#1301D6111yescharacteristic, complemented
163C2×C61—21no—
164C2×C61—21no—
165C2×C61#203611yescharacteristic
166C2×C61#204611yescharacteristic
167C2×C61#204811yescharacteristic
168D61—82no—
169C2×C4⋊Q8—611no—
170C2×C4×D4—611no—
171#476—1222no—
172#476—1222no—
173#431—611no—
174#431C2×D6111yescharacteristic
175C2×#106—1221no—
176#423—611no—
177#423C2×D6111yescharacteristic
178C2×C4⋊C8C2×D6111yescharacteristic
179C2×#102—1221no—
180C2×#100—1221no—
181#343—1221no—
182#335—1222no—
183#333—1222no—
184#332—1222no—
185D4⋊C8—1222no—
186C61#1045311yescharacteristic, complemented, Sylow
187C22×Q8—1221no—
188C22×D4—611no—
189C23×C4—1221no—
190C2×M16—21no—
191C2×M16—1221no—
192C22×C8—21no—
193C4⋊Q8—612no—
194C4⋊Q8—1221no—
195C4⋊Q8—1221no—
196C4⋊Q8—1221no—
197C4×D4—1221no—
198C4×D4—1221no—
199C4×D4—1221no—
200C4×D4—1221no—
201C2×C4⋊C4—611no—
202C2×C4⋊C4—1221no—
203C2×C4⋊C4—1221no—
204C2×C22⋊C4—1221no—
205C2×C42C22×D6111yescharacteristic
206#106—1224no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—612no—
212C4⋊C8—1221no—
213C4⋊C8—1221no—
214C4⋊C8D24412yes—
215#102—1224no—
216#100—2442no—
217C22⋊C8—21no—
218C22⋊C8—1221no—
219C4×C8—21no—
220C4×C8—1221no—
221C24—1221no—
222C2×Q8—1222no—
223C2×Q8—4882no—
224C2×Q8—2441no—
225C2×D4—612no—
226C2×D4—1221no—
227C2×D4—1221no—
228C2×D4—1221no—
229C2×D4—1221no—
230C2×D4—1221no—
231C22×C4—1221no—
232C22×C4—4882no—
233C22×C4—2441no—
234C22×C4—2441no—
235C22×C4—2441no—
236C22×C4#1175822211yescharacteristic
237C22×C4D4×D6111yescharacteristic
238C22×C4#1175822611yescharacteristic
239M16—41no—
240M16—41no—
241M16—2441no—
242M16—2441no—
243C2×C8—22no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—1222no—
249C2×C8—2441no—
250C4⋊C4—1222no—
251C4⋊C4—1221no—
252C4⋊C4—1221no—
253C4⋊C4—1222no—
254C4⋊C4—2441no—
255C4⋊C4—2441no—
256C22⋊C4—2441no—
257C22⋊C4—2441no—
258C42C2×D24411yescharacteristic
259C42#1175822211yescharacteristic
260C42D4×D6111yescharacteristic
261C42D4×D6111yescharacteristic
262C23—1222no—
263C23—4881no—
264C23—4881no—
265C23—2441no—
266C23—2441no—
267C23#1451976211yescharacteristic
268Q8—4882no—
269Q8—4882no—
270D4—1222no—
271D4—1222no—
272D4—2441no—
273D4—2441no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—4882no—
277C2×C4—4882no—
278C2×C4—4882no—
279C2×C4—4881no—
280C2×C4—2441no—
281C2×C4—2442no—
282C2×C4#1451974711yescharacteristic
283C2×C4#1451975011yescharacteristic
284C2×C4#1451975011yescharacteristic
285C2×C4#1451975211yescharacteristic
286C2×C4#1451976211yescharacteristic
287C2×C4#1451976511yescharacteristic
288C2×C4Q16×D6111yescharacteristic
289C2×C4#1451979811yescharacteristic
290C2×C4#1451980111yescharacteristic
291C2×C4#1451980611yescharacteristic
292C8—42no—
293C8—81no—
294C8—81no—
295C8—4881no—
296C8—2442no—
297C22—21no—
298C22—21no—
299C22—4882no—
300C22—4882no—
301C22—4881no—
302C22—4881no—
303C22—2442no—
304C22#1683968011yescharacteristic
305C22#1684009911yescharacteristic
306C22#1684010611yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—4881no—
311C4—4882no—
312C4#1683990811yescharacteristic
313C4#1683991911yescharacteristic
314C4#1684000211yescharacteristic
315C4#1684000711yescharacteristic
316C4#1684002612yes—
317C2—21no—
318C2—21no—
319C2—4882no—
320C2#2144702511yescharacteristic
321C2#2144708311yescharacteristic
322C2#2144709511yescharacteristic
subgroupG/H
centerC22#16839680
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1301C2
socleC22×C61#394
prime radicalC22×D244D4
hypercenter#1301D61
hypocenterC61#10453
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
3. #21444593C2semidirect
4. #21444503C2semidirect
5. #21444501C2semidirect
7. C61×#1301C2semidirect
162. #1301D61semidirect
186. C61#10453semidirect

Maximal subgroups

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

subgroupindex
C2×#168400532
#214458082
#214445932
#214445032
#214445012
#214442112
C61×#13012
#1045361