Finite Group Databasebeta

← #13147050

Subgroups — #13147050

Order 1728.

448 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 43 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
1D4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C4⋊#6258C211yesmaximal, characteristic, complemented
3#11720736C211yesmaximal, characteristic, complemented
4C3×#5415—33nomaximal
5C3×#5217—31nomaximal
6C22×C22⋊H(3,𝔽3)—21no—
7C2×#6258—21no—
8C4×C22⋊H(3,𝔽3)C2211yescharacteristic
9#88545—41nomaximal
10C22×C32×D4D311yescharacteristic, complemented
11C3×D4×A4D313yescomplemented
12C3×C4⋊S4—33no—
13C2×C3×#213—31no—
14C2×C3×#213—31no—
15C3×#930—31no—
16C3×A4⋊C8—33no—
17C3×#832—31no—
18C3×#832—31no—
19C3×#830—31no—
20C2×C22⋊H(3,𝔽3)—41no—
21C2×C22⋊H(3,𝔽3)D411yescharacteristic
22#6258—41no—
23D4×H(3,𝔽3)—41no—
24C4⋊#272—41no—
25#6180—41no—
26#5415—33no—
27C3×#447—91no—
28#5217—31no—
29C24×C32—21no—
30C22×C3×A4—23no—
31C2×C3×S4—63no—
32C2×C32×D4—31no—
33C2×C32×D4—31no—
34C2×C32×D4—61no—
35C22×C32×C4C2×D311yescharacteristic
36C2×C3×#67—61no—
37C2×C3×D12—31no—
38C3×C4×A4C2×D313yes—
39C3×#213—62no—
40C3×#213—61no—
41C3×#213—61no—
42C3×#213—123no—
43C3×(C2×D3)⋊C4—61no—
44C3×C4⋊Dic3—31no—
45C2×C3×C3⋊C8—31no—
46C2×C3×C3⋊C8—31no—
47C22×H(3,𝔽3)—81no—
48C2×#272—81no—
49C22⋊H(3,𝔽3)D811yescharacteristic, complemented
50C4×H(3,𝔽3)—41no—
51C22×C3×D4—31no—
52C22×C3×D4C3⋊D311yescharacteristic
53D4×A4—33no—
54C4⋊S4—33no—
55C2×C3×D8—91no—
56C2×C3×D8—91no—
57C3×#121—91no—
58C2×#213—31no—
59C2×#213—31no—
60#930—31no—
61A4⋊C8—33no—
62C3×#102—91no—
63C3×#102—91no—
64C3×C22⋊C8—91no—
65#832—31no—
66#832—31no—
67#830—31no—
68C23×C32—41no—
69C23×C32—61no—
70C23×C32—61no—
71C23×C32—121no—
72C23×C32#6711yescharacteristic
73C22×C3×D3—61no—
74C2×C3×A4—43no—
75C2×C3×A4#6713yes—
76C3×S4—123no—
77C32×D4—32no—
78C32×D4—31no—
79C32×D4—43no—
80C32×D4—61no—
81C32×D4S411yescharacteristic, complemented
82C2×C32×C4—31no—
83C2×C32×C4—31no—
84C3×#67—121no—
85C3×#67—121no—
86C2×C3×Dic3—61no—
87C3×D12—61no—
88C3×D12—61no—
89C3×D12—123no—
90C3×C3⋊C8—61no—
91C3×C3⋊C8—61no—
92C3×C3⋊C8—123no—
93#447—91noSylow
94C2×H(3,𝔽3)—41no—
95C2×H(3,𝔽3)—161no—
96#272—161no—
97C24×C3—21no—
98C24×C3—61no—
99C22×A4—63no—
100C2×S4—63no—
101C2×C3×D4—31no—
102C2×C3×D4—31no—
103C2×C3×D4—31no—
104C2×C3×D4—31no—
105C2×C3×D4—61no—
106C2×C3×D4—61no—
107C2×C3×D4—61no—
108C2×C3×D4—61no—
109C2×C3×D4—62no—
110C2×C3×D4—91no—
111C2×C3×D4—181no—
112C22×C3×C4—31no—
113C22×C3×C4C2×C3⋊D311yescharacteristic
114C2×#67—61no—
115C2×D12—31no—
116C4×A4—33no—
117C3×D8—182no—
118C3×D8—181no—
119C3×D8—181no—
120C2×C3×C8—91no—
121C2×C3×C8—91no—
122C3×C4⋊C4—91no—
123C3×C22⋊C4—181no—
124#213—62no—
125#213—61no—
126#213—61no—
127#213—123no—
128(C2×D3)⋊C4—61no—
129C4⋊Dic3—31no—
130C2×C3⋊C8—31no—
131C2×C3⋊C8—31no—
132C22×C32—21no—
133C22×C32—31no—
134C22×C32—31no—
135C22×C32—61no—
136C22×C32—83no—
137C22×C32—121no—
138C22×C32—121no—
139C22×C32—121no—
140C22×C32—121no—
141C22×C32#21311yescharacteristic, complemented
142C2×C3×D3—121no—
143C2×C3×D3—121no—
144C2×C3×D3—121no—
145C2×C3×D3—243no—
146C3×A4#21313yescomplemented
147C32×C4—31no—
148C32×C4—43no—
149C32×C4C2×S411yescharacteristic
150C3×Dic3—121no—
151C22×D4#27211yescharacteristic, complemented
152C2×D8—91no—
153C2×D8—91no—
154#121—91no—
155#102—91no—
156#102—91no—
157C22⋊C8—91no—
158H(3,𝔽3)—41noSylow
159C23×C3—31no—
160C23×C3—41no—
161C23×C3—61no—
162C23×C3—61no—
163C23×C3—61no—
164C23×C3—61no—
165C23×C3—62no—
166C23×C3—62no—
167C23×C3—121no—
168C23×C3—121no—
169C23×C3—121no—
170C23×C3—122no—
171C23×C3—181no—
172C23×C3#63811yescharacteristic
173C22×D3—61no—
174C2×A4—33no—
175C2×A4—123no—
176S4—123no—
177C3×D4—32no—
178C3×D4—31no—
179C3×D4—32no—
180C3×D4—31no—
181C3×D4—31no—
182C3×D4—61no—
183C3×D4—62no—
184C3×D4—61no—
185C3×D4—61no—
186C3×D4—62no—
187C3×D4—123no—
188C3×D4—181no—
189C3×D4—181no—
190C3×D4—361no—
191C3×D4—361no—
192C3×D4C3⋊S411yescharacteristic
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—181no—
200#67—121no—
201#67—121no—
202C2×Dic3—61no—
203D12—61no—
204D12—61no—
205D12—123no—
206C3×C8—181no—
207C3×C8—181no—
208C3⋊C8—61no—
209C3⋊C8—61no—
210C3⋊C8—123no—
211C2×C32—31no—
212C2×C32—31no—
213C2×C32—41no—
214C2×C32—43no—
215C2×C32—121no—
216C2×C32—163no—
217C2×C32GL(2,ℤ/4)11yescharacteristic
218C3×D3—241no—
219C3×D3—483no—
220C24—21no—
221C2×D4—31no—
222C2×D4—31no—
223C2×D4—61no—
224C2×D4—91no—
225C2×D4—181no—
226C22×C4C2×#27211yescharacteristic
227D8—182no—
228D8—181no—
229D8—181no—
230C2×C8—91no—
231C2×C8—91no—
232C4⋊C4—91no—
233C22⋊C4—181no—
234C22×C3—21no—
235C22×C3—31no—
236C22×C3—31no—
237C22×C3—31no—
238C22×C3—31no—
239C22×C3—31no—
240C22×C3—61no—
241C22×C3—61no—
242C22×C3—61no—
243C22×C3—62no—
244C22×C3—61no—
245C22×C3—61no—
246C22×C3—121no—
247C22×C3—121no—
248C22×C3—121no—
249C22×C3—121no—
250C22×C3—121no—
251C22×C3—121no—
252C22×C3—121no—
253C22×C3—121no—
254C22×C3—122no—
255C22×C3—122no—
256C22×C3—122no—
257C22×C3—122no—
258C22×C3—243no—
259C22×C3—361no—
260C22×C3—361no—
261C22×C3—361no—
262C22×C3#375311yescharacteristic
263C2×D3—121no—
264C2×D3—121no—
265C2×D3—121no—
266C2×D3—243no—
267A4—33no—
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—61no—
272C3×C4—123no—
273C3×C4—361no—
274C3×C4C2×C3⋊S411yescharacteristic
275Dic3—121no—
276C32—43no—
277C32#541511yescharacteristic, complemented
278C23—41no—
279C23—61no—
280C23—61no—
281C23—121no—
282C23—181no—
283C23#623511yescharacteristic
284D4—32no—
285D4—31no—
286D4—61no—
287D4—181no—
288D4—181no—
289D4—361no—
290D4—361no—
291D4#625811yescharacteristic, complemented
292C2×C4—31no—
293C2×C4—31no—
294C2×C4—181no—
295C8—181no—
296C8—181no—
297C2×C3—31no—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—31no—
302C2×C3—41no—
303C2×C3—61no—
304C2×C3—61no—
305C2×C3—121no—
306C2×C3—122no—
307C2×C3—121no—
308C2×C3—121no—
309C2×C3—123no—
310C2×C3—483no—
311C2×C3—721no—
312C2×C3#6426911yescharacteristic
313D3—241no—
314D3—483no—
315C22—21no—
316C22—31no—
317C22—31no—
318C22—61no—
319C22—121no—
320C22—121no—
321C22—121no—
322C22—121no—
323C22—361no—
324C22—361no—
325C22—361no—
326C22#8854511yescharacteristic, complemented
327C4—31no—
328C4—361no—
329C4C2×#625811yescharacteristic
330C3—31no—
331C3—123no—
332C3#1059364911yescharacteristic
333C2—31no—
334C2—31no—
335C2—41no—
336C2—121no—
337C2—721no—
338C2#1172239811yescharacteristic
subgroupG/H
centerC2×C3#64269
commutator, cosocleC4×C22⋊H(3,𝔽3)C22
FrattiniC3×C4C2×C3⋊S4
FittingC22×C32×D4D3
socleC23×C3#638
hypercenterC3×D4C3⋊S4
hypocenterC22⋊H(3,𝔽3)D8
LCSG ▷ C4×C22⋊H(3,𝔽3) ▷ C2×C22⋊H(3,𝔽3) ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. D4×C22⋊H(3,𝔽3)C2semidirect
2. C4⋊#6258C2semidirect
3. #11720736C2semidirect
10. C22×C32×D4D3semidirect
11. C3×D4×A4D3semidirect
49. C22⋊H(3,𝔽3)D8semidirect
81. C32×D4S4semidirect
141. C22×C32#213semidirect
146. C3×A4#213semidirect
151. C22×D4#272semidirect
277. C32#5415semidirect
291. D4#6258semidirect
326. C22#88545semidirect

Maximal subgroups

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

subgroupindex
D4×C22⋊H(3,𝔽3)2
C4⋊#62582
#117207362
C3×#54153
C3×#52173
#885454