Finite Group Databasebeta

← #10592645⋊C3 #13142495

Subgroups — #10592645⋊C3 #13142495

Order 1728.

951 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
3C2×#3757⋊C3C211yesmaximal, characteristic
4C2×#3750⋊C3C212yesmaximal, complemented
5#63658⋊C3C212yesmaximal, complemented
6#10592645C311yesmaximal, characteristic, complemented
7C3×#4946—31nomaximal
8C24×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3—22no—
12C22×(C3⋊Dic3)⋊C3—24no—
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C22×(C3⋊Dic3)⋊C3C2212yes—
15C22⋊C4×H(3,𝔽3)—22no—
16#3757⋊C3C414yescomplemented
17#3750⋊C3—24no—
18C23×C3⋊Dic3C2×C311yescharacteristic
19C23×C3×Dic3—31no—
20C2×C32×C22⋊C4—21no—
21C2×C32×C22⋊C4D311yescharacteristic
22C2×C32×C22⋊C4C2×C311yescharacteristic
23C2×#3757C2×C311yescharacteristic
24C2×#3750C2×C312yescomplemented
25C2×C3×#217—31no—
26C2×C3×Dic3⋊C4—32no—
27#63658C2×C312yescomplemented
28C3×C3⋊#95—32no—
29C23×H(3,𝔽3)—22no—
30C23×H(3,𝔽3)—22no—
31C23×H(3,𝔽3)C2×C412yes—
32C23×H(3,𝔽3)D412yes—
33C23×H(3,𝔽3)Q812yes—
34C23×H(3,𝔽3)C2311yescharacteristic
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)—22no—
38C2×C4×H(3,𝔽3)—22no—
39C2×(C3⋊Dic3)⋊C3—28no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—22no—
43C2×(C3⋊Dic3)⋊C3—22no—
44C2×(C3⋊Dic3)⋊C3C2×C414yescomplemented
45C2×(C3⋊Dic3)⋊C3D414yes—
46C3×#385—91no—
47#4946—31no—
48#4946—91no—
49C24×C32—21no—
50C24×C32C2×D311yescharacteristic
51C24×C32C22×C311yescharacteristic
52C22×C32×C4—21no—
53C22×C32×C4—21no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C22×C311yescharacteristic
57C22×C32×C4C22×C311yescharacteristic
58C22×C3⋊Dic3—22no—
59C22×C3⋊Dic3—24no—
60C22×C3⋊Dic3C22×C312yes—
61C22×C3⋊Dic3C22×C312yes—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—64no—
65C22×C3×Dic3—62no—
66C32×C22⋊C4—24no—
67C32×C22⋊C4—22no—
68C32×C22⋊C4—22no—
69#3757C3×C414yescomplemented
70#3750—24no—
71C3×#217—34no—
72C3×Dic3⋊C4—64no—
73C22×H(3,𝔽3)—24no—
74C22×H(3,𝔽3)—22no—
75C22×H(3,𝔽3)—24no—
76C22×H(3,𝔽3)—22no—
77C22×H(3,𝔽3)C4⋊C414yescomplemented
78C22×H(3,𝔽3)C22×C411yescharacteristic
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C22×H(3,𝔽3)C2×D412yes—
81C22×H(3,𝔽3)C2×Q811yescharacteristic
82C22×H(3,𝔽3)#4112yes—
83C4×H(3,𝔽3)—41no—
84C4×H(3,𝔽3)—41no—
85(C3⋊Dic3)⋊C3—24no—
86(C3⋊Dic3)⋊C3—24no—
87C23×C3×C4—91no—
88C23×Dic3—31no—
89C23×Dic3—91no—
90C2×C3×C4⋊C4—92no—
91C2×C3×C22⋊C4—31no—
92C2×C3×C22⋊C4—31no—
93C2×C3×C22⋊C4—61no—
94C2×C3×C22⋊C4—91no—
95C2×C3×C22⋊C4C3×D311yescharacteristic
96C2×#217—31no—
97C2×#217—91no—
98C2×Dic3⋊C4—32no—
99C2×Dic3⋊C4—92no—
100C3×#95—92no—
101C3⋊#95—32no—
102C3⋊#95—92no—
103C23×C32—22no—
104C23×C32—22no—
105C23×C32—22no—
106C23×C32—22no—
107C23×C32—22no—
108C23×C32—22no—
109C23×C32—22no—
110C23×C32—21no—
111C23×C32—42no—
112C23×C32—42no—
113C23×C32Dic612yes—
114C23×C32C4×D312yes—
115C23×C32#6712yes—
116C23×C32C2×C3×C412yes—
117C23×C32C3×D412yes—
118C23×C32C3×Q812yes—
119C23×C32C22×D311yescharacteristic
120C23×C32C23×C311yescharacteristic
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—22no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—22no—
131C2×C32×C4—42no—
132C2×C32×C4—42no—
133C2×C3⋊Dic3—28no—
134C2×C3⋊Dic3—22no—
135C2×C3⋊Dic3—22no—
136C2×C3⋊Dic3—22no—
137C2×C3⋊Dic3—22no—
138C2×C3⋊Dic3C2×C3×C414yescomplemented
139C2×C3⋊Dic3C3×D414yes—
140C2×C3×Dic3—34no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—68no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147#385—91noSylow
148C2×H(3,𝔽3)—24no—
149C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
150C2×H(3,𝔽3)C4×D412yes—
151C2×H(3,𝔽3)#12011yescharacteristic
152C2×H(3,𝔽3)C22⋊Q812yes—
153C2×H(3,𝔽3)#12311yescharacteristic
154C24×C3—31no—
155C24×C3—31no—
156C24×C3—61no—
157C24×C3C2×C3×D311yescharacteristic
158C22×C3×C4—31no—
159C22×C3×C4—31no—
160C22×C3×C4—31no—
161C22×C3×C4—31no—
162C22×C3×C4—61no—
163C22×C3×C4—61no—
164C22×C3×C4—92no—
165C22×C3×C4—92no—
166C22×C3×C4—184no—
167C22×C3×C4—182no—
168C22×C3×C4C2×C3×D311yescharacteristic
169C22×C3×C4C2×C3×D311yescharacteristic
170C22×Dic3—32no—
171C22×Dic3—32no—
172C22×Dic3—64no—
173C22×Dic3—62no—
174C22×Dic3—92no—
175C22×Dic3—92no—
176C22×Dic3—184no—
177C22×Dic3—182no—
178C3×C4⋊C4—184no—
179C3×C22⋊C4—22no—
180C3×C22⋊C4—62no—
181C3×C22⋊C4—62no—
182C3×C22⋊C4—64no—
183C3×C22⋊C4—94no—
184#217—34no—
185#217—94no—
186Dic3⋊C4—64no—
187Dic3⋊C4—184no—
188C22×C32—24no—
189C22×C32—24no—
190C22×C32—22no—
191C22×C32—22no—
192C22×C32—24no—
193C22×C32—22no—
194C22×C32—24no—
195C22×C32—22no—
196C22×C32—24no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—22no—
200C22×C32—22no—
201C22×C32—21no—
202C22×C32—44no—
203C22×C32—44no—
204C22×C32—42no—
205C22×C32—42no—
206C22×C32Dic3⋊C414yescomplemented
207C22×C32C3×C4⋊C414yescomplemented
208C22×C32C2×Dic611yescharacteristic
209C22×C32C2×C4×D311yescharacteristic
210C22×C32D3×D412yes—
211C22×C32#23712yes—
212C22×C32C2×#6711yescharacteristic
213C22×C32C22×C3×C411yescharacteristic
214C22×C32C2×C3×D412yes—
215C22×C32C2×C3×D411yescharacteristic
216C22×C32C2×C3×Q811yescharacteristic
217C22×C32C3×#4112yes—
218C32×C4—42no—
219C32×C4—42no—
220C32×C4—41no—
221C32×C4—41no—
222C32×C4—41no—
223C32×C4—41no—
224C3⋊Dic3—24no—
225C3⋊Dic3—24no—
226C3×Dic3—64no—
227C3×Dic3—64no—
228C23×C4—91no—
229C2×C4⋊C4—92no—
230C2×C22⋊C4—91no—
231C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic
232#95—92no—
233H(3,𝔽3)#38511yescharacteristic, complemented, Sylow
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—32no—
237C23×C3—32no—
238C23×C3—32no—
239C23×C3—32no—
240C23×C3—32no—
241C23×C3—32no—
242C23×C3—31no—
243C23×C3—31no—
244C23×C3—62no—
245C23×C3—62no—
246C23×C3—62no—
247C23×C3—62no—
248C23×C3—62no—
249C23×C3—62no—
250C23×C3—62no—
251C23×C3—61no—
252C23×C3—122no—
253C23×C3—122no—
254C23×C3C3×Dic612yes—
255C23×C3C3×C4×D312yes—
256C23×C3C3×#6712yes—
257C23×C3C22×C3×D311yescharacteristic
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—62no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—62no—
268C2×C3×C4—62no—
269C2×C3×C4—62no—
270C2×C3×C4—62no—
271C2×C3×C4—62no—
272C2×C3×C4—94no—
273C2×C3×C4—94no—
274C2×C3×C4—122no—
275C2×C3×C4—122no—
276C2×C3×C4—182no—
277C2×C3×C4—182no—
278C2×C3×C4—182no—
279C2×C3×C4—182no—
280C2×C3×C4—188no—
281C2×Dic3—34no—
282C2×Dic3—34no—
283C2×Dic3—68no—
284C2×Dic3—62no—
285C2×Dic3—62no—
286C2×Dic3—62no—
287C2×Dic3—62no—
288C2×Dic3—94no—
289C2×Dic3—94no—
290C2×Dic3—182no—
291C2×Dic3—182no—
292C2×Dic3—182no—
293C2×Dic3—182no—
294C2×Dic3—188no—
295C2×C32—24no—
296C2×C32—24no—
297C2×C32—22no—
298C2×C32—22no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—44no—
303C2×C32#87812yes—
304C2×C32#67⋊C412yes—
305C2×C32C2×Dic3⋊C411yescharacteristic
306C2×C32#93511yescharacteristic
307C2×C32#93711yescharacteristic
308C2×C32C2×C3×C4⋊C411yescharacteristic
309C2×C32C3×C4×D412yes—
310C2×C32C3×#12011yescharacteristic
311C2×C32C3×C22⋊Q812yes—
312C2×C32C3×#12311yescharacteristic
313C24C2×(C3⋊D3)⋊C311yescharacteristic
314C22×C4—92no—
315C22×C4—92no—
316C22×C4—184no—
317C22×C4—182no—
318C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
319C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
320C4⋊C4—184no—
321C22⋊C4—22no—
322C22⋊C4—94no—
323C22×C3—24no—
324C22×C3—24no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—31no—
328C22×C3—31no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—32no—
332C22×C3—32no—
333C22×C3—32no—
334C22×C3—32no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—34no—
338C22×C3—34no—
339C22×C3—62no—
340C22×C3—62no—
341C22×C3—62no—
342C22×C3—62no—
343C22×C3—64no—
344C22×C3—64no—
345C22×C3—64no—
346C22×C3—64no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—62no—
350C22×C3—62no—
351C22×C3—61no—
352C22×C3—64no—
353C22×C3—122no—
354C22×C3—122no—
355C22×C3—124no—
356C22×C3—124no—
357C22×C3C3×Dic3⋊C414yes—
358C22×C3C2×C3×Dic611yescharacteristic
359C22×C3C2×C3×C4×D311yescharacteristic
360C22×C3C3×D3×D412yes—
361C22×C3C3×#23712yes—
362C22×C3C2×C3×#6711yescharacteristic
363C3×C4—41no—
364C3×C4—41no—
365C3×C4—121no—
366C3×C4—121no—
367C3×C4—121no—
368C3×C4—121no—
369C3×C4—122no—
370C3×C4—122no—
371C3×C4—184no—
372C3×C4—184no—
373Dic3—64no—
374Dic3—64no—
375Dic3—184no—
376Dic3—184no—
377C32—21no—
378C32#494611yescharacteristic, complemented
379C32C3×#38511yescharacteristic, complemented
380C23—22no—
381C23—22no—
382C23#634⋊C312yes—
383C23C4×(C3⋊D3)⋊C312yes—
384C23#638⋊C312yes—
385C23C22×(C3⋊D3)⋊C311yescharacteristic
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—94no—
391C2×C4—94no—
392C2×C4—188no—
393C2×C4—182no—
394C2×C4—182no—
395C2×C4—182no—
396C2×C4—182no—
397C2×C3—24no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—32no—
405C2×C3—32no—
406C2×C3—32no—
407C2×C3—32no—
408C2×C3—64no—
409C2×C3—64no—
410C2×C3—61no—
411C2×C3—61no—
412C2×C3—61no—
413C2×C3—62no—
414C2×C3—62no—
415C2×C3—124no—
416C2×C3C3×#87812yes—
417C2×C3C3×#67⋊C412yes—
418C2×C3C2×C3×Dic3⋊C411yescharacteristic
419C2×C3C3×#93511yescharacteristic
420C2×C3C3×#93711yescharacteristic
421C22—22no—
422C22—22no—
423C22—24no—
424C22—24no—
425C22#3750⋊C314yescomplemented
426C22C2×#634⋊C311yescharacteristic
427C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
428C22D4×(C3⋊D3)⋊C312yes—
429C22#3830⋊C312yes—
430C22C2×#638⋊C311yescharacteristic
431C4—41no—
432C4—41no—
433C4—184no—
434C4—184no—
435C3—31no—
436C3—31no—
437C3—61no—
438C3C3×#494611yescharacteristic
439C2—24no—
440C2#64087⋊C312yes—
441C2#64090⋊C312yes—
442C2C2×#3750⋊C311yescharacteristic
443C2#64144⋊C311yescharacteristic
444C2#64146⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC24×H(3,𝔽3)C22
cosocleC23×C32C23×C3
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#385
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×#3750⋊C3C2semidirect
5. #63658⋊C3C2semidirect
6. #10592645C3semidirect
16. #3757⋊C3C4semidirect
24. C2×#3750C2×C3semidirect
27. #63658C2×C3semidirect
44. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
69. #3757C3×C4semidirect
77. C22×H(3,𝔽3)C4⋊C4semidirect
138. C2×C3⋊Dic3C2×C3×C4semidirect
206. C22×C32Dic3⋊C4semidirect
207. C22×C32C3×C4⋊C4semidirect
233. H(3,𝔽3)#385semidirect
378. C32#4946semidirect
379. C32C3×#385semidirect
425. C22#3750⋊C3semidirect

Maximal subgroups

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

subgroupindex
C23×(C3⋊Dic3)⋊C32
C2×C22⋊C4×H(3,𝔽3)2
C2×#3757⋊C32
C2×#3750⋊C32
#63658⋊C32
#105926453
C3×#49463