Finite Group Databasebeta

← #10592372⋊C3 #13141343

Subgroups — #10592372⋊C3 #13141343

Order 1728.

783 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#634⋊C3C211yesmaximal, characteristic, complemented
2C2×#3752⋊C3C211yesmaximal, characteristic
3C2×#3752⋊C3C212yesmaximal
4H(3,𝔽3)×#95C211yesmaximal, characteristic, complemented
5#63658⋊C3C212yesmaximal, complemented
6#10592372C311yesmaximal, characteristic, complemented
7C3×#4673—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2212yes—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C2×#634⋊C3—22no—
14C2×#634⋊C3—24no—
15#3752⋊C3—22no—
16#3752⋊C3—24no—
17C22×#634C2×C311yescharacteristic, complemented
18C22×C3×Dic6—31no—
19C2×#3752C2×C311yescharacteristic
20C2×#3752C2×C312yes—
21C2×C3×(C2×D3)⋊C4—32no—
22C2×C3×(C2×D3)⋊C4—31no—
23C32×#95—21no—
24C32×#95D311yescharacteristic, complemented
25C32×#95C2×C311yescharacteristic, complemented
26#63658C2×C312yescomplemented
27C3×C3⋊#95—32no—
28C23×H(3,𝔽3)C2311yescharacteristic
29C22×(C3⋊D3)⋊C3—21no—
30C22×(C3⋊D3)⋊C3—22no—
31C22×(C3⋊D3)⋊C3—22no—
32C22×(C3⋊D3)⋊C3—21no—
33C22×(C3⋊D3)⋊C3Q812yescomplemented
34C2×C4×H(3,𝔽3)—22no—
35C2×C4×H(3,𝔽3)—22no—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)—22no—
38C2×C4×H(3,𝔽3)D412yes—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—22no—
43C2×(C3⋊Dic3)⋊C3D414yes—
44#634⋊C3—28no—
45C3×#393—91no—
46#4673—31no—
47#4673—91no—
48C23×C3⋊D3C22×C311yescharacteristic
49C23×C3×D3—31no—
50C22×C32×C4—22no—
51C22×C32×C4—21no—
52C22×C32×C4C2×D312yes—
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C22×C312yes—
55C22×C32×C4C22×C311yescharacteristic
56C22×C3⋊Dic3C22×C311yescharacteristic
57C22×C3⋊Dic3C22×C312yes—
58C2×#634—22no—
59C2×#634—24no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—31no—
62C2×C3×Dic6—64no—
63C2×C3×Dic6—62no—
64#3752—22no—
65#3752—24no—
66C3×(C2×D3)⋊C4—64no—
67C3×(C2×D3)⋊C4—62no—
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C22×H(3,𝔽3)C2×D412yes—
70C22×H(3,𝔽3)C2×Q811yescharacteristic
71C22×H(3,𝔽3)#4112yes—
72C22×H(3,𝔽3)#4111yescharacteristic
73C2×(C3⋊D3)⋊C3—24no—
74C2×(C3⋊D3)⋊C3—22no—
75C2×(C3⋊D3)⋊C3—41no—
76C2×(C3⋊D3)⋊C3—41no—
77C2×(C3⋊D3)⋊C3—42no—
78C4×H(3,𝔽3)—22no—
79C4×H(3,𝔽3)—42no—
80(C3⋊Dic3)⋊C3—24no—
81(C3⋊Dic3)⋊C3—41no—
82C22×C3×Q8—91no—
83C22×Dic6—31no—
84C22×Dic6—91no—
85C2×C3×C22⋊C4—91no—
86C2×C3×C22⋊C4—92no—
87C2×(C2×D3)⋊C4—32no—
88C2×(C2×D3)⋊C4—31no—
89C2×(C2×D3)⋊C4—92no—
90C2×(C2×D3)⋊C4—91no—
91C3×#95—31no—
92C3×#95—31no—
93C3×#95—61no—
94C3×#95—92no—
95C3×#95C3×D311yescharacteristic
96C3⋊#95—32no—
97C3⋊#95—92no—
98C23×C32—21no—
99C23×C32C22×D311yescharacteristic
100C23×C32C23×C311yescharacteristic
101C22×C3⋊D3—21no—
102C22×C3⋊D3—22no—
103C22×C3⋊D3—22no—
104C22×C3⋊D3—21no—
105C22×C3⋊D3C3×Q812yescomplemented
106C22×C3×D3—32no—
107C22×C3×D3—61no—
108C22×C3×D3—62no—
109C22×C3×D3—62no—
110C22×C3×D3—61no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—24no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—42no—
122C2×C32×C4—42no—
123C2×C32×C4—42no—
124C2×C32×C4D1212yes—
125C2×C32×C4C3×D412yes—
126C2×C3⋊Dic3—22no—
127C2×C3⋊Dic3—22no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—22no—
130C2×C3⋊Dic3C3×D414yes—
131#634—28no—
132C2×C3×Dic3—34no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—62no—
137C3×Dic6—68no—
138#393—91noSylow
139C2×H(3,𝔽3)#12011yescharacteristic
140C2×H(3,𝔽3)C22⋊Q812yes—
141C2×H(3,𝔽3)C22⋊Q811yescharacteristic
142C2×H(3,𝔽3)#12411yescharacteristic
143C2×H(3,𝔽3)#12412yes—
144(C3⋊D3)⋊C3—42no—
145C24×C3—91no—
146C23×D3—31no—
147C23×D3—91no—
148C2×C3×Q8—182no—
149C2×C3×Q8—184no—
150C22×C3×C4—31no—
151C22×C3×C4—31no—
152C22×C3×C4—32no—
153C22×C3×C4—32no—
154C22×C3×C4—62no—
155C22×C3×C4—61no—
156C22×C3×C4—92no—
157C22×C3×C4—91no—
158C22×C3×C4C2×C3×D312yes—
159C22×C3×C4C2×C3×D311yescharacteristic
160C22×Dic3—32no—
161C22×Dic3—31no—
162C22×Dic3—92no—
163C22×Dic3—91no—
164C2×Dic6—64no—
165C2×Dic6—62no—
166C2×Dic6—184no—
167C2×Dic6—182no—
168C3×C22⋊C4—182no—
169C3×C22⋊C4—184no—
170(C2×D3)⋊C4—64no—
171(C2×D3)⋊C4—62no—
172(C2×D3)⋊C4—184no—
173(C2×D3)⋊C4—182no—
174C22×C32—21no—
175C22×C32—22no—
176C22×C32—22no—
177C22×C32—21no—
178C22×C32—21no—
179C22×C32C2×D1211yescharacteristic
180C22×C32#23512yes—
181C22×C32D3×D412yes—
182C22×C32#23711yescharacteristic
183C22×C32D3×Q811yescharacteristic
184C22×C32C2×C3×D411yescharacteristic
185C22×C32C2×C3×D412yes—
186C22×C32C2×C3×Q811yescharacteristic
187C22×C32C3×#4112yes—
188C22×C32C3×#4111yescharacteristic
189C2×C3⋊D3—24no—
190C2×C3⋊D3—22no—
191C2×C3⋊D3—41no—
192C2×C3⋊D3—42no—
193C2×C3⋊D3—41no—
194C2×C3×D3—64no—
195C2×C3×D3—62no—
196C2×C3×D3—121no—
197C2×C3×D3—121no—
198C2×C3×D3—122no—
199C32×C4—22no—
200C32×C4—22no—
201C32×C4—42no—
202C32×C4—44no—
203C32×C4—42no—
204C32×C4—42no—
205C3⋊Dic3—24no—
206C3⋊Dic3—41no—
207C3×Dic3—64no—
208C3×Dic3—121no—
209C22×Q8—91no—
210C2×C22⋊C4—92no—
211C2×C22⋊C4—91no—
212#95—92no—
213#95(C3⋊D3)⋊C311yescharacteristic, complemented
214H(3,𝔽3)#39311yescharacteristic, complemented, Sylow
215C23×C3—31no—
216C23×C3—31no—
217C23×C3—61no—
218C23×C3—92no—
219C23×C3—181no—
220C23×C3—182no—
221C23×C3—182no—
222C23×C3—181no—
223C23×C3C22×C3×D311yescharacteristic
224C22×D3—32no—
225C22×D3—61no—
226C22×D3—62no—
227C22×D3—62no—
228C22×D3—61no—
229C22×D3—92no—
230C22×D3—181no—
231C22×D3—182no—
232C22×D3—182no—
233C22×D3—181no—
234C3×Q8—188no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—32no—
240C2×C3×C4—32no—
241C2×C3×C4—62no—
242C2×C3×C4—62no—
243C2×C3×C4—62no—
244C2×C3×C4—62no—
245C2×C3×C4—62no—
246C2×C3×C4—62no—
247C2×C3×C4—62no—
248C2×C3×C4—62no—
249C2×C3×C4—64no—
250C2×C3×C4—62no—
251C2×C3×C4—94no—
252C2×C3×C4—122no—
253C2×C3×C4—122no—
254C2×C3×C4—122no—
255C2×C3×C4—182no—
256C2×C3×C4—182no—
257C2×C3×C4—181no—
258C2×C3×C4—182no—
259C2×C3×C4C3×D1212yes—
260C2×Dic3—34no—
261C2×Dic3—62no—
262C2×Dic3—62no—
263C2×Dic3—61no—
264C2×Dic3—62no—
265C2×Dic3—94no—
266C2×Dic3—182no—
267C2×Dic3—182no—
268C2×Dic3—181no—
269C2×Dic3—182no—
270Dic6—68no—
271Dic6—188no—
272C2×C32—21no—
273C2×C32—22no—
274C2×C32—22no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32#87511yescharacteristic
278C2×C32#88211yescharacteristic
279C2×C32#88512yes—
280C2×C32#89612yes—
281C2×C32#89711yescharacteristic
282C2×C32C3×#12011yescharacteristic
283C2×C32C3×C22⋊Q812yes—
284C2×C32C3×C22⋊Q811yescharacteristic
285C2×C32C3×#12412yes—
286C2×C32C3×#12411yescharacteristic
287C3⋊D3—42no—
288C3×D3—122no—
289C24—91no—
290C2×Q8—184no—
291C2×Q8—182no—
292C22×C4—92no—
293C22×C4—91no—
294C22×C4C2×(C3⋊D3)⋊C312yes—
295C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
296C22⋊C4—184no—
297C22⋊C4—182no—
298C22×C3—31no—
299C22×C3—31no—
300C22×C3—32no—
301C22×C3—32no—
302C22×C3—32no—
303C22×C3—32no—
304C22×C3—31no—
305C22×C3—31no—
306C22×C3—31no—
307C22×C3—31no—
308C22×C3—61no—
309C22×C3—62no—
310C22×C3—62no—
311C22×C3—61no—
312C22×C3—61no—
313C22×C3—184no—
314C22×C3—182no—
315C22×C3—361no—
316C22×C3—361no—
317C22×C3—362no—
318C22×C3C2×C3×D1211yescharacteristic
319C22×C3C3×#23512yes—
320C22×C3C3×D3×D412yes—
321C22×C3C3×#23711yescharacteristic
322C22×C3C3×D3×Q811yescharacteristic
323C2×D3—64no—
324C2×D3—62no—
325C2×D3—121no—
326C2×D3—122no—
327C2×D3—121no—
328C2×D3—184no—
329C2×D3—182no—
330C2×D3—361no—
331C2×D3—361no—
332C2×D3—362no—
333C3×C4—22no—
334C3×C4—42no—
335C3×C4—62no—
336C3×C4—62no—
337C3×C4—122no—
338C3×C4—122no—
339C3×C4—122no—
340C3×C4—124no—
341C3×C4—184no—
342C3×C4—361no—
343Dic3—64no—
344Dic3—121no—
345Dic3—184no—
346Dic3—361no—
347C32—21no—
348C32#467311yescharacteristic, complemented
349C32C3×#39311yescharacteristic, complemented
350C23—92no—
351C23—181no—
352C23—182no—
353C23—182no—
354C23—181no—
355C23C22×(C3⋊D3)⋊C311yescharacteristic
356Q8—188no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—94no—
362C2×C4—182no—
363C2×C4—182no—
364C2×C4—181no—
365C2×C4—182no—
366C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—32no—
372C2×C3—32no—
373C2×C3—32no—
374C2×C3—32no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—61no—
378C2×C3—61no—
379C2×C3—62no—
380C2×C3—62no—
381C2×C3—61no—
382C2×C3—362no—
383C2×C3C3×#87511yescharacteristic
384C2×C3C3×#88211yescharacteristic
385C2×C3C3×#88512yes—
386C2×C3C3×#89612yes—
387C2×C3C3×#89711yescharacteristic
388D3—122no—
389D3—362no—
390C22—184no—
391C22—182no—
392C22—361no—
393C22—361no—
394C22—362no—
395C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
396C22#3828⋊C312yes—
397C22D4×(C3⋊D3)⋊C312yes—
398C22#3830⋊C311yescharacteristic
399C22Q8×(C3⋊D3)⋊C311yescharacteristic
400C4—22no—
401C4—42no—
402C4—184no—
403C4—361no—
404C3—31no—
405C3—31no—
406C3—61no—
407C3C3×#467311yescharacteristic
408C2—362no—
409C2#64084⋊C311yescharacteristic
410C2#64091⋊C311yescharacteristic
411C2#64094⋊C312yes—
412C2#64105⋊C312yes—
413C2#64106⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC23×C32C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingH(3,𝔽3)×#95C2
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenter#95(C3⋊D3)⋊C3
hypocenterC32C3×#393
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
1. C22×#634⋊C3C2semidirect
4. H(3,𝔽3)×#95C2semidirect
5. #63658⋊C3C2semidirect
6. #10592372C3semidirect
17. C22×#634C2×C3semidirect
24. C32×#95D3semidirect
25. C32×#95C2×C3semidirect
26. #63658C2×C3semidirect
33. C22×(C3⋊D3)⋊C3Q8semidirect
105. C22×C3⋊D3C3×Q8semidirect
213. #95(C3⋊D3)⋊C3semidirect
214. H(3,𝔽3)#393semidirect
348. C32#4673semidirect
349. C32C3×#393semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
C2×#3752⋊C32
H(3,𝔽3)×#952
#63658⋊C32
#105923723
C3×#46733