Finite Group Databasebeta

← #10592363⋊C3 #13141354

Subgroups — #10592363⋊C3 #13141354

Order 1728.

839 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#3752⋊C3C212yesmaximal
3H(3,𝔽3)×#95C211yesmaximal, characteristic, complemented
4#63658⋊C3C211yesmaximal, characteristic, complemented
5#63658⋊C3C212yesmaximal, complemented
6#10592363C311yesmaximal, characteristic, complemented
7C3×#4664—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2212yes—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3—24no—
14C2×C4×(C3⋊D3)⋊C3C414yescomplemented
15#3752⋊C3—24no—
16C22×C4×C3⋊D3C2×C311yescharacteristic
17C22×C3×C4×D3—31no—
18C2×#3752C2×C312yes—
19C2×C3×(C2×D3)⋊C4—32no—
20C32×#95—21no—
21C32×#95D311yescharacteristic, complemented
22C32×#95C2×C311yescharacteristic, complemented
23#63658C2×C311yescharacteristic, complemented
24#63658C2×C312yescomplemented
25C3×C3⋊#95—32no—
26C3×C3⋊#95—31no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C22×(C3⋊D3)⋊C3—22no—
29C22×(C3⋊D3)⋊C3—21no—
30C22×(C3⋊D3)⋊C3—21no—
31C22×(C3⋊D3)⋊C3C2×C412yes—
32C22×(C3⋊D3)⋊C3D414yes—
33C2×C4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)—22no—
35C2×C4×H(3,𝔽3)—22no—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)C2×C412yes—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3C2×C412yes—
43C4×(C3⋊D3)⋊C3—28no—
44C3×#381—91no—
45#4664—31no—
46#4664—91no—
47C23×C3⋊D3C22×C311yescharacteristic
48C23×C3×D3—31no—
49C22×C32×C4—22no—
50C22×C32×C4—21no—
51C22×C32×C4C2×D312yes—
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C22×C312yes—
54C22×C32×C4C22×C311yescharacteristic
55C22×C3⋊Dic3C22×C312yes—
56C22×C3⋊Dic3C22×C311yescharacteristic
57C2×C4×C3⋊D3—24no—
58C2×C4×C3⋊D3C3×C414yescomplemented
59C22×C3×Dic3—31no—
60C22×C3×Dic3—32no—
61C2×C3×C4×D3—34no—
62C2×C3×C4×D3—64no—
63#3752—24no—
64C3×(C2×D3)⋊C4—64no—
65C22×H(3,𝔽3)C22×C411yescharacteristic
66C22×H(3,𝔽3)C2×D412yes—
67C22×H(3,𝔽3)#4112yes—
68C22×H(3,𝔽3)#4111yescharacteristic
69C22×H(3,𝔽3)#4111yescharacteristic
70C2×(C3⋊D3)⋊C3—22no—
71C2×(C3⋊D3)⋊C3—22no—
72C2×(C3⋊D3)⋊C3—24no—
73C2×(C3⋊D3)⋊C3—24no—
74C2×(C3⋊D3)⋊C3C22⋊C414yes—
75C4×H(3,𝔽3)—22no—
76C4×H(3,𝔽3)—42no—
77(C3⋊Dic3)⋊C3—22no—
78(C3⋊Dic3)⋊C3—42no—
79C23×C3×C4—91no—
80C22×C4×D3—31no—
81C22×C4×D3—91no—
82C2×C3×C22⋊C4—92no—
83C2×(C2×D3)⋊C4—32no—
84C2×(C2×D3)⋊C4—92no—
85C3×#95—31no—
86C3×#95—31no—
87C3×#95—61no—
88C3×#95—91no—
89C3×#95—92no—
90C3×#95C3×D311yescharacteristic
91C3⋊#95—32no—
92C3⋊#95—31no—
93C3⋊#95—92no—
94C3⋊#95—91no—
95C23×C32—21no—
96C23×C32C22×D311yescharacteristic
97C23×C32C23×C311yescharacteristic
98C22×C3⋊D3—22no—
99C22×C3⋊D3—21no—
100C22×C3⋊D3—21no—
101C22×C3⋊D3C2×C3×C412yes—
102C22×C3⋊D3C3×D414yes—
103C22×C3×D3—32no—
104C22×C3×D3—34no—
105C22×C3×D3—62no—
106C22×C3×D3—61no—
107C22×C3×D3—61no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—24no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—42no—
119C2×C32×C4—42no—
120C2×C32×C4—42no—
121C2×C32×C4C4×D312yes—
122C2×C32×C4C2×C3×C412yes—
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3—22no—
126C2×C3⋊Dic3—22no—
127C2×C3⋊Dic3C2×C3×C412yes—
128C4×C3⋊D3—28no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—62no—
134C3×C4×D3—68no—
135#381—91noSylow
136C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
137C2×H(3,𝔽3)#11711yescharacteristic
138C2×H(3,𝔽3)#11711yescharacteristic
139C2×H(3,𝔽3)#12312yes—
140C2×H(3,𝔽3)#12312yes—
141(C3⋊D3)⋊C3—24no—
142C24×C3—91no—
143C23×D3—31no—
144C23×D3—91no—
145C22×C3×C4—31no—
146C22×C3×C4—31no—
147C22×C3×C4—32no—
148C22×C3×C4—32no—
149C22×C3×C4—62no—
150C22×C3×C4—61no—
151C22×C3×C4—94no—
152C22×C3×C4—91no—
153C22×C3×C4—92no—
154C22×C3×C4—184no—
155C22×C3×C4C2×C3×D312yes—
156C22×C3×C4C2×C3×D311yescharacteristic
157C22×Dic3—31no—
158C22×Dic3—32no—
159C22×Dic3—91no—
160C22×Dic3—92no—
161C2×C4×D3—34no—
162C2×C4×D3—64no—
163C2×C4×D3—94no—
164C2×C4×D3—184no—
165C3×C22⋊C4—184no—
166(C2×D3)⋊C4—64no—
167(C2×D3)⋊C4—184no—
168C22×C32—21no—
169C22×C32—22no—
170C22×C32—22no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32C2×C4×D311yescharacteristic
174C22×C32#23512yes—
175C22×C32D3×D412yes—
176C22×C32#23711yescharacteristic
177C22×C32#23911yescharacteristic
178C22×C32C22×C3×C411yescharacteristic
179C22×C32C2×C3×D412yes—
180C22×C32C3×#4112yes—
181C22×C32C3×#4111yescharacteristic
182C22×C32C3×#4111yescharacteristic
183C2×C3⋊D3—22no—
184C2×C3⋊D3—22no—
185C2×C3⋊D3—24no—
186C2×C3⋊D3—24no—
187C2×C3⋊D3C3×C22⋊C414yes—
188C2×C3×D3—34no—
189C2×C3×D3—62no—
190C2×C3×D3—64no—
191C2×C3×D3—62no—
192C2×C3×D3—64no—
193C32×C4—22no—
194C32×C4—22no—
195C32×C4—42no—
196C32×C4—44no—
197C32×C4—42no—
198C32×C4—42no—
199C3⋊Dic3—22no—
200C3⋊Dic3—42no—
201C3×Dic3—62no—
202C3×Dic3—122no—
203C23×C4—91no—
204C2×C22⋊C4—92no—
205#95—92no—
206#95—91no—
207#95(C3⋊D3)⋊C311yescharacteristic, complemented
208H(3,𝔽3)#38111yescharacteristic, complemented, Sylow
209C23×C3—31no—
210C23×C3—31no—
211C23×C3—61no—
212C23×C3—94no—
213C23×C3—92no—
214C23×C3—182no—
215C23×C3—181no—
216C23×C3—181no—
217C23×C3C22×C3×D311yescharacteristic
218C22×D3—32no—
219C22×D3—34no—
220C22×D3—62no—
221C22×D3—61no—
222C22×D3—61no—
223C22×D3—92no—
224C22×D3—94no—
225C22×D3—182no—
226C22×D3—181no—
227C22×D3—181no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—32no—
233C2×C3×C4—32no—
234C2×C3×C4—62no—
235C2×C3×C4—62no—
236C2×C3×C4—62no—
237C2×C3×C4—62no—
238C2×C3×C4—62no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—62no—
242C2×C3×C4—64no—
243C2×C3×C4—62no—
244C2×C3×C4—92no—
245C2×C3×C4—122no—
246C2×C3×C4—122no—
247C2×C3×C4—122no—
248C2×C3×C4—188no—
249C2×C3×C4—182no—
250C2×C3×C4—182no—
251C2×C3×C4—182no—
252C2×C3×C4—182no—
253C2×C3×C4C3×C4×D312yes—
254C2×Dic3—32no—
255C2×Dic3—62no—
256C2×Dic3—62no—
257C2×Dic3—62no—
258C2×Dic3—62no—
259C2×Dic3—92no—
260C2×Dic3—182no—
261C2×Dic3—182no—
262C2×Dic3—182no—
263C2×Dic3—182no—
264C4×D3—68no—
265C4×D3—188no—
266C2×C32—21no—
267C2×C32—22no—
268C2×C32—21no—
269C2×C32—22no—
270C2×C32—21no—
271C2×C32#87211yescharacteristic
272C2×C32D3×C22⋊C411yescharacteristic
273C2×C32#88312yes—
274C2×C32#89211yescharacteristic
275C2×C32#89412yes—
276C2×C32C2×C3×C22⋊C411yescharacteristic
277C2×C32C3×#11711yescharacteristic
278C2×C32C3×#11711yescharacteristic
279C2×C32C3×#12312yes—
280C2×C32C3×#12312yes—
281C3⋊D3—24no—
282C3×D3—64no—
283C24—91no—
284C22×C4—91no—
285C22×C4—94no—
286C22×C4—92no—
287C22×C4—184no—
288C22×C4C2×(C3⋊D3)⋊C312yes—
289C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
290C22⋊C4—184no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—32no—
294C22×C3—32no—
295C22×C3—32no—
296C22×C3—32no—
297C22×C3—31no—
298C22×C3—31no—
299C22×C3—31no—
300C22×C3—31no—
301C22×C3—61no—
302C22×C3—62no—
303C22×C3—62no—
304C22×C3—61no—
305C22×C3—61no—
306C22×C3—94no—
307C22×C3—182no—
308C22×C3—184no—
309C22×C3—182no—
310C22×C3—184no—
311C22×C3C2×C3×C4×D311yescharacteristic
312C22×C3C3×#23512yes—
313C22×C3C3×D3×D412yes—
314C22×C3C3×#23711yescharacteristic
315C22×C3C3×#23911yescharacteristic
316C2×D3—34no—
317C2×D3—64no—
318C2×D3—62no—
319C2×D3—64no—
320C2×D3—62no—
321C2×D3—94no—
322C2×D3—182no—
323C2×D3—184no—
324C2×D3—182no—
325C2×D3—184no—
326C3×C4—22no—
327C3×C4—42no—
328C3×C4—62no—
329C3×C4—62no—
330C3×C4—122no—
331C3×C4—122no—
332C3×C4—122no—
333C3×C4—124no—
334C3×C4—182no—
335C3×C4—362no—
336Dic3—62no—
337Dic3—122no—
338Dic3—182no—
339Dic3—362no—
340C32—21no—
341C32#466411yescharacteristic, complemented
342C32C3×#38111yescharacteristic, complemented
343C23—92no—
344C23—94no—
345C23—182no—
346C23—181no—
347C23—181no—
348C23C22×(C3⋊D3)⋊C311yescharacteristic
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—92no—
354C2×C4—188no—
355C2×C4—182no—
356C2×C4—182no—
357C2×C4—182no—
358C2×C4—182no—
359C2×C4C4×(C3⋊D3)⋊C312yes—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—32no—
365C2×C3—32no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—32no—
369C2×C3—32no—
370C2×C3—61no—
371C2×C3—61no—
372C2×C3—62no—
373C2×C3—61no—
374C2×C3—62no—
375C2×C3—184no—
376C2×C3C3×#87211yescharacteristic
377C2×C3C3×D3×C22⋊C411yescharacteristic
378C2×C3C3×#88312yes—
379C2×C3C3×#89211yescharacteristic
380C2×C3C3×#89412yes—
381D3—64no—
382D3—184no—
383C22—94no—
384C22—182no—
385C22—184no—
386C22—182no—
387C22—184no—
388C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
389C22#3828⋊C312yes—
390C22D4×(C3⋊D3)⋊C312yes—
391C22#3830⋊C311yescharacteristic
392C22#8872211yescharacteristic
393C4—22no—
394C4—42no—
395C4—182no—
396C4—362no—
397C3—31no—
398C3—31no—
399C3—61no—
400C3C3×#466411yescharacteristic
401C2—184no—
402C2#64081⋊C311yescharacteristic
403C2C22⋊C4×(C3⋊D3)⋊C311yescharacteristic
404C2#64092⋊C312yes—
405C2#64101⋊C311yescharacteristic
406C2#64103⋊C312yes—
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
FittingH(3,𝔽3)×#95C2
prime radicalC23×(C3⋊D3)⋊C3C22
cosocleC23×C32C23×C3
hypercenter#95(C3⋊D3)⋊C3
hypocenterC32C3×#381
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
3. H(3,𝔽3)×#95C2semidirect
4. #63658⋊C3C2semidirect
5. #63658⋊C3C2semidirect
6. #10592363C3semidirect
14. C2×C4×(C3⋊D3)⋊C3C4semidirect
21. C32×#95D3semidirect
22. C32×#95C2×C3semidirect
23. #63658C2×C3semidirect
24. #63658C2×C3semidirect
58. C2×C4×C3⋊D3C3×C4semidirect
207. #95(C3⋊D3)⋊C3semidirect
208. H(3,𝔽3)#381semidirect
341. C32#4664semidirect
342. C32C3×#381semidirect

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
C2×#3752⋊C32
H(3,𝔽3)×#952
#63658⋊C32
#105923633
C3×#46643