Finite Group Databasebeta

← #10592682⋊C3 #13142619

Subgroups — #10592682⋊C3 #13142619

Order 1728.

617 conjugacy classes of subgroups.

The subgroup lattice has 354 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×#3750⋊C3C211yesmaximal, characteristic
3C2×#3750⋊C3C212yesmaximal
4#63658⋊C3C212yesmaximal
5#63658⋊C3C211yesmaximal, characteristic
6#10592682C311yesmaximal, characteristic, complemented
7C3×#4983—31nomaximal
8C22×C4×H(3,𝔽3)C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C4⋊C4×H(3,𝔽3)—22no—
14#3750⋊C3—22no—
15#3750⋊C3—24no—
16C2×C32×C4⋊C4—21no—
17C2×C32×C4⋊C4D311yescharacteristic
18C2×C32×C4⋊C4C2×C311yescharacteristic
19C2×#3750C2×C311yescharacteristic
20C2×#3750C2×C312yes—
21C2×C3×Dic3⋊C4—32no—
22C2×C3×Dic3⋊C4—31no—
23#63658C2×C312yes—
24#63658C2×C311yescharacteristic
25C3×C3⋊#95—31no—
26C3×C3⋊#95—32no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C2×C4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)D412yes—
33C2×(C3⋊Dic3)⋊C3—21no—
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—21no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—21no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3D412yes—
40C2×(C3⋊Dic3)⋊C3Q814yes—
41C3×#398—91no—
42#4983—31no—
43#4983—91no—
44C22×C32×C4—21no—
45C22×C32×C4—22no—
46C22×C32×C4C2×D312yes—
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C22×C312yes—
49C22×C32×C4C22×C311yescharacteristic
50C22×C3⋊Dic3C22×C311yescharacteristic
51C22×C3⋊Dic3C22×C311yescharacteristic
52C22×C3⋊Dic3C22×C312yes—
53C22×C3×Dic3—31no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—31no—
56C32×C4⋊C4—24no—
57C32×C4⋊C4—22no—
58C32×C4⋊C4—22no—
59#3750—22no—
60#3750—24no—
61C3×Dic3⋊C4—64no—
62C3×Dic3⋊C4—62no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C22×H(3,𝔽3)C2×Q812yes—
66C22×H(3,𝔽3)#4111yescharacteristic
67C22×H(3,𝔽3)#4112yes—
68C4×H(3,𝔽3)—22no—
69C4×H(3,𝔽3)—42no—
70(C3⋊Dic3)⋊C3—22no—
71(C3⋊Dic3)⋊C3—24no—
72(C3⋊Dic3)⋊C3—41no—
73C2×C3×C4⋊C4—31no—
74C2×C3×C4⋊C4—31no—
75C2×C3×C4⋊C4—61no—
76C2×C3×C4⋊C4—91no—
77C2×C3×C4⋊C4—92no—
78C2×C3×C4⋊C4C3×D311yescharacteristic
79C2×Dic3⋊C4—32no—
80C2×Dic3⋊C4—31no—
81C2×Dic3⋊C4—92no—
82C2×Dic3⋊C4—91no—
83C3×#95—91no—
84C3×#95—92no—
85C3⋊#95—31no—
86C3⋊#95—32no—
87C3⋊#95—91no—
88C3⋊#95—92no—
89C23×C32—21no—
90C23×C32C22×D311yescharacteristic
91C23×C32C23×C311yescharacteristic
92C2×C32×C4—24no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—24no—
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—22no—
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4—42no—
104C2×C32×C4—42no—
105C2×C32×C4#6712yes—
106C2×C32×C4C3×D412yes—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3—21no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—22no—
113C2×C3⋊Dic3C3×D412yes—
114C2×C3⋊Dic3C3×Q814yes—
115C2×C3×Dic3—32no—
116C2×C3×Dic3—34no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—61no—
123#398—91noSylow
124C2×H(3,𝔽3)#12111yescharacteristic
125C2×H(3,𝔽3)C22⋊Q812yes—
126C2×H(3,𝔽3)#12311yescharacteristic
127C2×H(3,𝔽3)#12512yes—
128C2×H(3,𝔽3)C4⋊Q811yescharacteristic
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4—32no—
132C22×C3×C4—32no—
133C22×C3×C4—61no—
134C22×C3×C4—62no—
135C22×C3×C4—92no—
136C22×C3×C4—91no—
137C22×C3×C4—91no—
138C22×C3×C4C2×C3×D312yes—
139C22×C3×C4C2×C3×D311yescharacteristic
140C22×Dic3—32no—
141C22×Dic3—31no—
142C22×Dic3—31no—
143C22×Dic3—92no—
144C22×Dic3—91no—
145C22×Dic3—91no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—62no—
148C3×C4⋊C4—62no—
149C3×C4⋊C4—64no—
150C3×C4⋊C4—182no—
151C3×C4⋊C4—184no—
152Dic3⋊C4—64no—
153Dic3⋊C4—62no—
154Dic3⋊C4—184no—
155Dic3⋊C4—182no—
156C22×C32—22no—
157C22×C32—21no—
158C22×C32—21no—
159C22×C32—22no—
160C22×C32—21no—
161C22×C32#23512yes—
162C22×C32D3×D411yescharacteristic
163C22×C32#23711yescharacteristic
164C22×C32D3×Q812yes—
165C22×C32C2×#6711yescharacteristic
166C22×C32C2×C3×D411yescharacteristic
167C22×C32C2×C3×D411yescharacteristic
168C22×C32C2×C3×Q812yes—
169C22×C32C3×#4112yes—
170C22×C32C3×#4111yescharacteristic
171C32×C4—24no—
172C32×C4—22no—
173C32×C4—22no—
174C32×C4—44no—
175C32×C4—42no—
176C32×C4—42no—
177C3⋊Dic3—22no—
178C3⋊Dic3—24no—
179C3⋊Dic3—41no—
180C3×Dic3—64no—
181C3×Dic3—62no—
182C3×Dic3—121no—
183C2×C4⋊C4—92no—
184C2×C4⋊C4—91no—
185C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic
186#95—91no—
187#95—92no—
188H(3,𝔽3)#39811yescharacteristic, complemented, Sylow
189C23×C3—31no—
190C23×C3—31no—
191C23×C3—61no—
192C23×C3C22×C3×D311yescharacteristic
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—32no—
198C2×C3×C4—32no—
199C2×C3×C4—62no—
200C2×C3×C4—62no—
201C2×C3×C4—64no—
202C2×C3×C4—62no—
203C2×C3×C4—62no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4—62no—
208C2×C3×C4—64no—
209C2×C3×C4—62no—
210C2×C3×C4—92no—
211C2×C3×C4—94no—
212C2×C3×C4—122no—
213C2×C3×C4—122no—
214C2×C3×C4—182no—
215C2×C3×C4—181no—
216C2×C3×C4—182no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×C3×C4—182no—
220C2×C3×C4C3×#6712yes—
221C2×Dic3—32no—
222C2×Dic3—34no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—62no—
228C2×Dic3—61no—
229C2×Dic3—92no—
230C2×Dic3—94no—
231C2×Dic3—182no—
232C2×Dic3—181no—
233C2×Dic3—182no—
234C2×Dic3—181no—
235C2×Dic3—181no—
236C2×Dic3—182no—
237C2×C32—22no—
238C2×C32—22no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32#88912yes—
243C2×C32#89612yes—
244C2×C32#92911yescharacteristic
245C2×C32#93811yescharacteristic
246C2×C32#94411yescharacteristic
247C2×C32C3×#12111yescharacteristic
248C2×C32C3×C22⋊Q812yes—
249C2×C32C3×#12311yescharacteristic
250C2×C32C3×#12512yes—
251C2×C32C3×C4⋊Q811yescharacteristic
252C22×C4—92no—
253C22×C4—91no—
254C22×C4—91no—
255C22×C4C2×(C3⋊D3)⋊C312yes—
256C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
257C4⋊C4—22no—
258C4⋊C4—184no—
259C4⋊C4—182no—
260C22×C3—32no—
261C22×C3—32no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—32no—
267C22×C3—32no—
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—62no—
271C22×C3—61no—
272C22×C3—61no—
273C22×C3—62no—
274C22×C3—61no—
275C22×C3C3×#23512yes—
276C22×C3C3×D3×D411yescharacteristic
277C22×C3C3×#23711yescharacteristic
278C22×C3C3×D3×Q812yes—
279C22×C3C2×C3×#6711yescharacteristic
280C3×C4—22no—
281C3×C4—42no—
282C3×C4—62no—
283C3×C4—62no—
284C3×C4—64no—
285C3×C4—122no—
286C3×C4—122no—
287C3×C4—124no—
288C3×C4—182no—
289C3×C4—184no—
290C3×C4—361no—
291Dic3—64no—
292Dic3—62no—
293Dic3—121no—
294Dic3—182no—
295Dic3—184no—
296Dic3—361no—
297C32—21no—
298C32#498311yescharacteristic, complemented
299C32C3×#39811yescharacteristic, complemented
300C23C22×(C3⋊D3)⋊C311yescharacteristic
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—92no—
306C2×C4—94no—
307C2×C4—182no—
308C2×C4—182no—
309C2×C4—181no—
310C2×C4—181no—
311C2×C4—182no—
312C2×C4—181no—
313C2×C4#638⋊C312yes—
314C2×C3—31no—
315C2×C3—31no—
316C2×C3—31no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—32no—
321C2×C3—32no—
322C2×C3—32no—
323C2×C3—32no—
324C2×C3—61no—
325C2×C3—61no—
326C2×C3—61no—
327C2×C3—62no—
328C2×C3—62no—
329C2×C3C3×#88912yes—
330C2×C3C3×#89612yes—
331C2×C3C3×#92911yescharacteristic
332C2×C3C3×#93811yescharacteristic
333C2×C3C3×#94411yescharacteristic
334C22#3828⋊C312yes—
335C22D4×(C3⋊D3)⋊C311yescharacteristic
336C22#3830⋊C311yescharacteristic
337C22Q8×(C3⋊D3)⋊C312yes—
338C22C2×#638⋊C311yescharacteristic
339C4—22no—
340C4—42no—
341C4—182no—
342C4—184no—
343C4—361no—
344C3—31no—
345C3—31no—
346C3—61no—
347C3C3×#498311yescharacteristic
348C2#64098⋊C312yes—
349C2#64105⋊C312yes—
350C2#64138⋊C311yescharacteristic
351C2#64147⋊C311yescharacteristic
352C2#64153⋊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×#398
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #10592682C3semidirect
188. H(3,𝔽3)#398semidirect
298. C32#4983semidirect
299. C32C3×#398semidirect

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×#3750⋊C32
#63658⋊C32
#105926823
C3×#49833