Finite Group Databasebeta

← #10592646⋊C3 #13142496

Subgroups — #10592646⋊C3 #13142496

Order 1728.

839 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#634⋊C3C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
3C2×#3757⋊C3C212yesmaximal
4#63658⋊C3C211yesmaximal, characteristic, complemented
5#63658⋊C3C212yesmaximal, complemented
6#10592646C311yesmaximal, characteristic, complemented
7C3×#4947—31nomaximal
8C24×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C2×#634⋊C3—22no—
14C2×#634⋊C3—24no—
15C22⋊C4×H(3,𝔽3)—22no—
16#3757⋊C3—24no—
17C22×#634C2×C311yescharacteristic, complemented
18C22×C3×Dic6—31no—
19C2×C32×C22⋊C4—21no—
20C2×C32×C22⋊C4D311yescharacteristic
21C2×C32×C22⋊C4C2×C311yescharacteristic
22C2×#3757C2×C312yes—
23C2×C3×#217—32no—
24#63658C2×C311yescharacteristic, complemented
25#63658C2×C312yescomplemented
26C3×C3⋊#95—32no—
27C3×C3⋊#95—31no—
28C23×H(3,𝔽3)—21no—
29C23×H(3,𝔽3)—22no—
30C23×H(3,𝔽3)—21no—
31C23×H(3,𝔽3)—22no—
32C23×H(3,𝔽3)Q812yescomplemented
33C23×H(3,𝔽3)C2311yescharacteristic
34C2×C4×H(3,𝔽3)—22no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)D412yes—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—22no—
43C2×(C3⋊Dic3)⋊C3D414yes—
44#634⋊C3—28no—
45C3×#393—91no—
46#4947—31no—
47#4947—91no—
48C24×C32—21no—
49C24×C32C2×D311yescharacteristic
50C24×C32C22×C311yescharacteristic
51C22×C32×C4—21no—
52C22×C32×C4—21no—
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C22×C311yescharacteristic
56C22×C32×C4C22×C311yescharacteristic
57C22×C3⋊Dic3C22×C312yes—
58C22×C3⋊Dic3C22×C312yes—
59C2×#634—22no—
60C2×#634—24no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—32no—
63C2×C3×Dic6—64no—
64C2×C3×Dic6—62no—
65C32×C22⋊C4—24no—
66C32×C22⋊C4—22no—
67C32×C22⋊C4—22no—
68#3757—24no—
69C3×#217—64no—
70C22×H(3,𝔽3)—22no—
71C22×H(3,𝔽3)—24no—
72C22×H(3,𝔽3)—41no—
73C22×H(3,𝔽3)—42no—
74C22×H(3,𝔽3)—41no—
75C22×H(3,𝔽3)C2×D412yes—
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C22×H(3,𝔽3)C2×Q811yescharacteristic
78C22×H(3,𝔽3)#4111yescharacteristic
79C22×H(3,𝔽3)#4112yes—
80C4×H(3,𝔽3)—22no—
81C4×H(3,𝔽3)—41no—
82(C3⋊Dic3)⋊C3—24no—
83(C3⋊Dic3)⋊C3—42no—
84C22×C3×Q8—91no—
85C22×Dic6—31no—
86C22×Dic6—91no—
87C2×C3×C22⋊C4—31no—
88C2×C3×C22⋊C4—31no—
89C2×C3×C22⋊C4—61no—
90C2×C3×C22⋊C4—92no—
91C2×C3×C22⋊C4C3×D311yescharacteristic
92C2×#217—32no—
93C2×#217—92no—
94C3×#95—91no—
95C3×#95—92no—
96C3⋊#95—32no—
97C3⋊#95—31no—
98C3⋊#95—92no—
99C3⋊#95—91no—
100C23×C32—22no—
101C23×C32—22no—
102C23×C32—21no—
103C23×C32—22no—
104C23×C32—22no—
105C23×C32—21no—
106C23×C32—22no—
107C23×C32—21no—
108C23×C32—21no—
109C23×C32—22no—
110C23×C32—22no—
111C23×C32—21no—
112C23×C32—42no—
113C23×C32—42no—
114C23×C32Dic612yescomplemented
115C23×C32C3×Q812yescomplemented
116C23×C32C22×D311yescharacteristic
117C23×C32C23×C311yescharacteristic
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—22no—
125C2×C32×C4—22no—
126C2×C32×C4—42no—
127C2×C32×C4—42no—
128C2×C32×C4#6712yes—
129C2×C32×C4C3×D412yes—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3—22no—
133C2×C3⋊Dic3—22no—
134C2×C3⋊Dic3—22no—
135C2×C3⋊Dic3C3×D414yes—
136#634—28no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—62no—
143C3×Dic6—68no—
144#393—91noSylow
145C2×H(3,𝔽3)—42no—
146C2×H(3,𝔽3)#12011yescharacteristic
147C2×H(3,𝔽3)C22⋊Q812yes—
148C2×H(3,𝔽3)C22⋊Q811yescharacteristic
149C2×H(3,𝔽3)#12412yes—
150C2×H(3,𝔽3)#12411yescharacteristic
151C24×C3—31no—
152C24×C3—31no—
153C24×C3—61no—
154C24×C3C2×C3×D311yescharacteristic
155C2×C3×Q8—182no—
156C2×C3×Q8—184no—
157C22×C3×C4—31no—
158C22×C3×C4—31no—
159C22×C3×C4—31no—
160C22×C3×C4—31no—
161C22×C3×C4—61no—
162C22×C3×C4—61no—
163C22×C3×C4—92no—
164C22×C3×C4—92no—
165C22×C3×C4C2×C3×D311yescharacteristic
166C22×C3×C4C2×C3×D311yescharacteristic
167C22×Dic3—32no—
168C22×Dic3—32no—
169C22×Dic3—92no—
170C22×Dic3—92no—
171C2×Dic6—64no—
172C2×Dic6—62no—
173C2×Dic6—184no—
174C2×Dic6—182no—
175C3×C22⋊C4—22no—
176C3×C22⋊C4—62no—
177C3×C22⋊C4—62no—
178C3×C22⋊C4—64no—
179C3×C22⋊C4—184no—
180#217—64no—
181#217—184no—
182C22×C32—24no—
183C22×C32—24no—
184C22×C32—22no—
185C22×C32—24no—
186C22×C32—22no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—22no—
190C22×C32—22no—
191C22×C32—21no—
192C22×C32—44no—
193C22×C32—44no—
194C22×C32—42no—
195C22×C32—42no—
196C22×C32—42no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—42no—
201C22×C32—41no—
202C22×C32C2×Dic611yescharacteristic
203C22×C32#23511yescharacteristic
204C22×C32D3×D412yes—
205C22×C32#23712yes—
206C22×C32C2×#6711yescharacteristic
207C22×C32C2×C3×D412yes—
208C22×C32C2×C3×D411yescharacteristic
209C22×C32C2×C3×Q811yescharacteristic
210C22×C32C3×#4111yescharacteristic
211C22×C32C3×#4112yes—
212C32×C4—22no—
213C32×C4—22no—
214C32×C4—42no—
215C32×C4—42no—
216C32×C4—41no—
217C32×C4—41no—
218C3⋊Dic3—24no—
219C3⋊Dic3—42no—
220C3×Dic3—64no—
221C3×Dic3—122no—
222C22×Q8—91no—
223C2×C22⋊C4—92no—
224C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic
225#95—92no—
226#95—91no—
227H(3,𝔽3)#39311yescharacteristic, complemented, Sylow
228C23×C3—21no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—21no—
232C23×C3—32no—
233C23×C3—32no—
234C23×C3—31no—
235C23×C3—31no—
236C23×C3—62no—
237C23×C3—61no—
238C23×C3—61no—
239C23×C3—62no—
240C23×C3—61no—
241C23×C3—61no—
242C23×C3—62no—
243C23×C3—62no—
244C23×C3—62no—
245C23×C3—62no—
246C23×C3—62no—
247C23×C3—61no—
248C23×C3—122no—
249C23×C3—122no—
250C23×C3C3×Dic612yes—
251C23×C3C22×C3×D311yescharacteristic
252C3×Q8—188no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—21no—
256C2×C3×C4—32no—
257C2×C3×C4—32no—
258C2×C3×C4—62no—
259C2×C3×C4—62no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—62no—
265C2×C3×C4—62no—
266C2×C3×C4—94no—
267C2×C3×C4—122no—
268C2×C3×C4—122no—
269C2×C3×C4—182no—
270C2×C3×C4—182no—
271C2×C3×C4—182no—
272C2×C3×C4—182no—
273C2×C3×C4—182no—
274C2×C3×C4C3×#6712yes—
275C2×Dic3—34no—
276C2×Dic3—62no—
277C2×Dic3—62no—
278C2×Dic3—62no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281C2×Dic3—94no—
282C2×Dic3—182no—
283C2×Dic3—182no—
284C2×Dic3—182no—
285C2×Dic3—182no—
286C2×Dic3—182no—
287Dic6—68no—
288Dic6—188no—
289C2×C32—22no—
290C2×C32—22no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—44no—
295C2×C32—42no—
296C2×C32—42no—
297C2×C32#87812yes—
298C2×C32#88512yes—
299C2×C32#92411yescharacteristic
300C2×C32#93611yescharacteristic
301C2×C32#93711yescharacteristic
302C2×C32C3×#12011yescharacteristic
303C2×C32C3×C22⋊Q812yes—
304C2×C32C3×C22⋊Q811yescharacteristic
305C2×C32C3×#12412yes—
306C2×C32C3×#12411yescharacteristic
307C24C2×(C3⋊D3)⋊C311yescharacteristic
308C2×Q8—184no—
309C2×Q8—182no—
310C22×C4—92no—
311C22×C4—92no—
312C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
313C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
314C22⋊C4—22no—
315C22⋊C4—184no—
316C22×C3—24no—
317C22×C3—22no—
318C22×C3—31no—
319C22×C3—31no—
320C22×C3—31no—
321C22×C3—31no—
322C22×C3—32no—
323C22×C3—32no—
324C22×C3—32no—
325C22×C3—32no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—42no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—62no—
332C22×C3—62no—
333C22×C3—64no—
334C22×C3—64no—
335C22×C3—64no—
336C22×C3—61no—
337C22×C3—61no—
338C22×C3—62no—
339C22×C3—62no—
340C22×C3—61no—
341C22×C3—124no—
342C22×C3—121no—
343C22×C3—121no—
344C22×C3—121no—
345C22×C3—121no—
346C22×C3—122no—
347C22×C3—122no—
348C22×C3—122no—
349C22×C3—122no—
350C22×C3—124no—
351C22×C3C2×C3×Dic611yescharacteristic
352C22×C3C3×#23511yescharacteristic
353C22×C3C3×D3×D412yes—
354C22×C3C3×#23712yes—
355C22×C3C2×C3×#6711yescharacteristic
356C3×C4—22no—
357C3×C4—41no—
358C3×C4—62no—
359C3×C4—62no—
360C3×C4—122no—
361C3×C4—121no—
362C3×C4—121no—
363C3×C4—122no—
364C3×C4—184no—
365C3×C4—362no—
366Dic3—64no—
367Dic3—122no—
368Dic3—184no—
369Dic3—362no—
370C32—21no—
371C32#494711yescharacteristic, complemented
372C32C3×#39311yescharacteristic, complemented
373C23—21no—
374C23—22no—
375C23—22no—
376C23—21no—
377C23#634⋊C312yescomplemented
378C23C22×(C3⋊D3)⋊C311yescharacteristic
379Q8—188no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—94no—
384C2×C4—182no—
385C2×C4—182no—
386C2×C4—182no—
387C2×C4—182no—
388C2×C4—182no—
389C2×C4#638⋊C312yes—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—32no—
397C2×C3—32no—
398C2×C3—32no—
399C2×C3—32no—
400C2×C3—42no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—61no—
404C2×C3—62no—
405C2×C3—62no—
406C2×C3—122no—
407C2×C3—122no—
408C2×C3—124no—
409C2×C3C3×#87812yes—
410C2×C3C3×#88512yes—
411C2×C3C3×#92411yescharacteristic
412C2×C3C3×#93611yescharacteristic
413C2×C3C3×#93711yescharacteristic
414C22—22no—
415C22—24no—
416C22—41no—
417C22—41no—
418C22—42no—
419C22C2×#634⋊C311yescharacteristic
420C22#3828⋊C311yescharacteristic
421C22D4×(C3⋊D3)⋊C312yes—
422C22#3830⋊C312yes—
423C22C2×#638⋊C311yescharacteristic
424C4—22no—
425C4—41no—
426C4—184no—
427C4—362no—
428C3—31no—
429C3—31no—
430C3—61no—
431C3C3×#494711yescharacteristic
432C2—42no—
433C2#64087⋊C312yes—
434C2#64094⋊C312yes—
435C2#64133⋊C311yescharacteristic
436C2#64145⋊C311yescharacteristic
437C2#64146⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC23×C32C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC24×H(3,𝔽3)C22
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#393
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#634⋊C3C2semidirect
4. #63658⋊C3C2semidirect
5. #63658⋊C3C2semidirect
6. #10592646C3semidirect
17. C22×#634C2×C3semidirect
24. #63658C2×C3semidirect
25. #63658C2×C3semidirect
32. C23×H(3,𝔽3)Q8semidirect
114. C23×C32Dic6semidirect
115. C23×C32C3×Q8semidirect
227. H(3,𝔽3)#393semidirect
371. C32#4947semidirect
372. C32C3×#393semidirect
377. C23#634⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
C2×C22⋊C4×H(3,𝔽3)2
C2×#3757⋊C32
#63658⋊C32
#105926463
C3×#49473