Finite Group Databasebeta

← #30607412

Subgroups — #30607412

Order 15616.

402 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#21462472C211yesmaximal, characteristic
2#21462167C212yesmaximal, complemented
3#21461700C211yesmaximal, characteristic
4#21460900C212yesmaximal, complemented
5#349×D61C211yesmaximal, characteristic
6C2×#14519856—21no—
7C2×#14519848C2211yescharacteristic
8C2×#14519847C2211yescharacteristic
9#16840773—22no—
10#16840773—22no—
11#16840772—21no—
12#16840772C2212yes—
13#16840733—21no—
14#16840687—22no—
15#16840686—22no—
16#16840638—21no—
17C2×M16×D61C2211yescharacteristic
18M32×D61C2212yes—
19#16839544C411yescharacteristic, complemented
20#16839543C412yescomplemented
21C61×#349C411yescharacteristic, complemented
22C2×#11758213—21no—
23C2×#11758213—21no—
24#14519856—22no—
25#14519856—22no—
26#14519856—41no—
27#14519856—41no—
28C2×C4⋊#6923—21no—
29C2×C4⋊#6923—21no—
30C2×C22⋊#6923—41no—
31C2×C4×#6923—41no—
32#14519848—22no—
33#14519847—22no—
34#14519846—22no—
35#14519846—22no—
36#14519845—21no—
37#14519845—21no—
38#14519845D412yes—
39#14519845D412yes—
40#14519834—41no—
41C22⋊#92420—41no—
42C22×C4×D61D411yescharacteristic
43C2×#11758194Q811yescharacteristic
44C2×#11758189C2×C411yescharacteristic
45C2×#11758189Q811yescharacteristic
46M16×D61—22no—
47M16×D61—22no—
48C2×C8×D61D411yescharacteristic
49C2×C8×D61C2311yescharacteristic
50#14519717C2×C412yescomplemented
51#14519716—22no—
52C16×D61—22no—
53C2×M16×C61C2×C411yescharacteristic
54M32×C61C2×C412yescomplemented
55C23×D61SD1611yescharacteristic
56C22×#6923—41no—
57C22×#6923—41no—
58C22×C4×C61C4⋊C411yescharacteristic
59C22×Dic61SD1611yescharacteristic
60C2×C4×D61—21no—
61C2×C4×D61—21no—
62C2×C4×D61—21no—
63C2×C4×D61—21no—
64C2×C4×D61C4⋊C411yescharacteristic
65C2×C4×D61SD1611yescharacteristic
66C2×C4×D61SD1611yescharacteristic
67C2×C4×D61C2×D411yescharacteristic
68C22⋊#6923—42no—
69C22⋊#6923—42no—
70C2×#92420—41no—
71C2×#92420—41no—
72C4⋊#6923—22no—
73C4⋊#6923—22no—
74C4⋊#6923—41no—
75C4⋊#6923—41no—
76C4×#6923—42no—
77C4×#6923—42no—
78#11758213—41no—
79#11758213—41no—
80#11758213—41no—
81#11758213—41no—
82M16×C61—22no—
83C2×C8×C61C4⋊C411yescharacteristic
84C2×C8×C61C22×C411yescharacteristic
85#11758194—22no—
86C2×#92418C4⋊C411yescharacteristic
87C2×#92418C2×Q811yescharacteristic
88#11758189—22no—
89#11758189—22no—
90C8×D61—21no—
91C8×D61—21no—
92C8×D61C2×D412yes—
93C8×D61#4112yes—
94C16×C61—22no—
95#11758185—22no—
96C23×C61#10611yescharacteristic
97C22×D61—21no—
98C22×D61—21no—
99C22×D61—81no—
100C22×D61#10611yescharacteristic
101C22×D61C2×SD1611yescharacteristic
102C2×#6923—42no—
103C2×#6923—42no—
104C2×#6923—82no—
105C2×#6923—82no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61#10611yescharacteristic
109C2×C4×C61C2×C4⋊C411yescharacteristic
110C2×Dic61—21no—
111C2×Dic61—21no—
112C2×Dic61#10611yescharacteristic
113C2×Dic61C2×SD1611yescharacteristic
114C4×D61—21no—
115C4×D61—21no—
116C4×D61—21no—
117C4×D61—21no—
118C4×D61—21no—
119C4×D61—21no—
120C4×D61#12011yescharacteristic
121C4×D61#12311yescharacteristic
122C4×D61AGL(1,ℤ/8)11yescharacteristic
123C4×D61#13711yescharacteristic
124#92420—81no—
125#92420—81no—
126C8×C61—21no—
127C8×C61C4×D412yes—
128#92418—21no—
129#92418C22⋊Q812yes—
130#12932—611nomaximal, Sylow
131C22×C61—21no—
132C22×C61—41no—
133C22×C61C2×#10611yescharacteristic
134C2×D61—21no—
135C2×D61—41no—
136C2×D61—41no—
137C2×D61—41no—
138C2×D61—41no—
139C2×D61—81no—
140C2×D61#45011yescharacteristic
141C2×D61#48311yescharacteristic
142#6923—82no—
143#6923—82no—
144C4×C61—21no—
145C4×C61#38511yescharacteristic
146C4×C61#42811yescharacteristic
147Dic61—21no—
148Dic61#44811yescharacteristic
149Dic61#48111yescharacteristic
150#2911—611no—
151#2155—612no—
152C2×#349—611no—
153#1811—611no—
154#1389—612no—
155C2×C61—21no—
156C2×C61—41no—
157C2×C61#185311yescharacteristic
158D61—21no—
159D61—21no—
160D61—41no—
161C22×M16—611no—
162C2×#117—1221no—
163C2×M32—612no—
164C2×#108—611no—
165C2×#108—611no—
166#428—1222no—
167#428—1222no—
168#427—612no—
169#427—1221no—
170#407—1221no—
171#365—1222no—
172#365—1222no—
173#349—611no—
174#349—612no—
175#349#692311yescharacteristic, complemented
176#341—1221no—
177C61#1293211yescharacteristic, complemented, Sylow
178C23×C4—611no—
179C2×M16—611no—
180C2×M16—611no—
181C2×M16—611no—
182C2×M16—1222no—
183C2×M16—1222no—
184C2×M16—1221no—
185C2×M16—1221no—
186C2×M16C2×#692311yescharacteristic
187C22×C8—611no—
188C22×C8—611no—
189#117—1222no—
190#117—1222no—
191#117—2441no—
192#117—2441no—
193C2×C4⋊C4—1221no—
194C2×C4⋊C4—1221no—
195C2×C22⋊C4—2441no—
196C2×C42—2441no—
197M32—612no—
198M32—1222no—
199M32C2×#692312yescomplemented
200C2×C16—1222no—
201#108—1222no—
202#108—1222no—
203#107—1222no—
204#107—1222no—
205#106—612no—
206#106—612no—
207#106—1221no—
208#106—1221no—
209C22⋊C8—2441no—
210C22⋊C8—2441no—
211C24—611no—
212C22×C4—611no—
213C22×C4—611no—
214C22×C4—611no—
215C22×C4—611no—
216C22×C4—611no—
217C22×C4—1221no—
218C22×C4—1221no—
219C22×C4—1221no—
220C22×C4—1221no—
221C22×C4—2441no—
222C22×C4—2441no—
223C22×C4C4⋊#692311yescharacteristic
224M16—22no—
225M16—1222no—
226M16—1222no—
227M16—1222no—
228M16—2441no—
229M16—2441no—
230M16—2441no—
231M16—2441no—
232C2×C8—611no—
233C2×C8—612no—
234C2×C8—612no—
235C2×C8—611no—
236C2×C8—1221no—
237C2×C8—1221no—
238C2×C8—2441no—
239C2×C8—2441no—
240C2×C8C4⋊#692311yescharacteristic
241C2×C8C22×#692311yescharacteristic
242C4⋊C4—1222no—
243C4⋊C4—1222no—
244C4⋊C4—2441no—
245C4⋊C4—2441no—
246C22⋊C4—2442no—
247C22⋊C4—2442no—
248C42—2442no—
249C42—2442no—
250C16—22no—
251C16—1222no—
252C23—611no—
253C23—611no—
254C23—1221no—
255C23—1221no—
256C23—4881no—
257C23#1451984511yescharacteristic
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—611no—
261C2×C4—611no—
262C2×C4—611no—
263C2×C4—611no—
264C2×C4—611no—
265C2×C4—611no—
266C2×C4—1221no—
267C2×C4—1221no—
268C2×C4—1221no—
269C2×C4—1221no—
270C2×C4—1221no—
271C2×C4—1221no—
272C2×C4—1221no—
273C2×C4—1221no—
274C2×C4—4882no—
275C2×C4—4882no—
276C2×C4—2442no—
277C2×C4—2442no—
278C2×C4#1451984511yescharacteristic
279C2×C4C2×C4⋊#692311yescharacteristic
280C8—21no—
281C8—612no—
282C8—1221no—
283C8—4881no—
284C8—4881no—
285C8D4×#692312yes—
286C22—21no—
287C22—41no—
288C22—611no—
289C22—611no—
290C22—1221no—
291C22—4881no—
292C22—2441no—
293C22—2441no—
294C22—2441no—
295C22—2441no—
296C22C2×#1451984511yescharacteristic
297C4—21no—
298C4—611no—
299C4—611no—
300C4—1221no—
301C4—4882no—
302C4—4882no—
303C4#1684070911yescharacteristic
304C4#1684077311yescharacteristic
305C2—21no—
306C2—41no—
307C2—1221no—
308C2—1221no—
309C2—2441no—
310C2#2146176911yescharacteristic
subgroupG/H
centerC2#21461769
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#349C4
socleC2×C61#1853
prime radicalC23×D61SD16
cosocleC2×C8×D61C23
hypercenter#349#6923
hypocenterC61#12932
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
2. #21462167C2semidirect
4. #21460900C2semidirect
19. #16839544C4semidirect
20. #16839543C4semidirect
21. C61×#349C4semidirect
50. #14519717C2×C4semidirect
54. M32×C61C2×C4semidirect
175. #349#6923semidirect
177. C61#12932semidirect
199. M32C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214624722
#214621672
#214617002
#214609002
#349×D612
#1293261