Finite Group Databasebeta

← #10592680⋊C3 #13142624

Subgroups — #10592680⋊C3 #13142624

Order 1728.

617 conjugacy classes of subgroups.

The subgroup lattice has 362 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
3C2×#3750⋊C3C212yesmaximal
4#63658⋊C3C212yesmaximal
5#63658⋊C3C211yesmaximal, characteristic
6#10592680C311yesmaximal, characteristic, complemented
7C3×#4981—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—
15#3750⋊C3—24no—
16C2×C32×C4⋊C4—21no—
17C2×C32×C4⋊C4D311yescharacteristic
18C2×C32×C4⋊C4C2×C311yescharacteristic
19C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
20C2×#3750C2×C312yes—
21C2×C3×C4⋊Dic3—31no—
22C2×C3×Dic3⋊C4—32no—
23#63658C2×C312yes—
24#63658C2×C311yescharacteristic
25C3×C3⋊#95—31no—
26C3×C3⋊#95—32no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—22no—
33C2×C4×H(3,𝔽3)D412yes—
34C2×C4×H(3,𝔽3)D412yes—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3Q814yes—
41C3×#398—91no—
42#4981—31no—
43#4981—91no—
44C22×C32×C4—21no—
45C22×C32×C4—21no—
46C22×C32×C4—21no—
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C2×D311yescharacteristic
49C22×C32×C4C2×D311yescharacteristic
50C22×C32×C4C22×C311yescharacteristic
51C22×C32×C4C22×C311yescharacteristic
52C22×C32×C4C22×C311yescharacteristic
53C22×C3⋊Dic3C22×C312yes—
54C22×C3⋊Dic3C22×C312yes—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—32no—
57C32×C4⋊C4—24no—
58C32×C4⋊C4—22no—
59C32×C4⋊C4—22no—
60C4⋊(C3⋊Dic3)—22no—
61#3750—24no—
62C3×C4⋊Dic3—62no—
63C3×Dic3⋊C4—64no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C22×H(3,𝔽3)C2×Q812yes—
67C22×H(3,𝔽3)#4111yescharacteristic
68C22×H(3,𝔽3)#4112yes—
69C4×H(3,𝔽3)—22no—
70C4×H(3,𝔽3)—22no—
71C4×H(3,𝔽3)—41no—
72(C3⋊Dic3)⋊C3—24no—
73(C3⋊Dic3)⋊C3—42no—
74C2×C3×C4⋊C4—31no—
75C2×C3×C4⋊C4—31no—
76C2×C3×C4⋊C4—61no—
77C2×C3×C4⋊C4—92no—
78C2×C3×C4⋊C4—91no—
79C2×C3×C4⋊C4C3×D311yescharacteristic
80C2×C4⋊Dic3—31no—
81C2×C4⋊Dic3—91no—
82C2×Dic3⋊C4—32no—
83C2×Dic3⋊C4—92no—
84C3×#95—91no—
85C3×#95—92no—
86C3⋊#95—31no—
87C3⋊#95—32no—
88C3⋊#95—91no—
89C3⋊#95—92no—
90C23×C32—21no—
91C23×C32C22×D311yescharacteristic
92C23×C32C23×C311yescharacteristic
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—21no—
96C2×C32×C4—22no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4—22no—
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—42no—
109C2×C32×C4—42no—
110C2×C32×C4D1212yes—
111C2×C32×C4#6712yes—
112C2×C32×C4C3×D412yes—
113C2×C32×C4C3×D412yes—
114C2×C3⋊Dic3—22no—
115C2×C3⋊Dic3—22no—
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—22no—
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3C3×Q814yes—
120C2×C3×Dic3—34no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126#398—91noSylow
127C2×H(3,𝔽3)#12111yescharacteristic
128C2×H(3,𝔽3)C22⋊Q812yes—
129C2×H(3,𝔽3)#12311yescharacteristic
130C2×H(3,𝔽3)#12512yes—
131C2×H(3,𝔽3)C4⋊Q811yescharacteristic
132C22×C3×C4—31no—
133C22×C3×C4—31no—
134C22×C3×C4—31no—
135C22×C3×C4—31no—
136C22×C3×C4—31no—
137C22×C3×C4—31no—
138C22×C3×C4—61no—
139C22×C3×C4—61no—
140C22×C3×C4—61no—
141C22×C3×C4—92no—
142C22×C3×C4—92no—
143C22×C3×C4C2×C3×D311yescharacteristic
144C22×C3×C4C2×C3×D311yescharacteristic
145C22×C3×C4C2×C3×D311yescharacteristic
146C22×Dic3—32no—
147C22×Dic3—32no—
148C22×Dic3—92no—
149C22×Dic3—92no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—62no—
152C3×C4⋊C4—62no—
153C3×C4⋊C4—64no—
154C3×C4⋊C4—184no—
155C3×C4⋊C4—182no—
156C4⋊Dic3—62no—
157C4⋊Dic3—182no—
158Dic3⋊C4—64no—
159Dic3⋊C4—184no—
160C22×C32—21no—
161C22×C32—21no—
162C22×C32—22no—
163C22×C32—22no—
164C22×C32—21no—
165C22×C32C2×D1211yescharacteristic
166C22×C32#23511yescharacteristic
167C22×C32#23712yes—
168C22×C32D3×Q812yes—
169C22×C32C2×#6711yescharacteristic
170C22×C32C2×C3×D411yescharacteristic
171C22×C32C2×C3×D411yescharacteristic
172C22×C32C2×C3×Q812yes—
173C22×C32C3×#4111yescharacteristic
174C22×C32C3×#4112yes—
175C32×C4—24no—
176C32×C4—22no—
177C32×C4—22no—
178C32×C4—22no—
179C32×C4—22no—
180C32×C4—42no—
181C32×C4—42no—
182C32×C4—41no—
183C32×C4—41no—
184C3⋊Dic3—24no—
185C3⋊Dic3—42no—
186C3×Dic3—64no—
187C3×Dic3—122no—
188C2×C4⋊C4—92no—
189C2×C4⋊C4—91no—
190C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic
191#95—91no—
192#95—92no—
193H(3,𝔽3)#39811yescharacteristic, complemented, Sylow
194C23×C3—31no—
195C23×C3—31no—
196C23×C3—61no—
197C23×C3C22×C3×D311yescharacteristic
198C2×C3×C4—21no—
199C2×C3×C4—22no—
200C2×C3×C4—21no—
201C2×C3×C4—22no—
202C2×C3×C4—21no—
203C2×C3×C4—32no—
204C2×C3×C4—32no—
205C2×C3×C4—32no—
206C2×C3×C4—32no—
207C2×C3×C4—61no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—61no—
211C2×C3×C4—62no—
212C2×C3×C4—62no—
213C2×C3×C4—62no—
214C2×C3×C4—62no—
215C2×C3×C4—61no—
216C2×C3×C4—61no—
217C2×C3×C4—62no—
218C2×C3×C4—62no—
219C2×C3×C4—62no—
220C2×C3×C4—62no—
221C2×C3×C4—62no—
222C2×C3×C4—94no—
223C2×C3×C4—122no—
224C2×C3×C4—122no—
225C2×C3×C4—182no—
226C2×C3×C4—182no—
227C2×C3×C4—182no—
228C2×C3×C4—182no—
229C2×C3×C4—182no—
230C2×C3×C4C3×D1212yes—
231C2×C3×C4C3×#6712yes—
232C2×Dic3—34no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—94no—
239C2×Dic3—182no—
240C2×Dic3—182no—
241C2×Dic3—182no—
242C2×Dic3—182no—
243C2×Dic3—182no—
244C2×C32—22no—
245C2×C32—22no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32#88912yes—
250C2×C32#89712yes—
251C2×C32#93011yescharacteristic
252C2×C32#93511yescharacteristic
253C2×C32#94411yescharacteristic
254C2×C32C3×#12111yescharacteristic
255C2×C32C3×C22⋊Q812yes—
256C2×C32C3×#12311yescharacteristic
257C2×C32C3×#12512yes—
258C2×C32C3×C4⋊Q811yescharacteristic
259C22×C4—92no—
260C22×C4—92no—
261C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
262C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
263C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
264C4⋊C4—22no—
265C4⋊C4—184no—
266C4⋊C4—182no—
267C22×C3—31no—
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—31no—
271C22×C3—32no—
272C22×C3—32no—
273C22×C3—32no—
274C22×C3—32no—
275C22×C3—31no—
276C22×C3—31no—
277C22×C3—61no—
278C22×C3—61no—
279C22×C3—62no—
280C22×C3—62no—
281C22×C3—61no—
282C22×C3C2×C3×D1211yescharacteristic
283C22×C3C3×#23511yescharacteristic
284C22×C3C3×#23712yes—
285C22×C3C3×D3×Q812yes—
286C22×C3C2×C3×#6711yescharacteristic
287C3×C4—22no—
288C3×C4—22no—
289C3×C4—41no—
290C3×C4—62no—
291C3×C4—62no—
292C3×C4—62no—
293C3×C4—62no—
294C3×C4—64no—
295C3×C4—122no—
296C3×C4—121no—
297C3×C4—121no—
298C3×C4—122no—
299C3×C4—184no—
300C3×C4—362no—
301Dic3—64no—
302Dic3—122no—
303Dic3—184no—
304Dic3—362no—
305C32—21no—
306C32#498111yescharacteristic, complemented
307C32C3×#39811yescharacteristic, complemented
308C23C22×(C3⋊D3)⋊C311yescharacteristic
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—94no—
315C2×C4—182no—
316C2×C4—182no—
317C2×C4—182no—
318C2×C4—182no—
319C2×C4—182no—
320C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
321C2×C4#638⋊C312yes—
322C2×C3—31no—
323C2×C3—31no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—31no—
328C2×C3—32no—
329C2×C3—32no—
330C2×C3—32no—
331C2×C3—32no—
332C2×C3—61no—
333C2×C3—61no—
334C2×C3—61no—
335C2×C3—62no—
336C2×C3—62no—
337C2×C3C3×#88912yes—
338C2×C3C3×#89712yes—
339C2×C3C3×#93011yescharacteristic
340C2×C3C3×#93511yescharacteristic
341C2×C3C3×#94411yescharacteristic
342C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
343C22#3828⋊C311yescharacteristic
344C22#3830⋊C312yes—
345C22Q8×(C3⋊D3)⋊C312yes—
346C22C2×#638⋊C311yescharacteristic
347C4—22no—
348C4—22no—
349C4—41no—
350C4—184no—
351C4—362no—
352C3—31no—
353C3—31no—
354C3—61no—
355C3C3×#498111yescharacteristic
356C2#64098⋊C312yes—
357C2#64106⋊C312yes—
358C2#64139⋊C311yescharacteristic
359C2#64144⋊C311yescharacteristic
360C2#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. #10592680C3semidirect
193. H(3,𝔽3)#398semidirect
306. C32#4981semidirect
307. 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×C4⋊((C3⋊Dic3)⋊C3)2
C2×#3750⋊C32
#63658⋊C32
#105926803
C3×#49813