Finite Group Databasebeta

← #30605370

Subgroups — #30605370

Order 15616.

364 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 72 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#21462056C211yesmaximal, characteristic
2#21461701C211yesmaximal, characteristic
3#21460791C214yesmaximal, complemented
4#21455470C211yesmaximal, characteristic
5D61⋊M32—21no—
6#16840772—21no—
7#16840741—21no—
8#16840659C2212yes—
9#16840658—21no—
10#16840630C2212yes—
11C2×#14519783C2211yescharacteristic
12#16840336—21no—
13#16840017C412yescomplemented
14#16840017C2212yes—
15#16840000—21no—
16#16840000—21no—
17#16839800C411yescharacteristic, complemented
18#16839771—21no—
19C22×Q16×C61C411yescharacteristic, complemented
20#14519856—22no—
21#14519852—41no—
22#14519851—22no—
23D61⋊C16—41no—
24#14519846—41no—
25#14519845—41no—
26#14519836—41no—
27#14519835—41no—
28C2×#11758222D411yescharacteristic
29#14519783—41no—
30#14519783—41no—
31C2×#11758194D411yescharacteristic
32C2×#11758189C2×C411yescharacteristic
33C2×#11758189C2311yescharacteristic
34C2×C8×D61D411yescharacteristic
35C2×C8×D61D411yescharacteristic
36#14519775—41no—
37#14519766—41no—
38#14519760—41no—
39#14519760—41no—
40#14519753—22no—
41#14519753—22no—
42#14519736—21no—
43#14519711—21no—
44#14519709C2×C412yes—
45#14519701—22no—
46#14519701—22no—
47C22×Q8×C61—21no—
48C2×Q16×C61—21no—
49C2×Q16×C61—41no—
50C2×Q16×C61—41no—
51C2×Q16×C61C2×C412yes—
52C22×C8×C61C2×C411yescharacteristic
53C2×Q8×C61—22no—
54C2×Q8×C61—42no—
55C2×Q8×C61—42no—
56C2×Q8×C61—41no—
57C22×C4×C61—41no—
58C22×C4×C61C22⋊C411yescharacteristic
59C2×C22⋊D61D811yescharacteristic
60C2×C22⋊D61SD1611yescharacteristic
61#11758222—41no—
62#11758222—41no—
63C2×D244D811yescharacteristic
64C2×C4×D61C22⋊C411yescharacteristic
65C2×C4×D61C2×D411yescharacteristic
66C2×Dic122SD1611yescharacteristic
67C22⋊#6923—42no—
68C4⋊#6923—42no—
69C4×#6923—42no—
70Q16×C61—42no—
71Q16×C61—41no—
72Q16×C61—41no—
73C2×C8×C61—21no—
74C2×C8×C61—21no—
75C2×C8×C61C22⋊C411yescharacteristic
76C2×C8×C61C22×C411yescharacteristic
77C22⋊Dic61—41no—
78#11758198—42no—
79#11758198—42no—
80C4⋊Dic61—22no—
81#11758196—42no—
82#11758196—42no—
83C4×Dic61—41no—
84#11758194—41no—
85C2×#92418C22⋊C411yescharacteristic
86C2×#92418C2×D411yescharacteristic
87#11758189—41no—
88#11758189—41no—
89C8×D61—21no—
90C8×D61—21no—
91C8×D61—21no—
92C8×D61—21no—
93#11758187—42no—
94C23×C61#10211yescharacteristic
95C22×D61#10211yescharacteristic
96C22×D61C2×D811yescharacteristic
97C2×#6923—82no—
98Q8×C61—42no—
99Q8×C61—42no—
100Q8×C61—82no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—42no—
104C2×C4×C61—82no—
105C2×C4×C61#10211yescharacteristic
106C2×C4×C61C2×C22⋊C411yescharacteristic
107C22⋊D61—41no—
108C22⋊D61—41no—
109C2×Dic61—42no—
110C2×Dic61#10211yescharacteristic
111C2×Dic61C2×SD1611yescharacteristic
112D244—41no—
113C4×D61—21no—
114C4×D61—21no—
115C4×D61—21no—
116C4×D61—21no—
117Dic122—41no—
118C8×C61—21no—
119C8×C61#9911yescharacteristic
120C8×C61#9911yescharacteristic
121#92418—21no—
122#92418—21no—
123#11869—611nomaximal, Sylow
124C22×C61—21no—
125C22×C61—41no—
126C22×C61C2×#10211yescharacteristic
127C2×D61—21no—
128C2×D61—21no—
129C2×D61—41no—
130C2×D61—41no—
131#6923—162no—
132C4×C61—21no—
133C4×C61—82no—
134C4×C61C2×#9911yescharacteristic
135C4×C61#42111yescharacteristic
136Dic61—21no—
137Dic61—21no—
138Dic61—82no—
139#3059—611no—
140#2115—611no—
141#1812—611no—
142#1340—614no—
143C2×C61—21no—
144C2×C61—41no—
145C2×C61#151911yescharacteristic
146D61—41no—
147D61—41no—
148C22×Q16#692311yescharacteristic, complemented
149C2×#131—611no—
150#536—1221no—
151C2×M32—1221no—
152#470—612no—
153#470—612no—
154#467—1221no—
155#467—1221no—
156#427—611no—
157#427—1221no—
158#419—1221no—
159#410—1221no—
160#349—612no—
161#349—1221no—
162H(3,ℤ/4)—612no—
163C61#1186911yescharacteristic, complemented, Sylow
164C2×#41—611no—
165C22×Q8—21no—
166C2×Q16—21no—
167C2×Q16—41no—
168C2×Q16—41no—
169C2×Q16C2×#692312yes—
170#131—2441no—
171#131—2441no—
172C2×M16—611no—
173C2×M16—611no—
174C2×M16—611no—
175C22×C8—611no—
176C22×C8—611no—
177C22×C8C2×#692311yescharacteristic
178C4⋊Q8—2441no—
179#124—2441no—
180#123—2441no—
181#123—2441no—
182C22⋊Q8—1222no—
183C22⋊Q8—1222no—
184#117—1221no—
185#117—1222no—
186M32—1222no—
187M32—2441no—
188C2×C16—2441no—
189#107—1221no—
190#107—2441no—
191#106—612no—
192#106—2441no—
193#103—1222no—
194#103—1222no—
195#99—2441no—
196#99—2441no—
197#41—2441no—
198#41—2441no—
199C2×Q8—22no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—41no—
203C2×Q8—611no—
204C2×D4—611no—
205C2×D4—611no—
206C2×D4—611no—
207C22×C4—41no—
208C22×C4—611no—
209C22×C4—611no—
210C22×C4C22⋊#692311yescharacteristic
211Q16—42no—
212Q16—41no—
213Q16—41no—
214M16—2441no—
215M16—2441no—
216M16—2441no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—611no—
220C2×C8—611no—
221C2×C8—1221no—
222C2×C8—1221no—
223C2×C8—1221no—
224C2×C8—1221no—
225C2×C8C22⋊#692311yescharacteristic
226C2×C8C22×#692311yescharacteristic
227C4⋊C4—1222no—
228C4⋊C4—2442no—
229C4⋊C4—2442no—
230C4⋊C4—2442no—
231C22⋊C4—2442no—
232C22⋊C4—2442no—
233C22⋊C4—2442no—
234C22⋊C4—2441no—
235C42—2442no—
236C42—2441no—
237C16—2442no—
238C23—611no—
239C23—611no—
240C23#1451983911yescharacteristic
241Q8—42no—
242Q8—42no—
243Q8—82no—
244Q8—2441no—
245D4—2441no—
246D4—2441no—
247D4—2441no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—82no—
252C2×C4—611no—
253C2×C4—611no—
254C2×C4—1221no—
255C2×C4—1221no—
256C2×C4—1221no—
257C2×C4—1221no—
258C2×C4—4882no—
259C2×C4—2442no—
260C2×C4#1451983911yescharacteristic
261C2×C4C2×C22⋊#692311yescharacteristic
262C8—21no—
263C8—1221no—
264C8—1221no—
265C8#1451983511yescharacteristic
266C8#1451983511yescharacteristic
267C22—21no—
268C22—41no—
269C22—1221no—
270C22—1221no—
271C22—2441no—
272C22—2441no—
273C22C2×#1451983911yescharacteristic
274C4—21no—
275C4—82no—
276C4—1221no—
277C4—1221no—
278C4—9762no—
279C4—4882no—
280C4C2×#1451983511yescharacteristic
281C4#1684076011yescharacteristic
282C2—21no—
283C2—41no—
284C2—2441no—
285C2—2441no—
286C2#2146125511yescharacteristic
subgroupG/H
centerC2#21461255
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC22×Q16×C61C4
socleC2×C61#1519
prime radicalC2×#11758222D4
cosocleC2×#11758189C23
hypercenterC22×Q16#6923
hypocenterC61#11869
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
3. #21460791C2semidirect
13. #16840017C4semidirect
17. #16839800C4semidirect
19. C22×Q16×C61C4semidirect
148. C22×Q16#6923semidirect
163. C61#11869semidirect

Maximal subgroups

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

subgroupindex
#214620562
#214617012
#214607912
#214554702
#1186961