Finite Group Databasebeta

← #10592685⋊C3 #13142623

Subgroups — #10592685⋊C3 #13142623

Order 1728.

561 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
3#63658⋊C3C211yesmaximal, characteristic
4#63658⋊C3C212yesmaximal
5#63658⋊C3C212yesmaximal
6#10592685C311yesmaximal, characteristic, complemented
7C3×#4986—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C4⋊C4×H(3,𝔽3)—22no—
14C4⋊((C3⋊Dic3)⋊C3)—22no—
15C2×C32×C4⋊C4—21no—
16C2×C32×C4⋊C4D311yescharacteristic
17C2×C32×C4⋊C4C2×C311yescharacteristic
18C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
19C2×C3×C4⋊Dic3—31no—
20#63658C2×C312yes—
21#63658C2×C312yes—
22#63658C2×C311yescharacteristic
23C3×C3⋊#95—32no—
24C3×C3⋊#95—32no—
25C3×C3⋊#95—31no—
26C23×H(3,𝔽3)C2311yescharacteristic
27C2×C4×H(3,𝔽3)—21no—
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)D412yes—
33C2×C4×H(3,𝔽3)Q812yes—
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—24no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C3×#400—91no—
40#4986—31no—
41#4986—91no—
42C22×C32×C4—21no—
43C22×C32×C4—21no—
44C22×C32×C4—21no—
45C22×C32×C4C2×D311yescharacteristic
46C22×C32×C4C2×D311yescharacteristic
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C22×C311yescharacteristic
49C22×C32×C4C22×C311yescharacteristic
50C22×C32×C4C22×C311yescharacteristic
51C22×C3⋊Dic3C22×C312yes—
52C22×C3⋊Dic3C22×C312yes—
53C22×C3×Dic3—32no—
54C22×C3×Dic3—32no—
55C32×C4⋊C4—24no—
56C32×C4⋊C4—22no—
57C32×C4⋊C4—22no—
58C4⋊(C3⋊Dic3)—22no—
59C3×C4⋊Dic3—62no—
60C22×H(3,𝔽3)C2×D411yescharacteristic
61C22×H(3,𝔽3)C2×Q811yescharacteristic
62C22×H(3,𝔽3)#4111yescharacteristic
63C22×H(3,𝔽3)#4112yes—
64C22×H(3,𝔽3)#4112yes—
65C4×H(3,𝔽3)—22no—
66C4×H(3,𝔽3)—22no—
67C4×H(3,𝔽3)—41no—
68(C3⋊Dic3)⋊C3—42no—
69(C3⋊Dic3)⋊C3—42no—
70C2×C3×C4⋊C4—31no—
71C2×C3×C4⋊C4—31no—
72C2×C3×C4⋊C4—61no—
73C2×C3×C4⋊C4—91no—
74C2×C3×C4⋊C4C3×D311yescharacteristic
75C2×C4⋊Dic3—31no—
76C2×C4⋊Dic3—91no—
77C3×#95—92no—
78C3×#95—91no—
79C3×#95—92no—
80C3⋊#95—31no—
81C3⋊#95—32no—
82C3⋊#95—32no—
83C3⋊#95—91no—
84C3⋊#95—92no—
85C3⋊#95—92no—
86C23×C32—21no—
87C23×C32C22×D311yescharacteristic
88C23×C32C23×C311yescharacteristic
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—21no—
92C2×C32×C4—22no—
93C2×C32×C4—21no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—42no—
105C2×C32×C4—42no—
106C2×C32×C4Dic612yes—
107C2×C32×C4#6712yes—
108C2×C32×C4C3×D412yes—
109C2×C32×C4C3×Q812yes—
110C2×C3⋊Dic3—22no—
111C2×C3⋊Dic3—22no—
112C2×C3⋊Dic3—22no—
113C2×C3⋊Dic3—24no—
114C2×C3⋊Dic3—22no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—64no—
119C2×C3×Dic3—62no—
120#400—91noSylow
121C2×H(3,𝔽3)C22⋊Q811yescharacteristic
122C2×H(3,𝔽3)#12311yescharacteristic
123C2×H(3,𝔽3)#12411yescharacteristic
124C2×H(3,𝔽3)#12512yes—
125C2×H(3,𝔽3)#12612yes—
126C22×C3×C4—31no—
127C22×C3×C4—31no—
128C22×C3×C4—31no—
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4—31no—
132C22×C3×C4—61no—
133C22×C3×C4—61no—
134C22×C3×C4—61no—
135C22×C3×C4—92no—
136C22×C3×C4—92no—
137C22×C3×C4C2×C3×D311yescharacteristic
138C22×C3×C4C2×C3×D311yescharacteristic
139C22×C3×C4C2×C3×D311yescharacteristic
140C22×Dic3—32no—
141C22×Dic3—32no—
142C22×Dic3—92no—
143C22×Dic3—92no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—62no—
146C3×C4⋊C4—62no—
147C3×C4⋊C4—64no—
148C3×C4⋊C4—182no—
149C4⋊Dic3—62no—
150C4⋊Dic3—182no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32—22no—
154C22×C32—22no—
155C22×C32—21no—
156C22×C32C2×Dic611yescharacteristic
157C22×C32#23511yescharacteristic
158C22×C32#23712yes—
159C22×C32#23912yes—
160C22×C32C2×#6711yescharacteristic
161C22×C32C2×C3×D411yescharacteristic
162C22×C32C2×C3×Q811yescharacteristic
163C22×C32C3×#4111yescharacteristic
164C22×C32C3×#4112yes—
165C22×C32C3×#4112yes—
166C32×C4—24no—
167C32×C4—22no—
168C32×C4—22no—
169C32×C4—22no—
170C32×C4—22no—
171C32×C4—42no—
172C32×C4—42no—
173C32×C4—41no—
174C32×C4—41no—
175C3⋊Dic3—42no—
176C3⋊Dic3—42no—
177C3×Dic3—122no—
178C3×Dic3—122no—
179C2×C4⋊C4—91no—
180C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic
181#95—91no—
182#95—92no—
183#95—92no—
184H(3,𝔽3)#40011yescharacteristic, complemented, Sylow
185C23×C3—31no—
186C23×C3—31no—
187C23×C3—61no—
188C23×C3C22×C3×D311yescharacteristic
189C2×C3×C4—21no—
190C2×C3×C4—22no—
191C2×C3×C4—21no—
192C2×C3×C4—22no—
193C2×C3×C4—21no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—61no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—61no—
202C2×C3×C4—62no—
203C2×C3×C4—62no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4—61no—
207C2×C3×C4—61no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4—62no—
213C2×C3×C4—122no—
214C2×C3×C4—122no—
215C2×C3×C4—182no—
216C2×C3×C4—182no—
217C2×C3×C4—182no—
218C2×C3×C4—182no—
219C2×C3×C4—184no—
220C2×C3×C4C3×Dic612yes—
221C2×C3×C4C3×#6712yes—
222C2×Dic3—62no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225C2×Dic3—64no—
226C2×Dic3—62no—
227C2×Dic3—182no—
228C2×Dic3—182no—
229C2×Dic3—182no—
230C2×Dic3—182no—
231C2×Dic3—184no—
232C2×C32—22no—
233C2×C32—22no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32#89012yes—
238C2×C32#89812yes—
239C2×C32#92411yescharacteristic
240C2×C32#93511yescharacteristic
241C2×C32#94711yescharacteristic
242C2×C32C3×C22⋊Q811yescharacteristic
243C2×C32C3×#12311yescharacteristic
244C2×C32C3×#12411yescharacteristic
245C2×C32C3×#12512yes—
246C2×C32C3×#12612yes—
247C22×C4—92no—
248C22×C4—92no—
249C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
250C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
251C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
252C4⋊C4—22no—
253C4⋊C4—182no—
254C22×C3—31no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—32no—
259C22×C3—32no—
260C22×C3—32no—
261C22×C3—32no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—61no—
265C22×C3—61no—
266C22×C3—62no—
267C22×C3—62no—
268C22×C3—61no—
269C22×C3C2×C3×Dic611yescharacteristic
270C22×C3C3×#23511yescharacteristic
271C22×C3C3×#23712yes—
272C22×C3C3×#23912yes—
273C22×C3C2×C3×#6711yescharacteristic
274C3×C4—22no—
275C3×C4—22no—
276C3×C4—41no—
277C3×C4—62no—
278C3×C4—62no—
279C3×C4—62no—
280C3×C4—62no—
281C3×C4—64no—
282C3×C4—122no—
283C3×C4—121no—
284C3×C4—121no—
285C3×C4—122no—
286C3×C4—362no—
287C3×C4—362no—
288Dic3—122no—
289Dic3—122no—
290Dic3—362no—
291Dic3—362no—
292C32—21no—
293C32#498611yescharacteristic, complemented
294C32C3×#40011yescharacteristic, complemented
295C23C22×(C3⋊D3)⋊C311yescharacteristic
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—182no—
302C2×C4—182no—
303C2×C4—182no—
304C2×C4—184no—
305C2×C4—182no—
306C2×C4#634⋊C312yes—
307C2×C4#638⋊C312yes—
308C2×C3—32no—
309C2×C3—32no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—31no—
316C2×C3—32no—
317C2×C3—32no—
318C2×C3—62no—
319C2×C3—61no—
320C2×C3—61no—
321C2×C3—61no—
322C2×C3—62no—
323C2×C3C3×#89012yes—
324C2×C3C3×#89812yes—
325C2×C3C3×#92411yescharacteristic
326C2×C3C3×#93511yescharacteristic
327C2×C3C3×#94711yescharacteristic
328C22C2×#634⋊C311yescharacteristic
329C22#3828⋊C311yescharacteristic
330C22#3830⋊C312yes—
331C22#8872212yes—
332C22C2×#638⋊C311yescharacteristic
333C4—22no—
334C4—22no—
335C4—41no—
336C4—362no—
337C4—362no—
338C3—31no—
339C3—31no—
340C3—61no—
341C3C3×#498611yescharacteristic
342C2#64099⋊C312yes—
343C2#64107⋊C312yes—
344C2#64133⋊C311yescharacteristic
345C2#64144⋊C311yescharacteristic
346C2#64156⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC23×C32C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radicalC23×H(3,𝔽3)C23
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#400
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #10592685C3semidirect
184. H(3,𝔽3)#400semidirect
293. C32#4986semidirect
294. C32C3×#400semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
C2×C4⋊((C3⋊Dic3)⋊C3)2
#63658⋊C32
#105926853
C3×#49863