Finite Group Databasebeta

← #13138982

Subgroups — #13138982

Order 1728.

643 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 643 conjugacy classes of subgroups — too many to draw. Only its 91 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×#88529C211yesmaximal, characteristic, complemented
2C2×#3745⋊C3C211yesmaximal, characteristic, complemented
3#11721158C212yesmaximal, complemented
4C2×#88436C212yesmaximal, complemented
5C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6#10590886—31nomaximal
7#10590880—31nomaximal
8C2×C4⋊#272C2212yes—
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×#6200—24no—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12#88529—22no—
13C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15#3745⋊C3—22no—
16#88436—24no—
17C2×#3745D311yescharacteristic, complemented
18C2×C3×D24—31no—
19C2×C3×D24—31no—
20C2×C3×D24—31no—
21C2×C3×D24—61no—
22#63914—32no—
23#63911—32no—
24C2×#3714—32no—
25C2×#3713—32no—
26C8×C3⋊Dic3D311yescharacteristic, complemented
27C3×C8×Dic3—31no—
28C22×#272—22no—
29C22×(C3⋊D3)⋊C3—22no—
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C4⋊#272—22no—
32C4⋊#272—22no—
33C2×(C3⋊Dic3)⋊C3D412yes—
34C4⋊((C3⋊D3)⋊C3)—22no—
35C4⋊((C3⋊D3)⋊C3)—22no—
36#6200—48no—
37C8×H(3,𝔽3)D412yescomplemented
38(C3⋊(C3⋊C8))⋊C3D412yescomplemented
39#5151—91no—
40#5151—91no—
41#4695—91no—
42C2×C4⋊(C3⋊D3)C2×D312yes—
43C2×C3×D12—32no—
44C2×C3×D12—32no—
45C2×C3×D12—32no—
46C2×C3×D12—62no—
47C2×#626—64no—
48C2×#625—64no—
49C2×C32×C8—21no—
50C2×C32×C8C2×D311yescharacteristic
51C2×C32×C8C2×D311yescharacteristic
52C4×C3⋊Dic3C2×D311yescharacteristic
53C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
54#3745—22no—
55C3×C4×Dic3—31no—
56C2×C3×C3⋊C8—31no—
57C3×D24—64no—
58C3×D24—62no—
59C3×D24—62no—
60C3×D24—62no—
61#3714—64no—
62#3713—64no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×#272—42no—
65C2×#272—44no—
66C2×(C3⋊D3)⋊C3—42no—
67C2×(C3⋊D3)⋊C3—44no—
68C4×H(3,𝔽3)C2×D411yescharacteristic
69C4×H(3,𝔽3)C2×D411yescharacteristic
70(C3⋊Dic3)⋊C3D814yescomplemented
71C2×C3×D8—91no—
72C2×C3×D8—91no—
73#940—92no—
74#940—92no—
75C2×#213—92no—
76C2×#213—92no—
77C2×D24—31no—
78C2×D24—31no—
79C2×D24—31no—
80C2×D24—61no—
81C2×D24—91no—
82C2×D24—92no—
83#874—92no—
84C3×C4×C8—91no—
85C8×Dic3—31no—
86C8×Dic3—91no—
87C22×C3⋊D3—22no—
88C22×C3×D3—62no—
89C22×C3×D3—62no—
90C22×C3×D3—62no—
91C22×C3×D3—122no—
92C2×C32×C4—21no—
93C2×C32×C4C22×D311yescharacteristic
94C2×C32×C4C22×D311yescharacteristic
95C2×C3⋊Dic3D1212yes—
96C4⋊(C3⋊D3)—22no—
97C4⋊(C3⋊D3)—22no—
98C2×C3×Dic3—32no—
99C3×D12—64no—
100C3×D12—62no—
101C3×D12—62no—
102C3×D12—62no—
103C3×D12—62no—
104C3×D12—62no—
105C3×D12—62no—
106C3×D12—122no—
107#626—128no—
108#625—128no—
109C32×C8—22no—
110C32×C8D1212yescomplemented
111C32×C8#6712yescomplemented
112C3⋊(C3⋊C8)D1212yescomplemented
113C3×C3⋊C8—32no—
114#493—271noSylow
115C2×H(3,𝔽3)#12711yescharacteristic
116C2×H(3,𝔽3)C2×D812yes—
117#272—82no—
118(C3⋊D3)⋊C3—82no—
119C2×C3×D4—92no—
120C2×C3×D4—92no—
121C2×#67—184no—
122C2×#67—184no—
123C2×D12—32no—
124C2×D12—32no—
125C2×D12—32no—
126C2×D12—62no—
127C2×D12—92no—
128C2×D12—184no—
129C3×D8—182no—
130C3×D8—182no—
131C2×C3×C8—31no—
132C2×C3×C8—31no—
133C2×C3×C8—61no—
134C2×C3×C8—91no—
135C2×C3×C8D3211yescharacteristic
136C3×C42—91no—
137#213—184no—
138#213—184no—
139C4×Dic3—31no—
140C4×Dic3—91no—
141C2×C3⋊C8—31no—
142C2×C3⋊C8—91no—
143D24—62no—
144D24—62no—
145D24—64no—
146D24—62no—
147D24—182no—
148D24—184no—
149C22×C32—21no—
150C22×C32C2×D1211yescharacteristic
151C22×C32D3×D411yescharacteristic
152C2×C3⋊D3—44no—
153C2×C3⋊D3—42no—
154C2×C3×D3—124no—
155C2×C3×D3—122no—
156C2×C3×D3—122no—
157C2×C3×D3—124no—
158C2×C3×D3—124no—
159C2×C3×D3—124no—
160C2×C3×D3—122no—
161C2×C3×D3—244no—
162C32×C4—21no—
163C32×C4—21no—
164C32×C4C2×D1211yescharacteristic
165C32×C4C2×D1211yescharacteristic
166C32×C4D3×D411yescharacteristic
167C32×C4C2×#6711yescharacteristic
168C3⋊Dic3D2414yescomplemented
169C3×Dic3—34no—
170C2×D8—91no—
171C2×D8—91no—
172C2×D8—272no—
173#127—272no—
174C4×C8—91no—
175H(3,𝔽3)#49311yescharacteristic, complemented, Sylow
176C23×C3—182no—
177C23×C3—182no—
178C22×D3—62no—
179C22×D3—62no—
180C22×D3—62no—
181C22×D3—122no—
182C22×D3—182no—
183C3×D4—182no—
184C3×D4—182no—
185C3×D4—182no—
186C3×D4—182no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—61no—
190C2×C3×C4—92no—
191C2×C3×C4C2×D3211yescharacteristic
192#67—368no—
193#67—368no—
194C2×Dic3—32no—
195C2×Dic3—92no—
196D12—62no—
197D12—62no—
198D12—62no—
199D12—62no—
200D12—64no—
201D12—62no—
202D12—62no—
203D12—122no—
204D12—182no—
205D12—182no—
206D12—368no—
207C3×C8—32no—
208C3×C8—32no—
209C3×C8—62no—
210C3×C8—92no—
211C3×C8#62612yes—
212C3⋊C8—32no—
213C3⋊C8—92no—
214C2×C32—22no—
215C2×C32—21no—
216C2×C32#87411yescharacteristic
217C2×C32C2×D2412yes—
218C2×C32D3×D812yes—
219C2×C32#94011yescharacteristic
220C3⋊D3—82no—
221C3×D3—244no—
222C3×D3—242no—
223C3×D3—242no—
224C3×D3—242no—
225C2×D4—92no—
226C2×D4—92no—
227C2×D4—544no—
228D8—182no—
229D8—182no—
230D8—544no—
231C2×C8—91no—
232C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
233C42—91no—
234C22×C3—31no—
235C22×C3—31no—
236C22×C3—61no—
237C22×C3—364no—
238C22×C3—362no—
239C22×C3—364no—
240C22×C3—362no—
241C22×C3D3×D1211yescharacteristic
242C2×D3—122no—
243C2×D3—122no—
244C2×D3—124no—
245C2×D3—124no—
246C2×D3—124no—
247C2×D3—124no—
248C2×D3—122no—
249C2×D3—244no—
250C2×D3—362no—
251C2×D3—364no—
252C3×C4—31no—
253C3×C4—31no—
254C3×C4—31no—
255C3×C4—31no—
256C3×C4—61no—
257C3×C4—61no—
258C3×C4—94no—
259C3×C4D3×D1211yescharacteristic
260C3×C4C2×#62611yescharacteristic
261Dic3—34no—
262Dic3—94no—
263C32—21no—
264C32#469511yescharacteristic, complemented
265C32#515111yescharacteristic, complemented
266C23—182no—
267C23—182no—
268D4—182no—
269D4—182no—
270D4—182no—
271D4—182no—
272D4—1088no—
273C2×C4—92no—
274C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
275C8—92no—
276C8#620012yescomplemented
277C2×C3—32no—
278C2×C3—32no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—62no—
282C2×C3—61no—
283C2×C3—722no—
284C2×C3—722no—
285C2×C3D3×D2412yes—
286C2×C3#6391111yescharacteristic
287D3—242no—
288D3—242no—
289D3—244no—
290D3—242no—
291D3—722no—
292C22—364no—
293C22—364no—
294C22—362no—
295C22—362no—
296C22#8865411yescharacteristic
297C4—94no—
298C4#8865411yescharacteristic
299C4C2×#620011yescharacteristic
300C3—31no—
301C3—31no—
302C3—61no—
303C3#1059088011yescharacteristic
304C2—722no—
305C2—722no—
306C2#1172084712yes—
307C2#1172115811yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#493
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88529C2semidirect
2. C2×#3745⋊C3C2semidirect
3. #11721158C2semidirect
4. C2×#88436C2semidirect
5. C8×(C3⋊Dic3)⋊C3C2semidirect
17. C2×#3745D3semidirect
26. C8×C3⋊Dic3D3semidirect
37. C8×H(3,𝔽3)D4semidirect
38. (C3⋊(C3⋊C8))⋊C3D4semidirect
70. (C3⋊Dic3)⋊C3D8semidirect
110. C32×C8D12semidirect
111. C32×C8#67semidirect
112. C3⋊(C3⋊C8)D12semidirect
168. C3⋊Dic3D24semidirect
175. H(3,𝔽3)#493semidirect
264. C32#4695semidirect
265. C32#5151semidirect
276. C8#6200semidirect

Maximal subgroups

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

subgroupindex
C2×#885292
C2×#3745⋊C32
#117211582
C2×#884362
C8×(C3⋊Dic3)⋊C32
#105908863
#105908803