Finite Group Databasebeta

← #10592366⋊C3 #13141333

Subgroups — #10592366⋊C3 #13141333

Order 1728.

895 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×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#3752⋊C3C212yesmaximal
3C2×#3750⋊C3C212yesmaximal, complemented
4H(3,𝔽3)×#95C211yesmaximal, characteristic, complemented
5#63658⋊C3C211yesmaximal, characteristic, complemented
6#10592366C311yesmaximal, characteristic, complemented
7C3×#4667—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—
16#3750⋊C3—24no—
17C22×C4×C3⋊D3C2×C311yescharacteristic
18C22×C3×C4×D3—31no—
19C2×#3752C2×C312yes—
20C2×#3750C2×C312yescomplemented
21C2×C3×(C2×D3)⋊C4—32no—
22C2×C3×Dic3⋊C4—32no—
23C32×#95—21no—
24C32×#95D311yescharacteristic, complemented
25C32×#95C2×C311yescharacteristic, complemented
26#63658C2×C311yescharacteristic, complemented
27C3×C3⋊#95—31no—
28C23×H(3,𝔽3)C2311yescharacteristic
29C22×(C3⋊D3)⋊C3—22no—
30C22×(C3⋊D3)⋊C3—22no—
31C22×(C3⋊D3)⋊C3C2×C412yes—
32C22×(C3⋊D3)⋊C3D412yes—
33C22×(C3⋊D3)⋊C3Q812yes—
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)C2×C412yes—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3C2×C412yes—
43C2×(C3⋊Dic3)⋊C3D414yes—
44C4×(C3⋊D3)⋊C3—28no—
45C3×#380—91no—
46#4667—31no—
47#4667—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×C312yes—
57C22×C3⋊Dic3C22×C311yescharacteristic
58C2×C4×C3⋊D3—24no—
59C2×C4×C3⋊D3C3×C414yescomplemented
60C22×C3×Dic3—31no—
61C22×C3×Dic3—32no—
62C2×C3×C4×D3—34no—
63C2×C3×C4×D3—64no—
64#3752—24no—
65#3750—24no—
66C3×(C2×D3)⋊C4—64no—
67C3×Dic3⋊C4—64no—
68C22×H(3,𝔽3)C22×C411yescharacteristic
69C22×H(3,𝔽3)C2×D412yes—
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C22×H(3,𝔽3)C2×Q811yescharacteristic
72C22×H(3,𝔽3)#4112yes—
73C2×(C3⋊D3)⋊C3—22no—
74C2×(C3⋊D3)⋊C3—22no—
75C2×(C3⋊D3)⋊C3—24no—
76C2×(C3⋊D3)⋊C3—24no—
77C2×(C3⋊D3)⋊C3C4⋊C414yescomplemented
78C4×H(3,𝔽3)—22no—
79C4×H(3,𝔽3)—42no—
80(C3⋊Dic3)⋊C3—42no—
81(C3⋊Dic3)⋊C3C22⋊C414yescomplemented
82C23×C3×C4—91no—
83C22×C4×D3—31no—
84C22×C4×D3—91no—
85C2×C3×C4⋊C4—92no—
86C2×C3×C22⋊C4—92no—
87C2×(C2×D3)⋊C4—32no—
88C2×(C2×D3)⋊C4—92no—
89C2×Dic3⋊C4—32no—
90C2×Dic3⋊C4—92no—
91C3×#95—31no—
92C3×#95—31no—
93C3×#95—61no—
94C3×#95—91no—
95C3×#95C3×D311yescharacteristic
96C3⋊#95—31no—
97C3⋊#95—91no—
98C23×C32—21no—
99C23×C32C22×D311yescharacteristic
100C23×C32C23×C311yescharacteristic
101C22×C3⋊D3—22no—
102C22×C3⋊D3—22no—
103C22×C3⋊D3C2×C3×C412yes—
104C22×C3⋊D3C3×D412yes—
105C22×C3⋊D3C3×Q812yes—
106C22×C3×D3—32no—
107C22×C3×D3—32no—
108C22×C3×D3—32no—
109C22×C3×D3—62no—
110C22×C3×D3—62no—
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×C4C4×D312yes—
125C2×C32×C4C2×C3×C412yes—
126C2×C3⋊Dic3—22no—
127C2×C3⋊Dic3—22no—
128C2×C3⋊Dic3—22no—
129C2×C3⋊Dic3C2×C3×C412yes—
130C2×C3⋊Dic3C3×D414yes—
131C4×C3⋊D3—28no—
132C2×C3×Dic3—32no—
133C2×C3×Dic3—34no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—62no—
137C3×C4×D3—68no—
138#380—91noSylow
139C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
140C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
141C2×H(3,𝔽3)#11711yescharacteristic
142C2×H(3,𝔽3)#12112yes—
143C2×H(3,𝔽3)C22⋊Q812yes—
144(C3⋊D3)⋊C3—24no—
145C24×C3—91no—
146C23×D3—31no—
147C23×D3—91no—
148C22×C3×C4—31no—
149C22×C3×C4—31no—
150C22×C3×C4—32no—
151C22×C3×C4—32no—
152C22×C3×C4—62no—
153C22×C3×C4—61no—
154C22×C3×C4—94no—
155C22×C3×C4—91no—
156C22×C3×C4—92no—
157C22×C3×C4—184no—
158C22×C3×C4C2×C3×D312yes—
159C22×C3×C4C2×C3×D311yescharacteristic
160C22×Dic3—31no—
161C22×Dic3—32no—
162C22×Dic3—91no—
163C22×Dic3—92no—
164C2×C4×D3—34no—
165C2×C4×D3—64no—
166C2×C4×D3—94no—
167C2×C4×D3—184no—
168C3×C4⋊C4—184no—
169C3×C22⋊C4—184no—
170(C2×D3)⋊C4—64no—
171(C2×D3)⋊C4—184no—
172Dic3⋊C4—64no—
173Dic3⋊C4—184no—
174C22×C32—21no—
175C22×C32—22no—
176C22×C32—22no—
177C22×C32—21no—
178C22×C32—21no—
179C22×C32C2×C4×D311yescharacteristic
180C22×C32#23512yes—
181C22×C32D3×D412yes—
182C22×C32D3×D411yescharacteristic
183C22×C32D3×Q811yescharacteristic
184C22×C32C22×C3×C411yescharacteristic
185C22×C32C2×C3×D412yes—
186C22×C32C2×C3×D411yescharacteristic
187C22×C32C2×C3×Q811yescharacteristic
188C22×C32C3×#4112yes—
189C2×C3⋊D3—22no—
190C2×C3⋊D3—22no—
191C2×C3⋊D3—24no—
192C2×C3⋊D3—24no—
193C2×C3⋊D3C3×C4⋊C414yescomplemented
194C2×C3×D3—34no—
195C2×C3×D3—62no—
196C2×C3×D3—62no—
197C2×C3×D3—64no—
198C2×C3×D3—64no—
199C32×C4—22no—
200C32×C4—22no—
201C32×C4—42no—
202C32×C4—44no—
203C32×C4—42no—
204C32×C4—42no—
205C3⋊Dic3—42no—
206C3⋊Dic3C3×C22⋊C414yescomplemented
207C3×Dic3—34no—
208C3×Dic3—122no—
209C23×C4—91no—
210C2×C4⋊C4—92no—
211C2×C22⋊C4—92no—
212#95—91no—
213#95(C3⋊D3)⋊C311yescharacteristic, complemented
214H(3,𝔽3)#38011yescharacteristic, complemented, Sylow
215C23×C3—31no—
216C23×C3—31no—
217C23×C3—61no—
218C23×C3—92no—
219C23×C3—92no—
220C23×C3—92no—
221C23×C3—182no—
222C23×C3—182no—
223C23×C3C22×C3×D311yescharacteristic
224C22×D3—32no—
225C22×D3—32no—
226C22×D3—32no—
227C22×D3—62no—
228C22×D3—62no—
229C22×D3—92no—
230C22×D3—92no—
231C22×D3—92no—
232C22×D3—182no—
233C22×D3—182no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—32no—
239C2×C3×C4—32no—
240C2×C3×C4—62no—
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—64no—
249C2×C3×C4—62no—
250C2×C3×C4—94no—
251C2×C3×C4—92no—
252C2×C3×C4—122no—
253C2×C3×C4—122no—
254C2×C3×C4—122no—
255C2×C3×C4—188no—
256C2×C3×C4—182no—
257C2×C3×C4—182no—
258C2×C3×C4—182no—
259C2×C3×C4C3×C4×D312yes—
260C2×Dic3—32no—
261C2×Dic3—34no—
262C2×Dic3—62no—
263C2×Dic3—62no—
264C2×Dic3—62no—
265C2×Dic3—92no—
266C2×Dic3—94no—
267C2×Dic3—182no—
268C2×Dic3—182no—
269C2×Dic3—182no—
270C4×D3—68no—
271C4×D3—188no—
272C2×C32—21no—
273C2×C32—22no—
274C2×C32—21no—
275C2×C32—22no—
276C2×C32—21no—
277C2×C32#87211yescharacteristic
278C2×C32D3×C22⋊C411yescharacteristic
279C2×C32#88412yes—
280C2×C32D3×C4⋊C411yescharacteristic
281C2×C32#89612yes—
282C2×C32C2×C3×C22⋊C411yescharacteristic
283C2×C32C2×C3×C4⋊C411yescharacteristic
284C2×C32C3×#11711yescharacteristic
285C2×C32C3×#12112yes—
286C2×C32C3×C22⋊Q812yes—
287C3⋊D3—24no—
288C3×D3—64no—
289C24—91no—
290C22×C4—91no—
291C22×C4—94no—
292C22×C4—92no—
293C22×C4—184no—
294C22×C4C2×(C3⋊D3)⋊C312yes—
295C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
296C4⋊C4—184no—
297C22⋊C4—184no—
298C22×C3—31no—
299C22×C3—31no—
300C22×C3—32no—
301C22×C3—32no—
302C22×C3—32no—
303C22×C3—31no—
304C22×C3—32no—
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—94no—
314C22×C3—182no—
315C22×C3—182no—
316C22×C3—184no—
317C22×C3—184no—
318C22×C3C2×C3×C4×D311yescharacteristic
319C22×C3C3×#23512yes—
320C22×C3C3×D3×D412yes—
321C22×C3C3×D3×D411yescharacteristic
322C22×C3C3×D3×Q811yescharacteristic
323C2×D3—34no—
324C2×D3—62no—
325C2×D3—64no—
326C2×D3—64no—
327C2×D3—62no—
328C2×D3—94no—
329C2×D3—182no—
330C2×D3—182no—
331C2×D3—184no—
332C2×D3—184no—
333C3×C4—22no—
334C3×C4—42no—
335C3×C4—62no—
336C3×C4—62no—
337C3×C4—94no—
338C3×C4—122no—
339C3×C4—122no—
340C3×C4—122no—
341C3×C4—124no—
342C3×C4—362no—
343Dic3—34no—
344Dic3—94no—
345Dic3—122no—
346Dic3—362no—
347C32—21no—
348C32#466711yescharacteristic, complemented
349C32C3×#38011yescharacteristic, complemented
350C23—92no—
351C23—92no—
352C23—92no—
353C23—182no—
354C23—182no—
355C23C22×(C3⋊D3)⋊C311yescharacteristic
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—92no—
361C2×C4—94no—
362C2×C4—188no—
363C2×C4—182no—
364C2×C4—182no—
365C2×C4—182no—
366C2×C4C4×(C3⋊D3)⋊C312yes—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—32no—
372C2×C3—32no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—32no—
376C2×C3—32no—
377C2×C3—61no—
378C2×C3—61no—
379C2×C3—62no—
380C2×C3—61no—
381C2×C3—62no—
382C2×C3—184no—
383C2×C3C3×#87211yescharacteristic
384C2×C3C3×D3×C22⋊C411yescharacteristic
385C2×C3C3×#88412yes—
386C2×C3C3×D3×C4⋊C411yescharacteristic
387C2×C3C3×#89612yes—
388D3—64no—
389D3—184no—
390C22—94no—
391C22—182no—
392C22—182no—
393C22—184no—
394C22—184no—
395C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
396C22#3828⋊C312yes—
397C22D4×(C3⋊D3)⋊C312yes—
398C22D4×(C3⋊D3)⋊C311yescharacteristic
399C22Q8×(C3⋊D3)⋊C311yescharacteristic
400C4—22no—
401C4—42no—
402C4—94no—
403C4—362no—
404C3—31no—
405C3—31no—
406C3—61no—
407C3C3×#466711yescharacteristic
408C2—184no—
409C2#64081⋊C311yescharacteristic
410C2C22⋊C4×(C3⋊D3)⋊C311yescharacteristic
411C2#64093⋊C312yes—
412C2C4⋊C4×(C3⋊D3)⋊C311yescharacteristic
413C2#64105⋊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×#380
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
3. C2×#3750⋊C3C2semidirect
4. H(3,𝔽3)×#95C2semidirect
5. #63658⋊C3C2semidirect
6. #10592366C3semidirect
14. C2×C4×(C3⋊D3)⋊C3C4semidirect
20. C2×#3750C2×C3semidirect
24. C32×#95D3semidirect
25. C32×#95C2×C3semidirect
26. #63658C2×C3semidirect
59. C2×C4×C3⋊D3C3×C4semidirect
77. C2×(C3⋊D3)⋊C3C4⋊C4semidirect
81. (C3⋊Dic3)⋊C3C22⋊C4semidirect
193. C2×C3⋊D3C3×C4⋊C4semidirect
206. C3⋊Dic3C3×C22⋊C4semidirect
213. #95(C3⋊D3)⋊C3semidirect
214. H(3,𝔽3)#380semidirect
348. C32#4667semidirect
349. C32C3×#380semidirect

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
C2×#3750⋊C32
H(3,𝔽3)×#952
#63658⋊C32
#105923663
C3×#46673