Finite Group Databasebeta

← #10592711⋊C3 #13142740

Subgroups — #10592711⋊C3 #13142740

Order 1728.

521 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#3748⋊C3C212yesmaximal, complemented
3#63657⋊C3C212yesmaximal, complemented
4C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
5C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
6#10592711C311yesmaximal, characteristic, complemented
7C3×#5012—31nomaximal
8C2×D4×H(3,𝔽3)C411yescharacteristic
9C22×C4×H(3,𝔽3)C2212yes—
10C4⋊C4×H(3,𝔽3)C411yescharacteristic
11C22⋊C4×H(3,𝔽3)C412yes—
12C42×H(3,𝔽3)C2211yescharacteristic
13#3748⋊C3—24no—
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17C32×C4×D4—21no—
18C32×C4×D4D311yescharacteristic
19C32×C4×D4C2×C311yescharacteristic
20C2×#3748C2×C312yescomplemented
21C2×C3×#208—32no—
22#63657C2×C312yescomplemented
23C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
24C4×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
25C3×#830—32no—
26C3×C4⋊(C3⋊C8)—31no—
27C3×C4×C3⋊C8—31no—
28C23×H(3,𝔽3)C2×C412yes—
29D4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)C2×C412yes—
33C2×C4×H(3,𝔽3)C2×C411yescharacteristic
34C2×C4×H(3,𝔽3)C2×C411yescharacteristic
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36(C3⋊(C3⋊C8))⋊C3—22no—
37(C3⋊(C3⋊C8))⋊C3—21no—
38(C3⋊(C3⋊C8))⋊C3D412yescomplemented
39C3×#436—91no—
40#5012—31no—
41#5012—91no—
42C2×C32×D4—21no—
43C2×C32×D4Dic311yescharacteristic
44C2×C32×D4C3×C411yescharacteristic
45C22×C32×C4—22no—
46C22×C32×C4C2×D312yes—
47C22×C32×C4C22×C312yes—
48C32×C4⋊C4—21no—
49C32×C4⋊C4Dic311yescharacteristic
50C32×C4⋊C4C3×C411yescharacteristic
51C32×C22⋊C4—22no—
52C32×C22⋊C4Dic312yes—
53C32×C22⋊C4C3×C412yes—
54C32×C42—21no—
55C32×C42C2×D311yescharacteristic
56C32×C42C22×C311yescharacteristic
57#3748—24no—
58C2×C3⋊(C3⋊C8)C22×C312yes—
59C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
60C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
61C3×#208—64no—
62C2×C3×C3⋊C8—31no—
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—31no—
65C22×H(3,𝔽3)—22no—
66C22×H(3,𝔽3)—22no—
67C22×H(3,𝔽3)—22no—
68C22×H(3,𝔽3)C22×C411yescharacteristic
69C4×H(3,𝔽3)—22no—
70C4×H(3,𝔽3)—21no—
71C4×H(3,𝔽3)M1612yescomplemented
72C4×H(3,𝔽3)C2×D411yescharacteristic
73C4×H(3,𝔽3)#4111yescharacteristic
74C2×C3×M16—92no—
75C3×C4×D4—31no—
76C3×C4×D4—31no—
77C3×C4×D4—61no—
78C3×C4×D4C3×D311yescharacteristic
79C2×#208—32no—
80C2×#208—92no—
81C3×C4⋊C8—91no—
82C3×C22⋊C8—92no—
83C3×C4×C8—91no—
84#830—32no—
85#830—92no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—91no—
88C4×C3⋊C8—31no—
89C4×C3⋊C8—91no—
90C23×C32—22no—
91C23×C32C2×Dic312yes—
92C23×C32C2×C3×C412yes—
93C32×D4—24no—
94C32×D4—22no—
95C32×D4—22no—
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—22no—
103C2×C32×C4—21no—
104C2×C32×C4—42no—
105C2×C32×C4—42no—
106C2×C32×C4C2×Dic311yescharacteristic
107C2×C32×C4C2×Dic312yes—
108C2×C32×C4C2×Dic311yescharacteristic
109C2×C32×C4C2×C3×C411yescharacteristic
110C2×C32×C4C2×C3×C412yes—
111C2×C32×C4C2×C3×C411yescharacteristic
112C2×C32×C4C22×D311yescharacteristic
113C2×C32×C4C23×C311yescharacteristic
114C3⋊(C3⋊C8)—22no—
115C3⋊(C3⋊C8)—21no—
116C3⋊(C3⋊C8)C3×D412yescomplemented
117C3×C3⋊C8—32no—
118C3×C3⋊C8—62no—
119C3×C3⋊C8—61no—
120#436—91noSylow
121C2×H(3,𝔽3)—42no—
122C2×H(3,𝔽3)C4×D411yescharacteristic
123C2×H(3,𝔽3)C2×M1611yescharacteristic
124C2×H(3,𝔽3)#13111yescharacteristic
125C2×C3×D4—31no—
126C2×C3×D4—31no—
127C2×C3×D4—61no—
128C2×C3×D4C3×Dic311yescharacteristic
129C22×C3×C4—32no—
130C22×C3×C4—32no—
131C22×C3×C4—62no—
132C22×C3×C4C2×C3×D312yes—
133C3×M16—184no—
134C2×C3×C8—91no—
135C2×C3×C8—91no—
136C2×C3×C8—92no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4—61no—
140C3×C4⋊C4C3×Dic311yescharacteristic
141C3×C22⋊C4—32no—
142C3×C22⋊C4—32no—
143C3×C22⋊C4—62no—
144C3×C22⋊C4C3×Dic312yes—
145C3×C42—31no—
146C3×C42—31no—
147C3×C42—61no—
148C3×C42C2×C3×D311yescharacteristic
149#208—64no—
150#208—184no—
151C2×C3⋊C8—31no—
152C2×C3⋊C8—31no—
153C2×C3⋊C8—32no—
154C2×C3⋊C8—91no—
155C2×C3⋊C8—91no—
156C2×C3⋊C8—92no—
157C22×C32—24no—
158C22×C32—22no—
159C22×C32—22no—
160C22×C32—22no—
161C22×C32—22no—
162C22×C32—22no—
163C22×C32—22no—
164C22×C32—21no—
165C22×C32—42no—
166C22×C32—42no—
167C22×C32C22×Dic311yescharacteristic
168C22×C32C22×C3×C411yescharacteristic
169C32×C4—22no—
170C32×C4—21no—
171C32×C4—22no—
172C32×C4—21no—
173C32×C4—22no—
174C32×C4—21no—
175C32×C4—21no—
176C32×C4—41no—
177C32×C4—42no—
178C32×C4#20812yescomplemented
179C32×C4C3×M1612yescomplemented
180C32×C4D3×D411yescharacteristic
181C32×C4#23711yescharacteristic
182C32×C4C2×C3×D411yescharacteristic
183C32×C4C3×#4111yescharacteristic
184C2×M16—92no—
185C4×D4(C3⋊D3)⋊C311yescharacteristic
186C4⋊C8—91no—
187C22⋊C8—92no—
188C4×C8—91no—
189H(3,𝔽3)#43611yescharacteristic, complemented, Sylow
190C23×C3—32no—
191C23×C3—32no—
192C23×C3—62no—
193C23×C3C2×C3×Dic312yes—
194C3×D4—22no—
195C3×D4—62no—
196C3×D4—62no—
197C3×D4—64no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—32no—
204C2×C3×C4—31no—
205C2×C3×C4—32no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4—61no—
213C2×C3×C4—61no—
214C2×C3×C4—62no—
215C2×C3×C4—61no—
216C2×C3×C4—122no—
217C2×C3×C4—122no—
218C2×C3×C4C2×C3×Dic311yescharacteristic
219C2×C3×C4C2×C3×Dic312yes—
220C2×C3×C4C2×C3×Dic311yescharacteristic
221C2×C3×C4C22×C3×D311yescharacteristic
222C3×C8—92no—
223C3×C8—181no—
224C3×C8—182no—
225C3⋊C8—32no—
226C3⋊C8—62no—
227C3⋊C8—61no—
228C3⋊C8—92no—
229C3⋊C8—181no—
230C3⋊C8—182no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32—44no—
235C2×C32—42no—
236C2×C32—42no—
237C2×C32C2×#20811yescharacteristic
238C2×C32D4×Dic311yescharacteristic
239C2×C32#94811yescharacteristic
240C2×C32C3×C4×D411yescharacteristic
241C2×C32C2×C3×M1611yescharacteristic
242C2×C32C3×#13111yescharacteristic
243C2×D4(C3⋊Dic3)⋊C311yescharacteristic
244C22×C4C2×(C3⋊D3)⋊C312yes—
245M16—184no—
246C2×C8—91no—
247C2×C8—91no—
248C2×C8—92no—
249C4⋊C4(C3⋊Dic3)⋊C311yescharacteristic
250C22⋊C4(C3⋊Dic3)⋊C312yes—
251C42C2×(C3⋊D3)⋊C311yescharacteristic
252C22×C3—22no—
253C22×C3—22no—
254C22×C3—22no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—62no—
258C22×C3—62no—
259C22×C3—62no—
260C22×C3—62no—
261C22×C3—62no—
262C22×C3—62no—
263C22×C3—64no—
264C22×C3—61no—
265C22×C3—122no—
266C22×C3—122no—
267C22×C3C22×C3×Dic311yescharacteristic
268C3×C4—22no—
269C3×C4—21no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—32no—
275C3×C4—32no—
276C3×C4—62no—
277C3×C4—62no—
278C3×C4—61no—
279C3×C4—61no—
280C3×C4—61no—
281C3×C4—61no—
282C3×C4—62no—
283C3×C4—122no—
284C3×C4—121no—
285C3×C4C3×#20812yes—
286C3×C4C3×D3×D411yescharacteristic
287C3×C4C3×#23711yescharacteristic
288C32—21no—
289C32#501211yescharacteristic, complemented
290C32C3×#43611yescharacteristic, complemented
291C23C2×(C3⋊Dic3)⋊C312yes—
292D4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4C2×(C3⋊Dic3)⋊C311yescharacteristic
296C2×C4C2×(C3⋊Dic3)⋊C312yes—
297C2×C4C2×(C3⋊Dic3)⋊C311yescharacteristic
298C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
299C8—92no—
300C8—182no—
301C8—181no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—42no—
309C2×C3—61no—
310C2×C3—61no—
311C2×C3—61no—
312C2×C3—122no—
313C2×C3—122no—
314C2×C3—124no—
315C2×C3C2×C3×#20811yescharacteristic
316C2×C3C3×D4×Dic311yescharacteristic
317C2×C3C3×#94811yescharacteristic
318C22—22no—
319C22—22no—
320C22—22no—
321C22C22×(C3⋊Dic3)⋊C311yescharacteristic
322C4—22no—
323C4—21no—
324C4#3748⋊C312yescomplemented
325C4D4×(C3⋊D3)⋊C311yescharacteristic
326C4#3830⋊C311yescharacteristic
327C3—31no—
328C3—31no—
329C3—61no—
330C3C3×#501211yescharacteristic
331C2—42no—
332C2C2×#3748⋊C311yescharacteristic
333C2D4×(C3⋊Dic3)⋊C311yescharacteristic
334C2#1172168911yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC4×D4×H(3,𝔽3)C2
socleC22×C3C22×C3×Dic3
prime radicalC2×D4×H(3,𝔽3)C4
cosocleC2×C32×C4C23×C3
hypercenterC4×D4(C3⋊D3)⋊C3
hypocenterC32C3×#436
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#3748⋊C3C2semidirect
3. #63657⋊C3C2semidirect
4. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
5. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
6. #10592711C3semidirect
20. C2×#3748C2×C3semidirect
22. #63657C2×C3semidirect
23. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
24. C4×C3⋊(C3⋊C8)C2×C3semidirect
38. (C3⋊(C3⋊C8))⋊C3D4semidirect
71. C4×H(3,𝔽3)M16semidirect
116. C3⋊(C3⋊C8)C3×D4semidirect
178. C32×C4#208semidirect
179. C32×C4C3×M16semidirect
189. H(3,𝔽3)#436semidirect
289. C32#5012semidirect
290. C32C3×#436semidirect
324. C4#3748⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×D4×H(3,𝔽3)2
C2×#3748⋊C32
#63657⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
C4×(C3⋊(C3⋊C8))⋊C32
#105927113
C3×#50123