Finite Group Databasebeta

← #13137372

Subgroups — #13137372

Order 1728.

513 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 513 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C2×((C3⋊D3)⋊C3)⋊C8C212yesmaximal, complemented
3C4×#6180C211yesmaximal, characteristic, complemented
4C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
5#11720317C212yesmaximal, complemented
6#10590327—31nomaximal
7C3⋊C8×D12—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic
9C2×C4×(C3⋊D3)⋊C3C2212yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13#3752⋊C3C412yes—
14C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
16((C3⋊D3)⋊C3)⋊C8—24no—
17C4×C4⋊(C3⋊D3)D311yescharacteristic
18C3×C4×D12—31no—
19C2×#3710—32no—
20C2×D3×C3⋊C8—32no—
21C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
22C3×C4⋊(C3⋊C8)—31no—
23C3×C4×C3⋊C8—31no—
24C3×C4×C3⋊C8—31no—
25C3×C4×C3⋊C8—61no—
26#63557—32no—
27(C2×D3)⋊(C3⋊C8)—32no—
28C22×(C3⋊D3)⋊C3C2×C412yes—
29C2×C4×H(3,𝔽3)C2×C411yescharacteristic
30C2×C4×H(3,𝔽3)C2×C411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3C2×C412yes—
33C4⋊((C3⋊D3)⋊C3)C814yescomplemented
34C4×(C3⋊D3)⋊C3—22no—
35C4×(C3⋊D3)⋊C3—22no—
36#6180—21no—
37#6180D412yescomplemented
38(C3⋊(C3⋊C8))⋊C3—22no—
39D4×C3⋊C8—91no—
40#4834—91no—
41C8×D12—91no—
42C2×C4⋊(C3⋊D3)Dic311yescharacteristic
43C2×C4×C3⋊D3C2×D312yes—
44C2×C3×D12—31no—
45C2×C3×C4×D3—32no—
46C32×C42—21no—
47C32×C42C2×D311yescharacteristic
48C32×C42C2×D311yescharacteristic
49#3752Dic312yes—
50C4⋊(C3⋊Dic3)Dic311yescharacteristic
51C2×C3⋊(C3⋊C8)C2×D312yes—
52C3×(C2×D3)⋊C4—32no—
53C3×C4⋊Dic3—31no—
54C2×C3×C3⋊C8—31no—
55C2×C3×C3⋊C8—31no—
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8—31no—
58C2×C3×C3⋊C8—32no—
59C2×C3×C3⋊C8—61no—
60C2×C3×C3⋊C8—61no—
61#3710—64no—
62D3×C3⋊C8—64no—
63C22×H(3,𝔽3)C22×C411yescharacteristic
64C2×(C3⋊D3)⋊C3—22no—
65C2×(C3⋊D3)⋊C3—22no—
66C2×(C3⋊D3)⋊C3C2×C814yescomplemented
67C4×H(3,𝔽3)—21no—
68C4×H(3,𝔽3)C2×C812yescomplemented
69C4×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)#4111yescharacteristic
71(C3⋊Dic3)⋊C3—22no—
72C3×C4×D4—91no—
73C22×C3⋊C8—92no—
74C2×D3×C8—92no—
75C2×D3×C8—92no—
76C4×D12—31no—
77C4×D12—91no—
78C3×C4⋊C8—91no—
79C3×C4×C8—91no—
80#830—92no—
81(C2×D3)⋊C8—92no—
82(C2×D3)⋊C8—92no—
83C4⋊(C3⋊C8)—31no—
84C4⋊(C3⋊C8)—91no—
85C4×C3⋊C8—31no—
86C4×C3⋊C8—31no—
87C4×C3⋊C8—61no—
88C22×C3⋊D3C2×Dic312yes—
89C22×C3×D3—32no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4C4×D311yescharacteristic
94C2×C32×C4C4×D311yescharacteristic
95C2×C32×C4C2×Dic311yescharacteristic
96C2×C32×C4C2×Dic311yescharacteristic
97C2×C32×C4C22×D311yescharacteristic
98C2×C32×C4C22×D311yescharacteristic
99C2×C3⋊Dic3C2×Dic312yes—
100C4⋊(C3⋊D3)C3⋊C814yescomplemented
101C4×C3⋊D3—22no—
102C4×C3⋊D3—22no—
103C2×C3×Dic3—32no—
104C3×D12—34no—
105C3×C4×D3—62no—
106C3×C4×D3—62no—
107C3⋊(C3⋊C8)—22no—
108C3×C3⋊C8—32no—
109C3×C3⋊C8—32no—
110C3×C3⋊C8—62no—
111C3×C3⋊C8—62no—
112C3×C3⋊C8—62no—
113C3×C3⋊C8—61no—
114C3×C3⋊C8—61no—
115C8×D4—271noSylow
116C2×H(3,𝔽3)C4×D411yescharacteristic
117C2×H(3,𝔽3)C22×C811yescharacteristic
118C2×H(3,𝔽3)#13111yescharacteristic
119(C3⋊D3)⋊C3—24no—
120C2×C3×D4—91no—
121C22×C3×C4—92no—
122C2×D12—31no—
123C2×D12—91no—
124C2×C4×D3—32no—
125C2×C4×D3—92no—
126C2×C3×C8—91no—
127C2×C3×C8—92no—
128C2×C3×C8—91no—
129C3×C4⋊C4—91no—
130C3×C22⋊C4—92no—
131C3×C42—31no—
132C3×C42—31no—
133C3×C42—61no—
134C3×C42D3211yescharacteristic
135(C2×D3)⋊C4—32no—
136(C2×D3)⋊C4—92no—
137C4⋊Dic3—31no—
138C4⋊Dic3—91no—
139C2×C3⋊C8—31no—
140C2×C3⋊C8—31no—
141C2×C3⋊C8—31no—
142C2×C3⋊C8—31no—
143C2×C3⋊C8—32no—
144C2×C3⋊C8—61no—
145C2×C3⋊C8—61no—
146C2×C3⋊C8—92no—
147C2×C3⋊C8—184no—
148D3×C8—184no—
149D3×C8—184no—
150C22×C32—21no—
151C22×C32C2×C4×D311yescharacteristic
152C22×C32C22×Dic311yescharacteristic
153C2×C3⋊D3—22no—
154C2×C3⋊D3—22no—
155C2×C3⋊D3C2×C3⋊C814yescomplemented
156C2×C3×D3—34no—
157C2×C3×D3—62no—
158C2×C3×D3—62no—
159C32×C4—22no—
160C32×C4—21no—
161C32×C4—21no—
162C32×C4—22no—
163C32×C4—21no—
164C32×C4—21no—
165C32×C4D3×C812yescomplemented
166C32×C4C2×C3⋊C812yescomplemented
167C32×C4D3×D411yescharacteristic
168C32×C4D3×D411yescharacteristic
169C32×C4#23711yescharacteristic
170C32×C4#23911yescharacteristic
171C3⋊Dic3—22no—
172C3×Dic3—62no—
173C22×C8—272no—
174C4×D4—91no—
175C4⋊C8—91no—
176C22⋊C8—272no—
177C4×C8—91no—
178H(3,𝔽3)C8×D411yescharacteristic, complemented, Sylow
179C23×C3—92no—
180C22×D3—32no—
181C22×D3—92no—
182C3×D4—94no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—92no—
193C2×C3×C4—182no—
194C2×C3×C4—182no—
195C2×C3×C4D3×Dic311yescharacteristic
196C2×C3×C4D3×Dic311yescharacteristic
197C2×C3×C4C2×D3211yescharacteristic
198C2×Dic3—32no—
199C2×Dic3—92no—
200D12—34no—
201D12—94no—
202C4×D3—62no—
203C4×D3—62no—
204C4×D3—182no—
205C4×D3—182no—
206C3×C8—92no—
207C3×C8—182no—
208C3×C8—181no—
209C3⋊C8—32no—
210C3⋊C8—32no—
211C3⋊C8—62no—
212C3⋊C8—61no—
213C3⋊C8—61no—
214C3⋊C8—62no—
215C3⋊C8—62no—
216C3⋊C8—182no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32#89311yescharacteristic
221C2×C32C2×D3×C811yescharacteristic
222C2×C32#90711yescharacteristic
223C2×C32C22×C3⋊C811yescharacteristic
224C2×C32D4×Dic311yescharacteristic
225C2×C32#94811yescharacteristic
226C3⋊D3—24no—
227C3×D3—64no—
228C2×D4—91no—
229C22×C4—92no—
230C2×C8—91no—
231C2×C8—91no—
232C2×C8—92no—
233C2×C8—544no—
234C4⋊C4—91no—
235C22⋊C4—92no—
236C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
237C22×C3—31no—
238C22×C3—31no—
239C22×C3—61no—
240C22×C3—94no—
241C22×C3—182no—
242C22×C3—182no—
243C22×C3C2×D3×Dic311yescharacteristic
244C2×D3—34no—
245C2×D3—62no—
246C2×D3—62no—
247C2×D3—94no—
248C2×D3—182no—
249C2×D3—182no—
250C3×C4—21no—
251C3×C4—31no—
252C3×C4—31no—
253C3×C4—31no—
254C3×C4—31no—
255C3×C4—32no—
256C3×C4—32no—
257C3×C4—61no—
258C3×C4—61no—
259C3×C4—62no—
260C3×C4—61no—
261C3×C4—61no—
262C3×C4—62no—
263C3×C4—182no—
264C3×C4D3×C3⋊C812yes—
265C3×C4#379611yescharacteristic
266C3×C4#380211yescharacteristic
267Dic3—62no—
268Dic3—182no—
269C32—21no—
270C32#483411yescharacteristic, complemented
271C32D4×C3⋊C811yescharacteristic, complemented
272C23—92no—
273D4—94no—
274C2×C4—92no—
275C2×C4—182no—
276C2×C4—182no—
277C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
278C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
279C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
280C8—92no—
281C8—181no—
282C8—182no—
283C2×C3—31no—
284C2×C3—31no—
285C2×C3—31no—
286C2×C3—31no—
287C2×C3—31no—
288C2×C3—31no—
289C2×C3—61no—
290C2×C3—61no—
291C2×C3—61no—
292C2×C3—184no—
293C2×C3C2×D3×C3⋊C811yescharacteristic
294C2×C3#6381511yescharacteristic
295C2×C3#6389811yescharacteristic
296D3—64no—
297D3—184no—
298C22—94no—
299C22—182no—
300C22—182no—
301C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
302C4—21no—
303C4—182no—
304C4((C3⋊D3)⋊C3)⋊C812yescomplemented
305C4#8864911yescharacteristic
306C4#8865511yescharacteristic
307C3—31no—
308C3—31no—
309C3—61no—
310C3#1059032911yescharacteristic
311C2—184no—
312C2C2×((C3⋊D3)⋊C3)⋊C811yescharacteristic
313C2#1172105511yescharacteristic
314C2#1172113611yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C8
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C8×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
3. C4×#6180C2semidirect
4. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
5. #11720317C2semidirect
21. C4⋊(C3⋊(C3⋊C8))D3semidirect
33. C4⋊((C3⋊D3)⋊C3)C8semidirect
37. #6180D4semidirect
66. C2×(C3⋊D3)⋊C3C2×C8semidirect
68. C4×H(3,𝔽3)C2×C8semidirect
100. C4⋊(C3⋊D3)C3⋊C8semidirect
155. C2×C3⋊D3C2×C3⋊C8semidirect
165. C32×C4D3×C8semidirect
166. C32×C4C2×C3⋊C8semidirect
178. H(3,𝔽3)C8×D4semidirect
270. C32#4834semidirect
271. C32D4×C3⋊C8semidirect
304. C4((C3⋊D3)⋊C3)⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊((C3⋊D3)⋊C3)2
C2×((C3⋊D3)⋊C3)⋊C82
C4×#61802
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203172
#105903273
C3⋊C8×D123