Finite Group Databasebeta

← #29583633

Subgroups — #29583633

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×#16840039C211yesmaximal, characteristic
2#21445808C211yesmaximal, characteristic
3#21444591C211yesmaximal, characteristic, complemented
4#21444504C211yesmaximal, characteristic, complemented
5#21444211C211yesmaximal, characteristic
6#21443715C211yesmaximal, characteristic, complemented
7C61×#1548C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9#16840039—22no—
10#16840039—22no—
11#16839815C2211yescharacteristic
12C2×#14519710—21no—
13#16839786C2211yescharacteristic
14C2×C4⋊#92418C2211yescharacteristic
15C2×#14519697—21no—
16C2×#14519693—21no—
17#16839530—21no—
18#16839512—22no—
19#16839508—22no—
20#16839491—22no—
21C2×C4⋊Q8×C61C2211yescharacteristic
22C61×#431C2211yescharacteristic
23C61×#423C2211yescharacteristic
24C61×#333—22no—
25C22×D244D411yescharacteristic
26C22×C4×D61—21no—
27C2×#11758194—21no—
28C22×#92418—21no—
29C4×D244—21no—
30C4×D244—21no—
31C4×D244—21no—
32C4×D244—21no—
33C2×C4⋊Dic61D411yescharacteristic
34C2×#11758198—21no—
35#14519710—24no—
36C4⋊#92418—21no—
37C4⋊#92418—21no—
38C4⋊#92418—21no—
39C4⋊#92418D412yes—
40C4⋊#92418D412yes—
41#14519697—24no—
42#14519693—42no—
43C22⋊#92418—21no—
44C4×#92418—21no—
45C22×Q8×C61—21no—
46C2×M16×C61—21no—
47C4⋊Q8×C61—21no—
48C4⋊Q8×C61—21no—
49C4⋊Q8×C61—21no—
50C4⋊Q8×C61D412yes—
51C2×C4⋊C4×C61—21no—
52C2×C4⋊C4×C61—21no—
53C2×C42×C61C2311yescharacteristic
54C4⋊C8×C61—21no—
55C4⋊C8×C61—21no—
56C4⋊C8×C61—21no—
57C22⋊C8×C61—21no—
58C4×C8×C61—21no—
59C23×D61—21no—
60C2×Q8×C61—22no—
61C2×Q8×C61—41no—
62C2×Q8×C61—82no—
63C22×C4×C61—41no—
64C22×C4×C61C2×D411yescharacteristic
65C22×C4×C61#4111yescharacteristic
66C22×C4×C61#4111yescharacteristic
67C22×Dic61—21no—
68C2×D244—21no—
69C2×D244—21no—
70C2×D244—21no—
71C2×D244—21no—
72C2×D244—21no—
73C2×D244SD1612yes—
74C2×C4×D61—41no—
75C2×C4×D61—41no—
76C2×C4×D61—82no—
77M16×C61—41no—
78M16×C61—41no—
79C2×C8×C61—41no—
80C2×C8×C61—41no—
81C4⋊C4×C61—22no—
82C4⋊C4×C61—22no—
83C4⋊C4×C61—41no—
84C4⋊C4×C61—41no—
85C42×C61C2×D411yescharacteristic
86C42×C61C2×D411yescharacteristic
87C42×C61C2×D411yescharacteristic
88C42×C61#4111yescharacteristic
89#11758198—41no—
90#11758198—41no—
91C4⋊Dic61—21no—
92C4⋊Dic61—21no—
93#11758194—41no—
94#11758194—41no—
95C2×#92418—22no—
96C2×#92418—22no—
97C2×#92418—41no—
98C2×#92418—41no—
99C2×#92418—41no—
100C23×C61#12311yescharacteristic
101C22×D61—22no—
102C22×D61—41no—
103C22×D61—41no—
104C22×D61—81no—
105C22×D61—81no—
106Q8×C61—82no—
107Q8×C61—82no—
108C2×C4×C61—22no—
109C2×C4×C61—41no—
110C2×C4×C61—42no—
111C2×C4×C61—82no—
112C2×C4×C61#12011yescharacteristic
113C2×C4×C61#12111yescharacteristic
114C2×C4×C61#12111yescharacteristic
115C2×C4×C61#12111yescharacteristic
116C2×C4×C61#12311yescharacteristic
117C2×C4×C61#12411yescharacteristic
118C2×C4×C61C2×SD1611yescharacteristic
119C2×C4×C61#13511yescharacteristic
120C2×C4×C61#13711yescharacteristic
121C2×C4×C61#13711yescharacteristic
122C2×Dic61—41no—
123C2×Dic61—81no—
124D244—22no—
125D244—22no—
126D244—41no—
127D244—41no—
128C4×D61—82no—
129C4×D61—82no—
130C8×C61—81no—
131C8×C61—81no—
132#92418—42no—
133#92418—42no—
134#92418—81no—
135#10448—611nomaximal, Sylow
136C22×C61—21no—
137C22×C61—21no—
138C22×C61#39411yescharacteristic
139C22×C61#48311yescharacteristic
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—82no—
150C4×C61#44811yescharacteristic
151C4×C61#45211yescharacteristic
152C4×C61#46511yescharacteristic
153C4×C61#47011yescharacteristic
154C4×C61#47212yes—
155Dic61—81no—
156C2×#474—611no—
157#1800—611no—
158#1570—611no—
159#1549—611no—
160#1548D6111yescharacteristic, complemented
161#1478—611no—
162#1301—611no—
163C2×C61—21no—
164C2×C61—21no—
165C2×C61#203511yescharacteristic
166C2×C61#204511yescharacteristic
167C2×C61#204911yescharacteristic
168D61—82no—
169C2×C4⋊Q8C2×D6111yescharacteristic
170C2×C4×D4—611no—
171#474—1222no—
172#474—1222no—
173#431—611no—
174#431C2×D6111yescharacteristic
175C2×#107—1221no—
176#423—611no—
177#423C2×D6111yescharacteristic
178C2×C4⋊C8—611no—
179C2×#102—1221no—
180C2×#100—1221no—
181#343—1221no—
182#334—1222no—
183#333—22no—
184#332—1222no—
185D4⋊C8—1222no—
186C61#1044811yescharacteristic, complemented, Sylow
187C22×Q8—21no—
188C22×D4—611no—
189C23×C4—1221no—
190C2×M16—21no—
191C2×M16—1221no—
192C22×C8—1221no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8C22⋊D6112yes—
197C4×D4—1221no—
198C4×D4—1221no—
199C4×D4—1221no—
200C4×D4—1221no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—611no—
204C2×C22⋊C4—1221no—
205C2×C42C22×D6111yescharacteristic
206#107—1224no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—612no—
211C4⋊C8—612no—
212C4⋊C8—1221no—
213C4⋊C8—1221no—
214C4⋊C8—1221no—
215#102—1224no—
216#100—2442no—
217C22⋊C8—21no—
218C22⋊C8—1221no—
219C4×C8—21no—
220C4×C8—1221no—
221C24—1221no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—82no—
225C2×D4—612no—
226C2×D4—1221no—
227C2×D4—1221no—
228C2×D4—1221no—
229C2×D4—1221no—
230C2×D4—1221no—
231C22×C4—41no—
232C22×C4—1221no—
233C22×C4—4882no—
234C22×C4—2441no—
235C22×C4—2441no—
236C22×C4#1175822211yescharacteristic
237C22×C4D4×D6111yescharacteristic
238C22×C4#1175822411yescharacteristic
239M16—41no—
240M16—41no—
241M16—2441no—
242M16—2441no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—1222no—
246C2×C8—1222no—
247C2×C8—2441no—
248C2×C8—2441no—
249C2×C8—2441no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—1221no—
255C4⋊C4—1221no—
256C22⋊C4—2441no—
257C22⋊C4—2441no—
258C42#1175822211yescharacteristic
259C42D4×D6111yescharacteristic
260C42D4×D6111yescharacteristic
261C42C2×C22⋊D6111yescharacteristic
262C23—1222no—
263C23—4881no—
264C23—4881no—
265C23—2441no—
266C23—2441no—
267C23#1451975911yescharacteristic
268Q8—82no—
269Q8—82no—
270D4—1222no—
271D4—1222no—
272D4—2441no—
273D4—2441no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—4882no—
279C2×C4—4882no—
280C2×C4—4881no—
281C2×C4—2441no—
282C2×C4#1451974511yescharacteristic
283C2×C4#1451974911yescharacteristic
284C2×C4#1451975011yescharacteristic
285C2×C4#1451975011yescharacteristic
286C2×C4#1451976011yescharacteristic
287C2×C4#1451976311yescharacteristic
288C2×C4SD16×D6111yescharacteristic
289C2×C4#1451980011yescharacteristic
290C2×C4#1451980611yescharacteristic
291C2×C4#1451980911yescharacteristic
292C8—81no—
293C8—81no—
294C8—4881no—
295C8—2442no—
296C8—2442no—
297C22—21no—
298C22—21no—
299C22—4882no—
300C22—4882no—
301C22—4881no—
302C22—4881no—
303C22—2442no—
304C22#1683967711yescharacteristic
305C22#1684009111yescharacteristic
306C22#1684010211yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—82no—
311C4—4881no—
312C4#1683989911yescharacteristic
313C4#1683992311yescharacteristic
314C4#1683999411yescharacteristic
315C4#1684001911yescharacteristic
316C4#1684003112yes—
317C2—21no—
318C2—21no—
319C2—4882no—
320C2#2144701311yescharacteristic
321C2#2144707311yescharacteristic
322C2#2144709711yescharacteristic
subgroupG/H
centerC22#16839677
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1548C2
socleC22×C61#394
prime radicalC22×D244D4
hypercenter#1548D61
hypocenterC61#10448
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. #21444591C2semidirect
4. #21444504C2semidirect
6. #21443715C2semidirect
7. C61×#1548C2semidirect
160. #1548D61semidirect
186. C61#10448semidirect

Maximal subgroups

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

subgroupindex
C2×#168400392
#214458082
#214445912
#214445042
#214442112
#214437152
C61×#15482
#1044861