Finite Group Databasebeta

← #30601998

Subgroups — #30601998

Order 15616.

474 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 100 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#21462011C212yesmaximal
2#21461376C211yesmaximal, characteristic
3#21461332C212yesmaximal
4C8×#11758217C211yesmaximal, characteristic
5C2×#16840140C211yesmaximal, characteristic
6#16840779C2211yescharacteristic
7#16840765C2212yes—
8C4×#11758217C2211yescharacteristic
9#16840642—24no—
10#16840626—21no—
11#16840625—21no—
12#16840622—24no—
13#92420⋊C8—21no—
14C8×#92420—21no—
15C22×Dic244C411yescharacteristic
16C22×#11758202C412yes—
17C2×#14519775C2212yes—
18#16840140—21no—
19#16840140—21no—
20#16840140—21no—
21#16840140—21no—
22C2×C8×Dic61C2211yescharacteristic
23C22×Q16×C61C411yescharacteristic
24C2×#11758217—22no—
25C2×#11758217D412yes—
26#14519844—22no—
27#14519844D412yes—
28C4⋊#92420—22no—
29C22⋊#92420—21no—
30#14519832—21no—
31C4×#92420—21no—
32C4×#92420—21no—
33C22×Dic122C2×C412yes—
34C2×Dic244—21no—
35C2×Dic244—21no—
36C2×Dic244—42no—
37C2×#11758202—22no—
38C2×#11758202—22no—
39C2×#11758202—44no—
40C22×#92418C2×C411yescharacteristic
41#14519775—22no—
42#14519775—22no—
43#14519775—22no—
44#14519775—22no—
45C2×#11758196—22no—
46C2×#11758196—22no—
47C2×C4×Dic61C2311yescharacteristic
48C8×Dic61D411yescharacteristic
49C8×Dic61D411yescharacteristic
50C8×Dic61D411yescharacteristic
51C8×Dic61D411yescharacteristic
52C22×Q8×C61C2×C412yes—
53C2×Q16×C61—21no—
54C2×Q16×C61—21no—
55C2×Q16×C61—42no—
56C22×C8×C61C2×C411yescharacteristic
57C2×Q8×C61—22no—
58C2×Q8×C61—22no—
59C2×Q8×C61—42no—
60C2×Q8×C61—42no—
61C22×C4×C61—22no—
62C22×C4×C61C22×C411yescharacteristic
63C22×Dic61—22no—
64C22×Dic61C2×D411yescharacteristic
65C22×Dic61#4111yescharacteristic
66C2×Dic122—22no—
67C2×Dic122—22no—
68C2×Dic122—42no—
69C2×Dic122—42no—
70#11758217—22no—
71#11758217—44no—
72#11758217Q1614yescomplemented
73C2×#92420—22no—
74C2×#92420—22no—
75C2×#92420—42no—
76Q16×C61—42no—
77Q16×C61—42no—
78C2×C8×C61—22no—
79C2×C8×C61C22⋊C411yescharacteristic
80C2×C8×C61C22⋊C411yescharacteristic
81#11758202—44no—
82#11758202—44no—
83#11758196—42no—
84#11758196—42no—
85#11758196—42no—
86#11758196—42no—
87C4×Dic61C2×D411yescharacteristic
88C4×Dic61C2×D411yescharacteristic
89C4×Dic61C2×D411yescharacteristic
90C4×Dic61#4111yescharacteristic
91C2×#92418—22no—
92C2×#92418C22⋊C411yescharacteristic
93C2×#92418C22⋊C411yescharacteristic
94Dic244—42no—
95Dic244—42no—
96C23×C61C4×D411yescharacteristic
97Q8×C61—42no—
98Q8×C61—42no—
99Q8×C61—82no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—42no—
103C2×C4×C61—82no—
104C2×C4×C61C2×C22⋊C411yescharacteristic
105C2×C4×C61C4×D411yescharacteristic
106C2×Dic61—22no—
107C2×Dic61—42no—
108C2×Dic61—82no—
109C2×Dic61#12111yescharacteristic
110C2×Dic61#12111yescharacteristic
111C2×Dic61#12411yescharacteristic
112C2×Dic61#12711yescharacteristic
113C2×Dic61C2×Q1612yes—
114C2×Dic61#13512yes—
115Dic122—42no—
116Dic122—42no—
117Dic122—82no—
118#92420—24no—
119#92420—42no—
120#92420—82no—
121C8×C61—21no—
122C8×C61#10012yes—
123#92418—21no—
124#92418#10012yes—
125#9798—611nomaximal, Sylow
126C22×C61—21no—
127C22×C61—21no—
128C22×C61#39011yescharacteristic
129C22×C61C4×Q1612yes—
130C4×C61—21no—
131C4×C61—82no—
132C4×C61C2×#10011yescharacteristic
133C4×C61C2×#10011yescharacteristic
134Dic61—21no—
135Dic61—82no—
136Dic61#46712yes—
137Dic61#47312yes—
138Dic61#49411yescharacteristic
139Dic61#49511yescharacteristic
140C2×#494—611no—
141#1997—612no—
142#1567—611no—
143#1548—612no—
144C8×M16—611no—
145C2×C61—22no—
146C2×C61#196511yescharacteristic
147C2×C61#196911yescharacteristic
148C2×C61#197911yescharacteristic
149C22×Q16—612no—
150C22×Q16—611no—
151C22×Q16#692311yescharacteristic
152C2×C4⋊Q8—612no—
153#494—1221no—
154#494—1221no—
155#494—1221no—
156#494—1221no—
157#431—611no—
158#423—612no—
159C4×M16—611no—
160C2×C4×C8—611no—
161#343—1224no—
162#335—1221no—
163#335—1221no—
164#333—1224no—
165#322—1221no—
166C82—1221no—
167C61#979811yescharacteristic, complemented, Sylow
168C22×Q8—612no—
169C22×Q8C2×#692312yes—
170C2×Q16—21no—
171C2×Q16—21no—
172C2×Q16—42no—
173C2×Q16—1222no—
174C2×Q16—1222no—
175C2×Q16—1221no—
176C2×Q16—1221no—
177C2×Q16—2442no—
178C2×Q16—2444no—
179C2×M16—612no—
180C2×M16—1222no—
181C22×C8—611no—
182C22×C8C2×#692311yescharacteristic
183C4⋊Q8—1222no—
184C4⋊Q8—1222no—
185C4⋊Q8—1222no—
186C4⋊Q8—1222no—
187C2×C4⋊C4—1222no—
188C2×C4⋊C4—1222no—
189C2×C42—611no—
190C4⋊C8—612no—
191C4⋊C8—1222no—
192C4⋊C8—1222no—
193C22⋊C8—1221no—
194#97—1221no—
195C4×C8—611no—
196C4×C8—611no—
197C4×C8—611no—
198C4×C8—611no—
199C4×C8—1221no—
200C4×C8—1221no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—1222no—
206C2×Q8—1222no—
207C2×Q8—2442no—
208C2×Q8—2442no—
209C22×C4—22no—
210C22×C4—611no—
211C22×C4—611no—
212C22×C4—1222no—
213C22×C4C22×#692311yescharacteristic
214Q16—42no—
215Q16—42no—
216Q16—2442no—
217Q16—2444no—
218Q16—2444no—
219Q16—2442no—
220M16—614no—
221M16—1222no—
222M16—2444no—
223C2×C8—22no—
224C2×C8—611no—
225C2×C8—611no—
226C2×C8—1222no—
227C2×C8—1222no—
228C2×C8—1222no—
229C2×C8—2442no—
230C2×C8C22⋊#692311yescharacteristic
231C2×C8C22⋊#692311yescharacteristic
232C4⋊C4—2442no—
233C4⋊C4—2442no—
234C4⋊C4—2442no—
235C4⋊C4—2442no—
236C42—611no—
237C42—611no—
238C42—611no—
239C42—611no—
240C23D4×#692311yescharacteristic
241Q8—42no—
242Q8—42no—
243Q8—82no—
244Q8—4882no—
245Q8—2442no—
246Q8—2442no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—42no—
250C2×C4—82no—
251C2×C4—611no—
252C2×C4—611no—
253C2×C4—611no—
254C2×C4—612no—
255C2×C4—612no—
256C2×C4—611no—
257C2×C4—1222no—
258C2×C4—4882no—
259C2×C4—2442no—
260C2×C4C2×C22⋊#692311yescharacteristic
261C2×C4D4×#692311yescharacteristic
262C8—21no—
263C8—612no—
264C8—1224no—
265C8—1221no—
266C8—4882no—
267C8—2442no—
268C8#1451983712yes—
269C22—21no—
270C22—21no—
271C22#1684071411yescharacteristic
272C22Q16×#692312yes—
273C4—21no—
274C4—82no—
275C4—612no—
276C4—611no—
277C4—611no—
278C4—612no—
279C4—1221no—
280C4—4882no—
281C4C2×#1451983711yescharacteristic
282C4D61⋊#10011yescharacteristic
283C2—22no—
284C2#2146196111yescharacteristic
285C2#2146196511yescharacteristic
286C2#2146198111yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×Q16×C61C4
socleC22×C61#390
prime radicalC23×C61C4×D4
cosocleC2×C4×Dic61C23
hypercenterC22×Q16#6923
hypocenterC61#9798
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
72. #11758217Q16semidirect
167. C61#9798semidirect

Maximal subgroups

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

subgroupindex
#214620112
#214613762
#214613322
C8×#117582172
C2×#168401402
#979861