Finite Group Databasebeta

← #13140891

Subgroups — #13140891

Order 1728.

518 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 79 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
1Q8×#1084C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C212yesmaximal, complemented
3#11721144C211yesmaximal, characteristic
4#11720308C212yesmaximal, complemented
5#11720304C211yesmaximal, characteristic, complemented
6#10591617—32nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×#6198—21no—
10C2×#6198—21no—
11C4⋊#1084—21no—
12C4⋊#1084C2211yescharacteristic
13C4×#1084—21no—
14C4×#1084C2211yescharacteristic
15#3755⋊C3—24no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C2×#3755D312yescomplemented
18C3×Q8×Dic3—32no—
19C3×Q8×Dic3—61no—
20C2×C3×#215—32no—
21#63911—32no—
22Dic3⋊(C3⋊C8)—32no—
23#63564—32no—
24#63562—32no—
25C22×(C3⋊D3)⋊C3—22no—
26Q8×H(3,𝔽3)—21no—
27Q8×H(3,𝔽3)D412yescomplemented
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084—21no—
31C2×#1084D411yescharacteristic
32C2×#1084D411yescharacteristic
33C4⋊((C3⋊D3)⋊C3)—22no—
34C4⋊((C3⋊D3)⋊C3)—22no—
35#6198—42no—
36#6198—42no—
37(C3⋊(C3⋊C8))⋊C3—22no—
38#5163—92no—
39#5027—91no—
40C2×C32×Q8—21no—
41C2×C32×Q8C2×D312yes—
42C2×C4⋊(C3⋊D3)C2×D312yes—
43C2×C3×D12—32no—
44C2×#626—62no—
45C2×#626—62no—
46#3755—24no—
47C2×C3⋊(C3⋊C8)C2×D312yes—
48C3×#215—64no—
49C3×C4⋊Dic3—32no—
50C3×C4⋊Dic3—62no—
51C3×C4⋊Dic3—61no—
52C3×C4⋊Dic3—62no—
53C3×C4×Dic3—32no—
54C3×C4×Dic3—62no—
55C3×C4×Dic3—61no—
56C3×C4×Dic3—62no—
57C2×C3×C3⋊C8—32no—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C2×(C3⋊D3)⋊C3—42no—
60C2×(C3⋊D3)⋊C3—42no—
61C2×(C3⋊D3)⋊C3—42no—
62C4×H(3,𝔽3)—22no—
63C4×H(3,𝔽3)C2×D411yescharacteristic
64C4×H(3,𝔽3)#4111yescharacteristic
65#1084—21no—
66#1084—41no—
67#1084SD1612yescomplemented
68C2×C3×SD16—92no—
69C3×C4×Q8—91no—
70Q8×Dic3—32no—
71Q8×Dic3—61no—
72C2×#215—32no—
73C2×#215—92no—
74#940—92no—
75#874—91no—
76#832—92no—
77#821—92no—
78Dic3⋊C8—92no—
79#809—92no—
80C4⋊(C3⋊C8)—91no—
81C22×C3⋊D3—22no—
82C22×C3×D3—62no—
83C32×Q8—22no—
84C32×Q8—22no—
85C32×Q8—22no—
86C32×Q8#6714yescomplemented
87C2×C32×C4—22no—
88C2×C32×C4—22no—
89C2×C32×C4—21no—
90C2×C32×C4C22×D312yes—
91C4⋊(C3⋊D3)—22no—
92C4⋊(C3⋊D3)—22no—
93C2×C3×Dic3—32no—
94C2×C3×Dic3—32no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99C3×D12—62no—
100C3×D12—62no—
101#626—124no—
102#626—124no—
103C3⋊(C3⋊C8)—22no—
104C3×C3⋊C8—62no—
105#460—271noSylow
106C2×H(3,𝔽3)#12111yescharacteristic
107C2×H(3,𝔽3)C2×SD1611yescharacteristic
108C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
109(C3⋊D3)⋊C3—82no—
110C2×C3×Q8—32no—
111C2×C3×Q8—61no—
112C2×C3×Q8D3211yescharacteristic
113C2×C3×D4—92no—
114C2×#67—182no—
115C2×#67—182no—
116C2×D12—32no—
117C2×D12—92no—
118C2×D12—181no—
119C2×D12—181no—
120C3×SD16—184no—
121C2×C3×C8—92no—
122C3×C4⋊C4—91no—
123C3×C4⋊C4—181no—
124C3×C42—91no—
125C3×C42—181no—
126#215—64no—
127#215—184no—
128C4⋊Dic3—32no—
129C4⋊Dic3—61no—
130C4⋊Dic3—62no—
131C4⋊Dic3—62no—
132C4×Dic3—32no—
133C4×Dic3—61no—
134C4×Dic3—62no—
135C4×Dic3—62no—
136C2×C3⋊C8—32no—
137C2×C3⋊C8—92no—
138C22×C32—21no—
139C22×C32D3×D412yes—
140C2×C3⋊D3—42no—
141C2×C3⋊D3—42no—
142C2×C3⋊D3—42no—
143C2×C3×D3—122no—
144C2×C3×D3—122no—
145C2×C3×D3—122no—
146C32×C4—24no—
147C32×C4—24no—
148C32×C4—21no—
149C32×C4—21no—
150C32×C4#23712yes—
151C32×C4C2×#6712yes—
152C3×Dic3—34no—
153C3×Dic3—62no—
154C3×Dic3—62no—
155C3×Dic3—121no—
156C3×Dic3—122no—
157C3×Dic3—122no—
158C2×SD16—92no—
159#127—271no—
160C4×Q8—91no—
161C4⋊C8—271no—
162#102—272no—
163H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
164C23×C3—182no—
165C22×D3—62no—
166C22×D3—182no—
167C3×Q8—21no—
168C3×Q8—34no—
169C3×Q8—62no—
170C3×Q8—62no—
171C3×Q8—62no—
172C3×Q8#62512yes—
173C3×D4—182no—
174C3×D4—182no—
175C2×C3×C4—21no—
176C2×C3×C4—32no—
177C2×C3×C4—62no—
178C2×C3×C4—62no—
179C2×C3×C4—61no—
180C2×C3×C4—91no—
181C2×C3×C4—91no—
182C2×C3×C4—181no—
183C2×C3×C4C2×D3211yescharacteristic
184#67—364no—
185#67—364no—
186C2×Dic3—32no—
187C2×Dic3—32no—
188C2×Dic3—61no—
189C2×Dic3—61no—
190C2×Dic3—62no—
191C2×Dic3—62no—
192D12—62no—
193D12—62no—
194D12—182no—
195D12—182no—
196D12—362no—
197D12—362no—
198C3×C8—182no—
199C3⋊C8—62no—
200C3⋊C8—182no—
201C2×C32—21no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32D3×SD1612yes—
205C2×C32#91412yes—
206C2×C32#93912yes—
207C3⋊D3—82no—
208C3×D3—242no—
209C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
210C2×D4—92no—
211C2×D4—541no—
212C2×D4—541no—
213SD16—184no—
214C2×C8—92no—
215C4⋊C4—91no—
216C4⋊C4—181no—
217C42—91no—
218C42—181no—
219C22×C3—32no—
220C22×C3—61no—
221C22×C3—362no—
222C22×C3—362no—
223C22×C3—362no—
224C22×C3#381111yescharacteristic
225C2×D3—122no—
226C2×D3—122no—
227C2×D3—122no—
228C2×D3—362no—
229C2×D3—362no—
230C2×D3—362no—
231C3×C4—22no—
232C3×C4—32no—
233C3×C4—32no—
234C3×C4—64no—
235C3×C4—64no—
236C3×C4—61no—
237C3×C4—61no—
238C3×C4—92no—
239C3×C4—181no—
240C3×C4—361no—
241C3×C4#380511yescharacteristic
242C3×C4C2×#62511yescharacteristic
243Dic3—34no—
244Dic3—62no—
245Dic3—62no—
246Dic3—121no—
247Dic3—122no—
248Dic3—122no—
249C32—21no—
250C32#516312yescomplemented
251C23—182no—
252Q8—21no—
253Q8#619812yescomplemented
254D4—182no—
255D4—182no—
256D4—1082no—
257D4—1082no—
258C2×C4—21no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4—181no—
262C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
263C8—182no—
264C2×C3—32no—
265C2×C3—32no—
266C2×C3—32no—
267C2×C3—61no—
268C2×C3—61no—
269C2×C3—61no—
270C2×C3—722no—
271C2×C3#6394011yescharacteristic
272C2×C3#6394111yescharacteristic
273C2×C3#6398011yescharacteristic
274D3—242no—
275D3—722no—
276C22—362no—
277C22—362no—
278C22—362no—
279C22#8867711yescharacteristic
280C4—22no—
281C4—92no—
282C4—181no—
283C4—361no—
284C4#8867211yescharacteristic
285C4C2×#619811yescharacteristic
286C3—32no—
287C3—61no—
288C3#1059162211yescharacteristic
289C2—722no—
290C2#1172122811yescharacteristic
291C2#1172122911yescharacteristic
292C2#1172131811yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721144C2
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#460
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×#1084C2semidirect
2. C2×#3755⋊C3C2semidirect
4. #11720308C2semidirect
5. #11720304C2semidirect
17. C2×#3755D3semidirect
27. Q8×H(3,𝔽3)D4semidirect
67. #1084SD16semidirect
86. C32×Q8#67semidirect
163. H(3,𝔽3)#460semidirect
250. C32#5163semidirect
253. Q8#6198semidirect

Maximal subgroups

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

subgroupindex
Q8×#10842
C2×#3755⋊C32
#117211442
#117203082
#117203042
#105916173