Finite Group Databasebeta

← #29583671

Subgroups — #29583671

Order 15616.

408 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 99 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×#16840052C211yesmaximal, characteristic
2#21445811C211yesmaximal, characteristic
3#21444594C211yesmaximal, characteristic, complemented
4#21444503C211yesmaximal, characteristic, complemented
5#21444499C211yesmaximal, characteristic, complemented
6#21443715C211yesmaximal, characteristic, complemented
7C61×#1478C211yesmaximal, characteristic
8C2×#14519777C2211yescharacteristic
9#16840052—22no—
10#16840052—22no—
11C2×#14519708—21no—
12#16839786C2211yescharacteristic
13#16839786C2211yescharacteristic
14C2×C4⋊#92418C2211yescharacteristic
15C2×#14519698—21no—
16C2×#14519694—21no—
17#16839531—21no—
18#16839515—22no—
19#16839510—22no—
20#16839509—22no—
21C2×C4×D4×C61C2211yescharacteristic
22C61×#431C2211yescharacteristic
23C61×#431C2211yescharacteristic
24C61×D4⋊C8—22no—
25C22×Dic122—21no—
26C2×#11758194—21no—
27C2×#11758194—21no—
28C22×#92418—21no—
29#14519777—21no—
30#14519777—21no—
31#14519777—21no—
32#14519777D412yes—
33C2×C4⋊Dic61—21no—
34C2×C4⋊Dic61—21no—
35#14519708—24no—
36C4⋊#92418—21no—
37C4⋊#92418—21no—
38C4⋊#92418—21no—
39C4⋊#92418—21no—
40C4⋊#92418—21no—
41C4⋊#92418D412yes—
42#14519698—24no—
43#14519694—42no—
44C22×D4×C61D411yescharacteristic
45C23×C4×C61—21no—
46C4×D4×C61—21no—
47C4×D4×C61—21no—
48C4×D4×C61—21no—
49C4×D4×C61—21no—
50C2×C4⋊C4×C61D411yescharacteristic
51C2×C22⋊C4×C61—21no—
52C2×C42×C61C2311yescharacteristic
53C4⋊C8×C61—21no—
54C4⋊C8×C61D412yes—
55C22⋊C8×C61—21no—
56C22⋊C8×C61—21no—
57C4×C8×C61—21no—
58C4×C8×C61—21no—
59C24×C61—21no—
60C2×D4×C61—21no—
61C2×D4×C61—21no—
62C2×D4×C61—21no—
63C2×D4×C61—21no—
64C2×D4×C61—21no—
65C2×D4×C61Q1612yes—
66C22×C4×C61—21no—
67C22×C4×C61—41no—
68C22×C4×C61—41no—
69C22×C4×C61—82no—
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61#4111yescharacteristic
72C22×C4×C61#4111yescharacteristic
73C22×Dic61—41no—
74C2×Dic122—22no—
75C2×Dic122—41no—
76C2×Dic122—82no—
77C2×C8×C61—22no—
78C2×C8×C61—41no—
79C4⋊C4×C61—21no—
80C4⋊C4×C61—21no—
81C22⋊C4×C61—41no—
82C22⋊C4×C61—41no—
83C42×C61C2×D411yescharacteristic
84C42×C61C2×D411yescharacteristic
85C42×C61C2×D411yescharacteristic
86C42×C61#4111yescharacteristic
87C4⋊Dic61—22no—
88C4⋊Dic61—22no—
89C4⋊Dic61—41no—
90C4⋊Dic61—41no—
91#11758194—41no—
92#11758194—41no—
93#11758194—41no—
94#11758194—41no—
95C2×#92418—22no—
96C2×#92418—41no—
97C2×#92418—41no—
98C2×#92418—41no—
99C2×#92418—41no—
100C23×C61—22no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61—81no—
104C23×C61—81no—
105C23×C61#12311yescharacteristic
106D4×C61—22no—
107D4×C61—22no—
108D4×C61—41no—
109D4×C61—41no—
110C2×C4×C61—22no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—82no—
114C2×C4×C61—82no—
115C2×C4×C61—81no—
116C2×C4×C61#12011yescharacteristic
117C2×C4×C61#12111yescharacteristic
118C2×C4×C61#12111yescharacteristic
119C2×C4×C61#12111yescharacteristic
120C2×C4×C61#12311yescharacteristic
121C2×C4×C61#12411yescharacteristic
122C2×C4×C61C2×Q1611yescharacteristic
123C2×C4×C61#13511yescharacteristic
124C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C61#13711yescharacteristic
126C2×Dic61—42no—
127C2×Dic61—82no—
128Dic122—82no—
129Dic122—82no—
130C8×C61—42no—
131C8×C61—81no—
132#92418—42no—
133#92418—81no—
134#92418—81no—
135#10453—611nomaximal, Sylow
136C22×C61—21no—
137C22×C61—21no—
138C22×C61—42no—
139C22×C61—82no—
140C22×C61—82no—
141C22×C61—81no—
142C22×C61—81no—
143C22×C61#39411yescharacteristic
144C22×C61#48411yescharacteristic
145C22×C61#48511yescharacteristic
146C4×C61—21no—
147C4×C61—21no—
148C4×C61—21no—
149C4×C61—81no—
150C4×C61#44911yescharacteristic
151C4×C61#45111yescharacteristic
152C4×C61#46711yescharacteristic
153C4×C61#46811yescharacteristic
154C4×C61#47112yes—
155Dic61—82no—
156C2×#476—611no—
157#1800—611no—
158#1571—611no—
159#1549—611no—
160#1548—611no—
161#1478D6111yescharacteristic
162#1301—611no—
163C2×C61—21no—
164C2×C61—21no—
165C2×C61—82no—
166C2×C61#203611yescharacteristic
167C2×C61#204611yescharacteristic
168C2×C61#204811yescharacteristic
169C2×C4⋊Q8—611no—
170C2×C4×D4C2×D6111yescharacteristic
171#476—1222no—
172#476—1222no—
173#431C2×D6111yescharacteristic
174#431C2×D6111yescharacteristic
175C2×#106—1221no—
176#423—611no—
177#423—611no—
178C2×C4⋊C8—611no—
179C2×#102—1221no—
180C2×#100—1221no—
181#343—1221no—
182#335—1222no—
183#333—1222no—
184#332—1222no—
185D4⋊C8—22no—
186C61#1045311yescharacteristic, complemented, Sylow
187C22×Q8—1221no—
188C22×D4D24411yescharacteristic
189C23×C4—21no—
190C2×M16—1221no—
191C2×M16—1221no—
192C22×C8—1221no—
193C4⋊Q8—612no—
194C4⋊Q8—1221no—
195C4⋊Q8—1221no—
196C4⋊Q8—1221no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C2×C4⋊C4—1221no—
202C2×C4⋊C4—1221no—
203C2×C4⋊C4D24411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C42C22×D6111yescharacteristic
206#106—1224no—
207C4⋊C8—21no—
208C4⋊C8—612no—
209C4⋊C8—1221no—
210C4⋊C8—1221no—
211C4⋊C8—1221no—
212C4⋊C8—1221no—
213C4⋊C8—1221no—
214C4⋊C8C22⋊D6112yes—
215#102—1224no—
216#100—2442no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C24—21no—
222C2×Q8—1222no—
223C2×Q8—4882no—
224C2×Q8—2441no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4Dic24412yes—
231C22×C4—21no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—82no—
235C22×C4—2441no—
236C22×C4C2×D24411yescharacteristic
237C22×C4#1175822411yescharacteristic
238C22×C4#1175822411yescharacteristic
239M16—2441no—
240M16—2441no—
241M16—2441no—
242M16—2441no—
243C2×C8—22no—
244C2×C8—41no—
245C2×C8—1222no—
246C2×C8—2441no—
247C2×C8—2441no—
248C2×C8—2441no—
249C2×C8—2441no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—1222no—
253C4⋊C4—1222no—
254C4⋊C4—2441no—
255C4⋊C4—2441no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C42#1175822211yescharacteristic
259C42D4×D6111yescharacteristic
260C42D4×D6111yescharacteristic
261C42C2×C22⋊D6111yescharacteristic
262C23—22no—
263C23—41no—
264C23—41no—
265C23—81no—
266C23—81no—
267C23#1451976111yescharacteristic
268Q8—4882no—
269Q8—4882no—
270D4—22no—
271D4—22no—
272D4—41no—
273D4—41no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—4882no—
281C2×C4—2442no—
282C2×C4#1451974711yescharacteristic
283C2×C4#1451974811yescharacteristic
284C2×C4#1451974911yescharacteristic
285C2×C4#1451975111yescharacteristic
286C2×C4#1451975911yescharacteristic
287C2×C4#1451976311yescharacteristic
288C2×C4C2×Dic24411yescharacteristic
289C2×C4#1451979911yescharacteristic
290C2×C4#1451980511yescharacteristic
291C2×C4#1451981011yescharacteristic
292C8—42no—
293C8—81no—
294C8—4881no—
295C8—4881no—
296C8—2442no—
297C22—21no—
298C22—21no—
299C22—42no—
300C22—82no—
301C22—82no—
302C22—81no—
303C22—81no—
304C22#1683967811yescharacteristic
305C22#1684009811yescharacteristic
306C22#1684010411yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—81no—
311C4—4882no—
312C4#1683990911yescharacteristic
313C4#1683992011yescharacteristic
314C4#1684000311yescharacteristic
315C4#1684000811yescharacteristic
316C4#1684002312yes—
317C2—21no—
318C2—21no—
319C2—82no—
320C2#2144702311yescharacteristic
321C2#2144708111yescharacteristic
322C2#2144709311yescharacteristic
subgroupG/H
centerC22#16839678
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1478C2
socleC22×C61#394
prime radicalC22×D4×C61D4
hypercenter#1478D61
hypocenterC61#10453
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
3. #21444594C2semidirect
4. #21444503C2semidirect
5. #21444499C2semidirect
6. #21443715C2semidirect
186. C61#10453semidirect

Maximal subgroups

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

subgroupindex
C2×#168400522
#214458112
#214445942
#214445032
#214444992
#214437152
C61×#14782
#1045361