Finite Group Databasebeta

← #13138853

Subgroups — #13138853

Order 1728.

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

normal subgroups only

This group has 491 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
1C2×#3745⋊C3C211yesmaximal, characteristic, complemented
2C2×#3743⋊C3C211yesmaximal, characteristic, complemented
3#11721141C212yesmaximal, complemented
4#11720550C211yesmaximal, characteristic, complemented
5#11720308C212yesmaximal, complemented
6#10590846—31nomaximal
7#10590835—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
10C2×#6198—22no—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4⋊#1084C2212yes—
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14#3745⋊C3—22no—
15#3743⋊C3—22no—
16#88450—22no—
17C2×#3745D311yescharacteristic, complemented
18C2×#3743D311yescharacteristic, complemented
19C2×C3×D24—31no—
20C2×C3×ΓL(1,𝔽25)—31no—
21#63916—32no—
22#63909—32no—
23C3×#817—31no—
24C3×#817—31no—
25C3×#817—61no—
26#63564—32no—
27#63562—32no—
28C22×(C3⋊D3)⋊C3—22no—
29C22×(C3⋊D3)⋊C3D411yescharacteristic
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×#1084—22no—
32C2×(C3⋊Dic3)⋊C3D411yescharacteristic
33C4⋊((C3⋊D3)⋊C3)—22no—
34C4⋊((C3⋊D3)⋊C3)—22no—
35C4×(C3⋊D3)⋊C3—21no—
36C4×(C3⋊D3)⋊C3—21no—
37#6198—44no—
38C8×H(3,𝔽3)D412yescomplemented
39(C3⋊(C3⋊C8))⋊C3—21no—
40#5159—91no—
41#5135—91no—
42#4865—91no—
43C2×C4⋊(C3⋊D3)C2×D312yes—
44C2×C4×C3⋊D3C2×D311yescharacteristic
45C2×C3×D12—32no—
46C2×C3×C4×D3—31no—
47C2×#626—62no—
48C2×#626—62no—
49C2×C32×C8—21no—
50C2×C32×C8C2×D311yescharacteristic
51C2×C32×C8C2×D311yescharacteristic
52C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
53#3745—22no—
54#3743—22no—
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—62no—
58C2×C3×C3⋊C8—31no—
59C3×D24—62no—
60C3×ΓL(1,𝔽25)—62no—
61#3722—62no—
62(C2×D3)⋊Dic3—62no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×(C3⋊D3)⋊C3—21no—
65C2×(C3⋊D3)⋊C3—22no—
66C2×(C3⋊D3)⋊C3—42no—
67C2×(C3⋊D3)⋊C3—42no—
68C2×(C3⋊D3)⋊C3—42no—
69C4×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)#4111yescharacteristic
71#1084—42no—
72(C3⋊Dic3)⋊C3—21no—
73C2×C3×D8—91no—
74C2×C3×M16—91no—
75#938—92no—
76#930—92no—
77C2×D24—31no—
78C2×D24—91no—
79C2×ΓL(1,𝔽25)—31no—
80C2×ΓL(1,𝔽25)—91no—
81#895—92no—
82C3×#106—91no—
83#832—92no—
84#821—92no—
85#817—31no—
86#817—31no—
87#817—61no—
88#809—92no—
89C22×C3⋊D3—22no—
90C22×C3⋊D3D1211yescharacteristic
91C22×C3×D3—31no—
92C22×C3×D3—62no—
93C2×C32×C4—21no—
94C2×C32×C4C22×D311yescharacteristic
95C2×C32×C4C22×D311yescharacteristic
96C2×C3⋊Dic3D1211yescharacteristic
97C4⋊(C3⋊D3)—22no—
98C4⋊(C3⋊D3)—22no—
99C4×C3⋊D3—21no—
100C4×C3⋊D3—21no—
101C2×C3×Dic3—31no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—122no—
105C3×D12—62no—
106C3×D12—62no—
107C3×C4×D3—61no—
108C3×C4×D3—61no—
109#626—124no—
110#626—124no—
111C32×C8—22no—
112C32×C8#6712yescomplemented
113C32×C8#6712yescomplemented
114C3⋊(C3⋊C8)—21no—
115C3×C3⋊C8—61no—
116#469—271noSylow
117C2×H(3,𝔽3)#12111yescharacteristic
118C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
119(C3⋊D3)⋊C3—41no—
120(C3⋊D3)⋊C3—82no—
121C2×C3×D4—92no—
122C22×C3×C4—91no—
123C2×#67—182no—
124C2×#67—182no—
125C2×D12—32no—
126C2×D12—92no—
127C2×D12—182no—
128C2×C4×D3—31no—
129C2×C4×D3—91no—
130C3×D8—182no—
131C3×M16—182no—
132C2×C3×C8—31no—
133C2×C3×C8—31no—
134C2×C3×C8—61no—
135C2×C3×C8—91no—
136C2×C3×C8D3211yescharacteristic
137C3×C4⋊C4—92no—
138#217—182no—
139(C2×D3)⋊C4—182no—
140(C2×D3)⋊C4—182no—
141C4⋊Dic3—32no—
142C4⋊Dic3—32no—
143C4⋊Dic3—62no—
144C2×C3⋊C8—31no—
145C2×C3⋊C8—91no—
146D24—62no—
147D24—182no—
148ΓL(1,𝔽25)—62no—
149ΓL(1,𝔽25)—182no—
150C22×C32—21no—
151C22×C32C2×D1211yescharacteristic
152C22×C32D3×D411yescharacteristic
153C2×C3⋊D3—22no—
154C2×C3⋊D3—21no—
155C2×C3⋊D3—42no—
156C2×C3⋊D3—42no—
157C2×C3⋊D3—42no—
158C2×C3×D3—62no—
159C2×C3×D3—61no—
160C2×C3×D3—122no—
161C2×C3×D3—122no—
162C2×C3×D3—122no—
163C32×C4—21no—
164C32×C4—21no—
165C32×C4#23511yescharacteristic
166C32×C4#23711yescharacteristic
167C32×C4C2×#6711yescharacteristic
168C32×C4C2×#6711yescharacteristic
169C3⋊Dic3—21no—
170C3×Dic3—61no—
171C3×Dic3—122no—
172C3×Dic3—122no—
173C3×Dic3—242no—
174C2×D8—91no—
175C2×M16—91no—
176#121—272no—
177#106—91no—
178#102—272no—
179H(3,𝔽3)#46911yescharacteristic, complemented, Sylow
180C23×C3—91no—
181C23×C3—182no—
182C22×D3—31no—
183C22×D3—62no—
184C22×D3—91no—
185C22×D3—182no—
186C3×D4—182no—
187C3×D4—182no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—61no—
191C2×C3×C4—91no—
192C2×C3×C4—181no—
193C2×C3×C4—181no—
194C2×C3×C4—182no—
195C2×C3×C4C2×D3211yescharacteristic
196#67—364no—
197#67—364no—
198C2×Dic3—31no—
199C2×Dic3—62no—
200C2×Dic3—62no—
201C2×Dic3—91no—
202C2×Dic3—122no—
203D12—62no—
204D12—62no—
205D12—182no—
206D12—182no—
207D12—364no—
208C4×D3—61no—
209C4×D3—61no—
210C4×D3—181no—
211C4×D3—181no—
212C3×C8—32no—
213C3×C8—32no—
214C3×C8—62no—
215C3×C8—181no—
216C3×C8#62512yes—
217C3⋊C8—61no—
218C3⋊C8—181no—
219C2×C32—22no—
220C2×C32—21no—
221C2×C32#90812yes—
222C2×C32#91112yes—
223C2×C32#93011yescharacteristic
224C2×C32#93811yescharacteristic
225C3⋊D3—41no—
226C3⋊D3—82no—
227C3×D3—121no—
228C3×D3—242no—
229C2×D4—92no—
230C2×D4—542no—
231C22×C4—91no—
232D8—182no—
233M16—182no—
234C2×C8—91no—
235C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
236C4⋊C4—92no—
237C22⋊C4—542no—
238C22×C3—31no—
239C22×C3—31no—
240C22×C3—61no—
241C22×C3—182no—
242C22×C3—181no—
243C22×C3—362no—
244C22×C3—362no—
245C22×C3—362no—
246C22×C3D3×D1211yescharacteristic
247C2×D3—62no—
248C2×D3—61no—
249C2×D3—122no—
250C2×D3—122no—
251C2×D3—122no—
252C2×D3—181no—
253C2×D3—182no—
254C2×D3—362no—
255C2×D3—362no—
256C2×D3—362no—
257C3×C4—31no—
258C3×C4—31no—
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—61no—
262C3×C4—61no—
263C3×C4—181no—
264C3×C4—362no—
265C3×C4#379511yescharacteristic
266C3×C4C2×#62511yescharacteristic
267Dic3—61no—
268Dic3—122no—
269Dic3—122no—
270Dic3—181no—
271Dic3—242no—
272C32—21no—
273C32#513511yescharacteristic, complemented
274C32#515911yescharacteristic, complemented
275C23—91no—
276C23—182no—
277D4—182no—
278D4—182no—
279D4—1084no—
280C2×C4—91no—
281C2×C4—181no—
282C2×C4—181no—
283C2×C4—182no—
284C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
285C8—181no—
286C8#619812yescomplemented
287C2×C3—32no—
288C2×C3—32no—
289C2×C3—31no—
290C2×C3—31no—
291C2×C3—62no—
292C2×C3—61no—
293C2×C3—361no—
294C2×C3—722no—
295C2×C3#6379512yes—
296C2×C3#6390811yescharacteristic
297D3—121no—
298D3—242no—
299D3—361no—
300D3—722no—
301C22—182no—
302C22—181no—
303C22—362no—
304C22—362no—
305C22—362no—
306C22#8865411yescharacteristic
307C4—181no—
308C4—362no—
309C4#8864811yescharacteristic
310C4C2×#619811yescharacteristic
311C3—31no—
312C3—31no—
313C3—61no—
314C3#1059083611yescharacteristic
315C2—361no—
316C2—722no—
317C2#1172084912yes—
318C2#1172114111yescharacteristic
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)#469
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×#3745⋊C3C2semidirect
2. C2×#3743⋊C3C2semidirect
3. #11721141C2semidirect
4. #11720550C2semidirect
5. #11720308C2semidirect
17. C2×#3745D3semidirect
18. C2×#3743D3semidirect
38. C8×H(3,𝔽3)D4semidirect
112. C32×C8#67semidirect
113. C32×C8#67semidirect
179. H(3,𝔽3)#469semidirect
273. C32#5135semidirect
274. C32#5159semidirect
286. C8#6198semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
C2×#3743⋊C32
#117211412
#117205502
#117203082
#105908463
#105908353