Finite Group Databasebeta

← #13137376

Subgroups — #13137376

Order 1728.

507 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 121 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))⋊C3C211yesmaximal, characteristic, complemented
5#11720317C212yesmaximal, complemented
6#10590329—31nomaximal
7#10590327—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic
9C2×C4×(C3⋊D3)⋊C3C2212yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11C2×#6180C2212yes—
12#3752⋊C3C412yes—
13C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
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)—61no—
25C3×C4×C3⋊C8—31no—
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—22no—
37(C3⋊(C3⋊C8))⋊C3—21no—
38(C3⋊(C3⋊C8))⋊C3D412yescomplemented
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×D311yescharacteristic
52C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
53C3×(C2×D3)⋊C4—32no—
54C3×C4⋊Dic3—31no—
55C2×C3×C3⋊C8—31no—
56C2×C3×C3⋊C8—32no—
57C2×C3×C3⋊C8—31no—
58C2×C3×C3⋊C8—32no—
59C2×C3×C3⋊C8—62no—
60#3710—64no—
61D3×C3⋊C8—64no—
62C22×H(3,𝔽3)C22×C411yescharacteristic
63C2×(C3⋊D3)⋊C3—22no—
64C2×(C3⋊D3)⋊C3—22no—
65C2×(C3⋊D3)⋊C3C2×C814yescomplemented
66C4×H(3,𝔽3)—21no—
67C4×H(3,𝔽3)C2×C812yescomplemented
68C4×H(3,𝔽3)C2×D411yescharacteristic
69C4×H(3,𝔽3)#4111yescharacteristic
70(C3⋊Dic3)⋊C3—22no—
71C3×C4×D4—91no—
72C22×C3⋊C8—92no—
73C2×D3×C8—92no—
74C2×D3×C8—92no—
75C4×D12—31no—
76C4×D12—91no—
77C3×C4⋊C8—91no—
78C3×C4×C8—91no—
79#830—92no—
80(C2×D3)⋊C8—92no—
81(C2×D3)⋊C8—92no—
82C4⋊(C3⋊C8)—31no—
83C4⋊(C3⋊C8)—31no—
84C4⋊(C3⋊C8)—61no—
85C4×C3⋊C8—31no—
86C4×C3⋊C8—91no—
87C22×C3⋊D3C2×Dic312yes—
88C22×C3×D3—32no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4C4×D311yescharacteristic
93C2×C32×C4C4×D311yescharacteristic
94C2×C32×C4C2×Dic311yescharacteristic
95C2×C32×C4C2×Dic311yescharacteristic
96C2×C32×C4C22×D311yescharacteristic
97C2×C32×C4C22×D311yescharacteristic
98C2×C3⋊Dic3C2×Dic312yes—
99C4⋊(C3⋊D3)C3⋊C814yescomplemented
100C4×C3⋊D3—22no—
101C4×C3⋊D3—22no—
102C2×C3×Dic3—32no—
103C3×D12—34no—
104C3×C4×D3—62no—
105C3×C4×D3—62no—
106C3⋊(C3⋊C8)—21no—
107C3⋊(C3⋊C8)D1212yescomplemented
108C3×C3⋊C8—32no—
109C3×C3⋊C8—62no—
110C3×C3⋊C8—62no—
111C3×C3⋊C8—61no—
112C3×C3⋊C8—122no—
113C8×D4—271noSylow
114C2×H(3,𝔽3)C4×D411yescharacteristic
115C2×H(3,𝔽3)C22×C811yescharacteristic
116C2×H(3,𝔽3)#13111yescharacteristic
117(C3⋊D3)⋊C3—24no—
118C2×C3×D4—91no—
119C22×C3×C4—92no—
120C2×D12—31no—
121C2×D12—91no—
122C2×C4×D3—32no—
123C2×C4×D3—92no—
124C2×C3×C8—91no—
125C2×C3×C8—91no—
126C2×C3×C8—92no—
127C3×C4⋊C4—91no—
128C3×C22⋊C4—92no—
129C3×C42—31no—
130C3×C42—31no—
131C3×C42—61no—
132C3×C42D3211yescharacteristic
133(C2×D3)⋊C4—32no—
134(C2×D3)⋊C4—92no—
135C4⋊Dic3—31no—
136C4⋊Dic3—91no—
137C2×C3⋊C8—31no—
138C2×C3⋊C8—32no—
139C2×C3⋊C8—32no—
140C2×C3⋊C8—31no—
141C2×C3⋊C8—62no—
142C2×C3⋊C8—91no—
143C2×C3⋊C8—91no—
144C2×C3⋊C8—184no—
145D3×C8—184no—
146D3×C8—184no—
147C22×C32—21no—
148C22×C32C2×C4×D311yescharacteristic
149C22×C32C22×Dic311yescharacteristic
150C2×C3⋊D3—22no—
151C2×C3⋊D3—22no—
152C2×C3⋊D3C2×C3⋊C814yescomplemented
153C2×C3×D3—34no—
154C2×C3×D3—62no—
155C2×C3×D3—62no—
156C32×C4—21no—
157C32×C4—21no—
158C32×C4—22no—
159C32×C4—21no—
160C32×C4—21no—
161C32×C4—41no—
162C32×C4D3×C812yescomplemented
163C32×C4C2×C3⋊C812yescomplemented
164C32×C4C2×D1211yescharacteristic
165C32×C4#23511yescharacteristic
166C32×C4D3×D411yescharacteristic
167C32×C4#23711yescharacteristic
168C3⋊Dic3—22no—
169C3×Dic3—62no—
170C22×C8—272no—
171C4×D4—91no—
172C4⋊C8—91no—
173C22⋊C8—272no—
174C4×C8—91no—
175H(3,𝔽3)C8×D411yescharacteristic, complemented, Sylow
176C23×C3—92no—
177C22×D3—32no—
178C22×D3—92no—
179C3×D4—94no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—61no—
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—92no—
190C2×C3×C4—182no—
191C2×C3×C4—182no—
192C2×C3×C4D3×Dic311yescharacteristic
193C2×C3×C4D3×Dic311yescharacteristic
194C2×C3×C4C2×D3211yescharacteristic
195C2×Dic3—32no—
196C2×Dic3—92no—
197D12—34no—
198D12—94no—
199C4×D3—62no—
200C4×D3—62no—
201C4×D3—182no—
202C4×D3—182no—
203C3×C8—92no—
204C3×C8—181no—
205C3×C8—182no—
206C3⋊C8—32no—
207C3⋊C8—62no—
208C3⋊C8—62no—
209C3⋊C8—61no—
210C3⋊C8—92no—
211C3⋊C8—122no—
212C3⋊C8—181no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32C4×D1211yescharacteristic
217C2×C32C2×D3×C811yescharacteristic
218C2×C32#90111yescharacteristic
219C2×C32C22×C3⋊C811yescharacteristic
220C2×C32D4×Dic311yescharacteristic
221C2×C32#94811yescharacteristic
222C3⋊D3—24no—
223C3×D3—64no—
224C2×D4—91no—
225C22×C4—92no—
226C2×C8—91no—
227C2×C8—91no—
228C2×C8—92no—
229C2×C8—544no—
230C4⋊C4—91no—
231C22⋊C4—92no—
232C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
233C22×C3—31no—
234C22×C3—31no—
235C22×C3—61no—
236C22×C3—94no—
237C22×C3—182no—
238C22×C3—182no—
239C22×C3C2×D3×Dic311yescharacteristic
240C2×D3—34no—
241C2×D3—62no—
242C2×D3—62no—
243C2×D3—94no—
244C2×D3—182no—
245C2×D3—182no—
246C3×C4—21no—
247C3×C4—31no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—32no—
252C3×C4—32no—
253C3×C4—61no—
254C3×C4—61no—
255C3×C4—61no—
256C3×C4—61no—
257C3×C4—62no—
258C3×C4—121no—
259C3×C4—182no—
260C3×C4D3×C3⋊C812yes—
261C3×C4#379511yescharacteristic
262C3×C4D3×D1211yescharacteristic
263Dic3—62no—
264Dic3—182no—
265C32—21no—
266C32C8×D1211yescharacteristic, complemented
267C32D4×C3⋊C811yescharacteristic, complemented
268C23—92no—
269D4—94no—
270C2×C4—92no—
271C2×C4—182no—
272C2×C4—182no—
273C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
274C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
275C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
276C8—92no—
277C8—181no—
278C8—182no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—31no—
283C2×C3—31no—
284C2×C3—31no—
285C2×C3—61no—
286C2×C3—61no—
287C2×C3—61no—
288C2×C3—184no—
289C2×C3C2×D3×C3⋊C811yescharacteristic
290C2×C3#6381411yescharacteristic
291C2×C3Dic3×D1211yescharacteristic
292D3—64no—
293D3—184no—
294C22—94no—
295C22—182no—
296C22—182no—
297C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
298C4—21no—
299C4—182no—
300C4((C3⋊D3)⋊C3)⋊C812yescomplemented
301C4#8864811yescharacteristic
302C4#8865411yescharacteristic
303C3—31no—
304C3—31no—
305C3—61no—
306C3C3⋊C8×D1211yescharacteristic
307C2—184no—
308C2C2×((C3⋊D3)⋊C3)⋊C811yescharacteristic
309C2#1172105411yescharacteristic
310C2#1172113211yescharacteristic
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))⋊C3C2semidirect
5. #11720317C2semidirect
21. C4×C3⋊(C3⋊C8)D3semidirect
33. C4⋊((C3⋊D3)⋊C3)C8semidirect
38. (C3⋊(C3⋊C8))⋊C3D4semidirect
65. C2×(C3⋊D3)⋊C3C2×C8semidirect
67. C4×H(3,𝔽3)C2×C8semidirect
99. C4⋊(C3⋊D3)C3⋊C8semidirect
107. C3⋊(C3⋊C8)D12semidirect
152. C2×C3⋊D3C2×C3⋊C8semidirect
162. C32×C4D3×C8semidirect
163. C32×C4C2×C3⋊C8semidirect
175. H(3,𝔽3)C8×D4semidirect
266. C32C8×D12semidirect
267. C32D4×C3⋊C8semidirect
300. 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))⋊C32
#117203172
#105903293
#105903273