Finite Group Databasebeta

← #10592678⋊C3 #13142615

Subgroups — #10592678⋊C3 #13142615

Order 1728.

729 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#634⋊C3C211yesmaximal, characteristic
2C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
3C2×#3750⋊C3C212yesmaximal
4#63658⋊C3C212yesmaximal
5#63658⋊C3C211yesmaximal, characteristic
6#10592678C311yesmaximal, characteristic, complemented
7C3×#4979—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—
13C2×#634⋊C3—22no—
14C2×#634⋊C3—24no—
15C4⋊C4×H(3,𝔽3)—22no—
16#3750⋊C3—24no—
17C22×#634C2×C311yescharacteristic
18C22×C3×Dic6—31no—
19C2×C32×C4⋊C4—21no—
20C2×C32×C4⋊C4D311yescharacteristic
21C2×C32×C4⋊C4C2×C311yescharacteristic
22C2×#3750C2×C312yes—
23C2×C3×Dic3⋊C4—32no—
24#63658C2×C312yes—
25#63658C2×C311yescharacteristic
26C3×C3⋊#95—32no—
27C3×C3⋊#95—31no—
28C23×H(3,𝔽3)C2311yescharacteristic
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)D412yes—
35C2×C4×H(3,𝔽3)Q812yescomplemented
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3D414yes—
41C2×(C3⋊Dic3)⋊C3Q814yescomplemented
42#634⋊C3—28no—
43C3×#395—91no—
44#4979—31no—
45#4979—91no—
46C22×C32×C4—21no—
47C22×C32×C4—21no—
48C22×C32×C4—21no—
49C22×C32×C4C2×D311yescharacteristic
50C22×C32×C4C2×D311yescharacteristic
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C22×C311yescharacteristic
53C22×C32×C4C22×C311yescharacteristic
54C22×C32×C4C22×C311yescharacteristic
55C22×C3⋊Dic3C22×C312yes—
56C22×C3⋊Dic3C22×C312yes—
57C2×#634—22no—
58C2×#634—24no—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—32no—
61C2×C3×Dic6—64no—
62C2×C3×Dic6—62no—
63C32×C4⋊C4—24no—
64C32×C4⋊C4—22no—
65C32×C4⋊C4—22no—
66#3750—24no—
67C3×Dic3⋊C4—64no—
68C22×H(3,𝔽3)C2×D412yes—
69C22×H(3,𝔽3)C2×D411yescharacteristic
70C22×H(3,𝔽3)C2×Q811yescharacteristic
71C22×H(3,𝔽3)C2×Q812yes—
72C22×H(3,𝔽3)#4111yescharacteristic
73C4×H(3,𝔽3)—22no—
74C4×H(3,𝔽3)—22no—
75C4×H(3,𝔽3)—41no—
76(C3⋊Dic3)⋊C3—24no—
77(C3⋊Dic3)⋊C3—24no—
78C22×C3×Q8—91no—
79C22×Dic6—31no—
80C22×Dic6—91no—
81C2×C3×C4⋊C4—31no—
82C2×C3×C4⋊C4—31no—
83C2×C3×C4⋊C4—61no—
84C2×C3×C4⋊C4—92no—
85C2×C3×C4⋊C4C3×D311yescharacteristic
86C2×Dic3⋊C4—32no—
87C2×Dic3⋊C4—92no—
88C3×#95—92no—
89C3×#95—91no—
90C3⋊#95—31no—
91C3⋊#95—32no—
92C3⋊#95—91no—
93C3⋊#95—92no—
94C23×C32—21no—
95C23×C32C22×D311yescharacteristic
96C23×C32C23×C311yescharacteristic
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4—22no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—21no—
108C2×C32×C4—22no—
109C2×C32×C4—21no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—42no—
113C2×C32×C4—42no—
114C2×C32×C4Dic612yescomplemented
115C2×C32×C4#6712yes—
116C2×C32×C4C3×D412yes—
117C2×C32×C4C3×Q812yescomplemented
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3—22no—
120C2×C3⋊Dic3—22no—
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3C3×D414yes—
123C2×C3⋊Dic3C3×Q814yescomplemented
124#634—28no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—62no—
131C3×Dic6—68no—
132#395—91noSylow
133C2×H(3,𝔽3)#12011yescharacteristic
134C2×H(3,𝔽3)C22⋊Q812yes—
135C2×H(3,𝔽3)C22⋊Q811yescharacteristic
136C2×H(3,𝔽3)C4⋊Q812yes—
137C2×H(3,𝔽3)C4⋊Q811yescharacteristic
138C2×C3×Q8—182no—
139C2×C3×Q8—184no—
140C22×C3×C4—31no—
141C22×C3×C4—31no—
142C22×C3×C4—31no—
143C22×C3×C4—31no—
144C22×C3×C4—31no—
145C22×C3×C4—31no—
146C22×C3×C4—61no—
147C22×C3×C4—61no—
148C22×C3×C4—61no—
149C22×C3×C4—92no—
150C22×C3×C4—92no—
151C22×C3×C4C2×C3×D311yescharacteristic
152C22×C3×C4C2×C3×D311yescharacteristic
153C22×C3×C4C2×C3×D311yescharacteristic
154C22×Dic3—32no—
155C22×Dic3—32no—
156C22×Dic3—92no—
157C22×Dic3—92no—
158C2×Dic6—64no—
159C2×Dic6—62no—
160C2×Dic6—184no—
161C2×Dic6—182no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—62no—
164C3×C4⋊C4—62no—
165C3×C4⋊C4—64no—
166C3×C4⋊C4—184no—
167Dic3⋊C4—64no—
168Dic3⋊C4—184no—
169C22×C32—21no—
170C22×C32—21no—
171C22×C32—22no—
172C22×C32—22no—
173C22×C32—21no—
174C22×C32C2×Dic611yescharacteristic
175C22×C32#23511yescharacteristic
176C22×C32D3×D412yes—
177C22×C32D3×Q812yes—
178C22×C32C2×#6711yescharacteristic
179C22×C32C2×C3×D412yes—
180C22×C32C2×C3×D411yescharacteristic
181C22×C32C2×C3×Q811yescharacteristic
182C22×C32C2×C3×Q812yes—
183C22×C32C3×#4111yescharacteristic
184C32×C4—24no—
185C32×C4—22no—
186C32×C4—22no—
187C32×C4—22no—
188C32×C4—22no—
189C32×C4—42no—
190C32×C4—42no—
191C32×C4—41no—
192C32×C4—41no—
193C3⋊Dic3—24no—
194C3⋊Dic3—24no—
195C3×Dic3—64no—
196C3×Dic3—64no—
197C22×Q8—91no—
198C2×C4⋊C4—92no—
199C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic
200#95—91no—
201#95—92no—
202H(3,𝔽3)#39511yescharacteristic, complemented, Sylow
203C23×C3—31no—
204C23×C3—31no—
205C23×C3—61no—
206C23×C3C22×C3×D311yescharacteristic
207C3×Q8—188no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—21no—
211C2×C3×C4—22no—
212C2×C3×C4—21no—
213C2×C3×C4—32no—
214C2×C3×C4—32no—
215C2×C3×C4—32no—
216C2×C3×C4—32no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—62no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—62no—
229C2×C3×C4—62no—
230C2×C3×C4—62no—
231C2×C3×C4—62no—
232C2×C3×C4—94no—
233C2×C3×C4—94no—
234C2×C3×C4—122no—
235C2×C3×C4—122no—
236C2×C3×C4—182no—
237C2×C3×C4—182no—
238C2×C3×C4—182no—
239C2×C3×C4—182no—
240C2×C3×C4C3×Dic612yes—
241C2×C3×C4C3×#6712yes—
242C2×Dic3—34no—
243C2×Dic3—34no—
244C2×Dic3—62no—
245C2×Dic3—62no—
246C2×Dic3—62no—
247C2×Dic3—62no—
248C2×Dic3—94no—
249C2×Dic3—94no—
250C2×Dic3—182no—
251C2×Dic3—182no—
252C2×Dic3—182no—
253C2×Dic3—182no—
254Dic6—68no—
255Dic6—188no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32#88812yes—
262C2×C32#89612yes—
263C2×C32#92411yescharacteristic
264C2×C32#93711yescharacteristic
265C2×C32#94411yescharacteristic
266C2×C32C3×#12011yescharacteristic
267C2×C32C3×C22⋊Q812yes—
268C2×C32C3×C22⋊Q811yescharacteristic
269C2×C32C3×C4⋊Q812yes—
270C2×C32C3×C4⋊Q811yescharacteristic
271C2×Q8—184no—
272C2×Q8—182no—
273C22×C4—92no—
274C22×C4—92no—
275C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
276C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
277C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
278C4⋊C4—22no—
279C4⋊C4—184no—
280C22×C3—31no—
281C22×C3—31no—
282C22×C3—31no—
283C22×C3—31no—
284C22×C3—32no—
285C22×C3—32no—
286C22×C3—32no—
287C22×C3—32no—
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—61no—
291C22×C3—61no—
292C22×C3—62no—
293C22×C3—62no—
294C22×C3—61no—
295C22×C3C2×C3×Dic611yescharacteristic
296C22×C3C3×#23511yescharacteristic
297C22×C3C3×D3×D412yes—
298C22×C3C3×D3×Q812yes—
299C22×C3C2×C3×#6711yescharacteristic
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—41no—
303C3×C4—62no—
304C3×C4—62no—
305C3×C4—62no—
306C3×C4—62no—
307C3×C4—64no—
308C3×C4—122no—
309C3×C4—121no—
310C3×C4—121no—
311C3×C4—122no—
312C3×C4—184no—
313C3×C4—184no—
314Dic3—64no—
315Dic3—64no—
316Dic3—184no—
317Dic3—184no—
318C32—21no—
319C32#497911yescharacteristic, complemented
320C32C3×#39511yescharacteristic, complemented
321C23C22×(C3⋊D3)⋊C311yescharacteristic
322Q8—188no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—94no—
329C2×C4—94no—
330C2×C4—182no—
331C2×C4—182no—
332C2×C4—182no—
333C2×C4—182no—
334C2×C4#634⋊C312yescomplemented
335C2×C4#638⋊C312yes—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—32no—
343C2×C3—32no—
344C2×C3—32no—
345C2×C3—32no—
346C2×C3—61no—
347C2×C3—61no—
348C2×C3—61no—
349C2×C3—62no—
350C2×C3—62no—
351C2×C3C3×#88812yes—
352C2×C3C3×#89612yes—
353C2×C3C3×#92411yescharacteristic
354C2×C3C3×#93711yescharacteristic
355C2×C3C3×#94411yescharacteristic
356C22C2×#634⋊C311yescharacteristic
357C22#3828⋊C311yescharacteristic
358C22D4×(C3⋊D3)⋊C312yes—
359C22Q8×(C3⋊D3)⋊C312yes—
360C22C2×#638⋊C311yescharacteristic
361C4—22no—
362C4—22no—
363C4—41no—
364C4—184no—
365C4—184no—
366C3—31no—
367C3—31no—
368C3—61no—
369C3C3×#497911yescharacteristic
370C2#64097⋊C312yes—
371C2#64105⋊C312yes—
372C2#64133⋊C311yescharacteristic
373C2#64146⋊C311yescharacteristic
374C2#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×#395
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #10592678C3semidirect
35. C2×C4×H(3,𝔽3)Q8semidirect
41. C2×(C3⋊Dic3)⋊C3Q8semidirect
114. C2×C32×C4Dic6semidirect
117. C2×C32×C4C3×Q8semidirect
123. C2×C3⋊Dic3C3×Q8semidirect
202. H(3,𝔽3)#395semidirect
319. C32#4979semidirect
320. C32C3×#395semidirect
334. C2×C4#634⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
C2×C4⋊C4×H(3,𝔽3)2
C2×#3750⋊C32
#63658⋊C32
#105926783
C3×#49793