Finite Group Databasebeta

← #10592649⋊C3 #13142497

Subgroups — #10592649⋊C3 #13142497

Order 1728.

951 conjugacy classes of subgroups.

The subgroup lattice has 440 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×C4⋊((C3⋊Dic3)⋊C3)C212yesmaximal, complemented
5#63658⋊C3C212yesmaximal, complemented
6#10592649C311yesmaximal, characteristic, complemented
7C3×#4950—31nomaximal
8C24×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2212yes—
10C22×(C3⋊Dic3)⋊C3—22no—
11C22×(C3⋊Dic3)⋊C3—24no—
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2212yes—
15C22⋊C4×H(3,𝔽3)C414yescomplemented
16#3757⋊C3—22no—
17C4⋊((C3⋊Dic3)⋊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×C4⋊(C3⋊Dic3)C2×C312yescomplemented
25C2×C3×#217—31no—
26C2×C3×C4⋊Dic3—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)—22no—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)C2×C414yescomplemented
38C2×(C3⋊Dic3)⋊C3—28no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—22no—
45C2×(C3⋊Dic3)⋊C3D414yes—
46C3×#385—91no—
47#4950—31no—
48#4950—91no—
49C24×C32—21no—
50C24×C32C2×D311yescharacteristic
51C24×C32C22×C311yescharacteristic
52C22×C32×C4—22no—
53C22×C32×C4C2×D312yes—
54C22×C32×C4C22×C312yes—
55C22×C3⋊Dic3—22no—
56C22×C3⋊Dic3—24no—
57C22×C3⋊Dic3C22×C311yescharacteristic
58C22×C3⋊Dic3C22×C311yescharacteristic
59C22×C3⋊Dic3C22×C312yes—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—31no—
62C22×C3×Dic3—31no—
63C22×C3×Dic3—64no—
64C22×C3×Dic3—62no—
65C32×C22⋊C4—24no—
66C32×C22⋊C4Dic314yescomplemented
67C32×C22⋊C4C3×C414yescomplemented
68#3757—22no—
69C4⋊(C3⋊Dic3)—24no—
70C3×#217—62no—
71C3×C4⋊Dic3—64no—
72C22×H(3,𝔽3)—24no—
73C22×H(3,𝔽3)—22no—
74C22×H(3,𝔽3)—24no—
75C22×H(3,𝔽3)—22no—
76C22×H(3,𝔽3)C4⋊C414yescomplemented
77C22×H(3,𝔽3)C22×C411yescharacteristic
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C22×H(3,𝔽3)C2×D412yes—
80C22×H(3,𝔽3)C2×Q811yescharacteristic
81C22×H(3,𝔽3)#4112yes—
82C4×H(3,𝔽3)—24no—
83(C3⋊Dic3)⋊C3—24no—
84(C3⋊Dic3)⋊C3—41no—
85(C3⋊Dic3)⋊C3—41no—
86C23×C3×C4—91no—
87C23×Dic3—31no—
88C23×Dic3—91no—
89C2×C3×C4⋊C4—92no—
90C2×C3×C22⋊C4—31no—
91C2×C3×C22⋊C4—31no—
92C2×C3×C22⋊C4—61no—
93C2×C3×C22⋊C4—91no—
94C2×C3×C22⋊C4C3×D311yescharacteristic
95C2×#217—31no—
96C2×#217—91no—
97C2×C4⋊Dic3—32no—
98C2×C4⋊Dic3—92no—
99C3×#95—92no—
100C3⋊#95—32no—
101C3⋊#95—92no—
102C23×C32—22no—
103C23×C32—22no—
104C23×C32—22no—
105C23×C32—22no—
106C23×C32—22no—
107C23×C32—22no—
108C23×C32—22no—
109C23×C32—21no—
110C23×C32—42no—
111C23×C32—42no—
112C23×C32Dic612yes—
113C23×C32D1212yes—
114C23×C32C2×Dic312yes—
115C23×C32C2×C3×C412yes—
116C23×C32C3×D412yes—
117C23×C32C3×Q812yes—
118C23×C32C22×D311yescharacteristic
119C23×C32C23×C311yescharacteristic
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—24no—
125C2×C32×C4—42no—
126C2×C32×C4—42no—
127C2×C32×C4C2×Dic314yescomplemented
128C2×C32×C4C2×C3×C414yescomplemented
129C2×C3⋊Dic3—28no—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3—22no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—22no—
136C2×C3⋊Dic3C3×D414yes—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—68no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—62no—
145#385—91noSylow
146C2×H(3,𝔽3)—24no—
147C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
148C2×H(3,𝔽3)C4×D412yes—
149C2×H(3,𝔽3)#12011yescharacteristic
150C2×H(3,𝔽3)C22⋊Q812yes—
151C2×H(3,𝔽3)#12311yescharacteristic
152C24×C3—31no—
153C24×C3—31no—
154C24×C3—61no—
155C24×C3C2×C3×D311yescharacteristic
156C22×C3×C4—32no—
157C22×C3×C4—32no—
158C22×C3×C4—62no—
159C22×C3×C4—92no—
160C22×C3×C4—91no—
161C22×C3×C4—91no—
162C22×C3×C4—184no—
163C22×C3×C4—182no—
164C22×C3×C4C2×C3×D312yes—
165C22×Dic3—32no—
166C22×Dic3—31no—
167C22×Dic3—31no—
168C22×Dic3—64no—
169C22×Dic3—62no—
170C22×Dic3—92no—
171C22×Dic3—91no—
172C22×Dic3—91no—
173C22×Dic3—184no—
174C22×Dic3—182no—
175C3×C4⋊C4—184no—
176C3×C22⋊C4—34no—
177C3×C22⋊C4—34no—
178C3×C22⋊C4—64no—
179C3×C22⋊C4—182no—
180C3×C22⋊C4C3×Dic314yes—
181#217—62no—
182#217—182no—
183C4⋊Dic3—64no—
184C4⋊Dic3—184no—
185C22×C32—24no—
186C22×C32—22no—
187C22×C32—22no—
188C22×C32—24no—
189C22×C32—24no—
190C22×C32—22no—
191C22×C32—24no—
192C22×C32—22no—
193C22×C32—24no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—22no—
197C22×C32—22no—
198C22×C32—21no—
199C22×C32—44no—
200C22×C32—42no—
201C22×C32—42no—
202C22×C32—44no—
203C22×C32C4⋊Dic314yescomplemented
204C22×C32C3×C4⋊C414yescomplemented
205C22×C32C2×Dic611yescharacteristic
206C22×C32C2×D1211yescharacteristic
207C22×C32D3×D412yes—
208C22×C32#23712yes—
209C22×C32C22×Dic311yescharacteristic
210C22×C32C22×C3×C411yescharacteristic
211C22×C32C2×C3×D411yescharacteristic
212C22×C32C2×C3×D412yes—
213C22×C32C2×C3×Q811yescharacteristic
214C22×C32C3×#4112yes—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4—44no—
218C3⋊Dic3—24no—
219C3⋊Dic3—41no—
220C3⋊Dic3—41no—
221C3×Dic3—64no—
222C3×Dic3—121no—
223C3×Dic3—121no—
224C23×C4—91no—
225C2×C4⋊C4—92no—
226C2×C22⋊C4—91no—
227C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic
228#95—92no—
229H(3,𝔽3)#38511yescharacteristic, complemented, Sylow
230C23×C3—22no—
231C23×C3—22no—
232C23×C3—32no—
233C23×C3—32no—
234C23×C3—32no—
235C23×C3—32no—
236C23×C3—32no—
237C23×C3—32no—
238C23×C3—31no—
239C23×C3—31no—
240C23×C3—62no—
241C23×C3—62no—
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×C3C3×D1212yes—
252C23×C3C2×C3×Dic312yes—
253C23×C3C22×C3×D311yescharacteristic
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—34no—
257C2×C3×C4—34no—
258C2×C3×C4—62no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—62no—
262C2×C3×C4—64no—
263C2×C3×C4—94no—
264C2×C3×C4—122no—
265C2×C3×C4—122no—
266C2×C3×C4—182no—
267C2×C3×C4—182no—
268C2×C3×C4—188no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4—182no—
272C2×C3×C4—182no—
273C2×C3×C4C2×C3×Dic314yes—
274C2×Dic3—34no—
275C2×Dic3—68no—
276C2×Dic3—62no—
277C2×Dic3—62no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—62no—
281C2×Dic3—62no—
282C2×Dic3—94no—
283C2×Dic3—182no—
284C2×Dic3—182no—
285C2×Dic3—188no—
286C2×Dic3—182no—
287C2×Dic3—181no—
288C2×Dic3—181no—
289C2×Dic3—182no—
290C2×C32—24no—
291C2×C32—24no—
292C2×C32—22no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—22no—
297C2×C32—44no—
298C2×C32#87812yes—
299C2×C32#88211yescharacteristic
300C2×C32#88611yescharacteristic
301C2×C32C2×C4⋊Dic311yescharacteristic
302C2×C32D4×Dic312yes—
303C2×C32C2×C3×C4⋊C411yescharacteristic
304C2×C32C3×C4×D412yes—
305C2×C32C3×#12011yescharacteristic
306C2×C32C3×C22⋊Q812yes—
307C2×C32C3×#12311yescharacteristic
308C24C2×(C3⋊D3)⋊C311yescharacteristic
309C22×C4—92no—
310C22×C4—91no—
311C22×C4—91no—
312C22×C4—184no—
313C22×C4—182no—
314C22×C4C2×(C3⋊D3)⋊C312yes—
315C4⋊C4—184no—
316C22⋊C4—182no—
317C22⋊C4(C3⋊Dic3)⋊C314yescomplemented
318C22×C3—24no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—24no—
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—31no—
325C22×C3—31no—
326C22×C3—32no—
327C22×C3—32no—
328C22×C3—32no—
329C22×C3—32no—
330C22×C3—31no—
331C22×C3—31no—
332C22×C3—34no—
333C22×C3—34no—
334C22×C3—62no—
335C22×C3—62no—
336C22×C3—64no—
337C22×C3—64no—
338C22×C3—64no—
339C22×C3—64no—
340C22×C3—62no—
341C22×C3—62no—
342C22×C3—61no—
343C22×C3—61no—
344C22×C3—62no—
345C22×C3—62no—
346C22×C3—61no—
347C22×C3—64no—
348C22×C3—124no—
349C22×C3—124no—
350C22×C3—122no—
351C22×C3—122no—
352C22×C3C3×C4⋊Dic314yes—
353C22×C3C2×C3×Dic611yescharacteristic
354C22×C3C2×C3×D1211yescharacteristic
355C22×C3C3×D3×D412yes—
356C22×C3C3×#23712yes—
357C22×C3C22×C3×Dic311yescharacteristic
358C3×C4—24no—
359C3×C4—64no—
360C3×C4—64no—
361C3×C4—124no—
362C3×C4—184no—
363C3×C4—361no—
364C3×C4—361no—
365Dic3—64no—
366Dic3—121no—
367Dic3—121no—
368Dic3—184no—
369Dic3—361no—
370Dic3—361no—
371C32—21no—
372C32#495011yescharacteristic, complemented
373C32C3×#38511yescharacteristic, complemented
374C23—22no—
375C23—22no—
376C23#634⋊C312yes—
377C23C4⋊((C3⋊D3)⋊C3)12yes—
378C23C2×(C3⋊Dic3)⋊C312yes—
379C23C22×(C3⋊D3)⋊C311yescharacteristic
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—94no—
383C2×C4—188no—
384C2×C4—182no—
385C2×C4—182no—
386C2×C4—181no—
387C2×C4—181no—
388C2×C4—182no—
389C2×C4—182no—
390C2×C4C2×(C3⋊Dic3)⋊C314yescomplemented
391C2×C3—24no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—32no—
395C2×C3—32no—
396C2×C3—32no—
397C2×C3—32no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—64no—
403C2×C3—64no—
404C2×C3—61no—
405C2×C3—62no—
406C2×C3—62no—
407C2×C3—61no—
408C2×C3—61no—
409C2×C3—124no—
410C2×C3C3×#87812yes—
411C2×C3C3×#88211yescharacteristic
412C2×C3C3×#88611yescharacteristic
413C2×C3C2×C3×C4⋊Dic311yescharacteristic
414C2×C3C3×D4×Dic312yes—
415C22—24no—
416C22—24no—
417C22—22no—
418C22—22no—
419C22C4⋊((C3⋊Dic3)⋊C3)14yescomplemented
420C22C2×#634⋊C311yescharacteristic
421C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
422C22D4×(C3⋊D3)⋊C312yes—
423C22#3830⋊C312yes—
424C22C22×(C3⋊Dic3)⋊C311yescharacteristic
425C4—24no—
426C4—184no—
427C4—361no—
428C4—361no—
429C3—31no—
430C3—31no—
431C3—61no—
432C3C3×#495011yescharacteristic
433C2—24no—
434C2#64087⋊C312yes—
435C2#64091⋊C311yescharacteristic
436C2#64095⋊C311yescharacteristic
437C2C2×C4⋊((C3⋊Dic3)⋊C3)11yescharacteristic
438C2D4×(C3⋊Dic3)⋊C312yes—
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×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. #63658⋊C3C2semidirect
6. #10592649C3semidirect
15. C22⋊C4×H(3,𝔽3)C4semidirect
24. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
27. #63658C2×C3semidirect
37. C2×C4×H(3,𝔽3)C2×C4semidirect
66. C32×C22⋊C4Dic3semidirect
67. C32×C22⋊C4C3×C4semidirect
76. C22×H(3,𝔽3)C4⋊C4semidirect
127. C2×C32×C4C2×Dic3semidirect
128. C2×C32×C4C2×C3×C4semidirect
203. C22×C32C4⋊Dic3semidirect
204. C22×C32C3×C4⋊C4semidirect
229. H(3,𝔽3)#385semidirect
317. C22⋊C4(C3⋊Dic3)⋊C3semidirect
372. C32#4950semidirect
373. C32C3×#385semidirect
390. C2×C4C2×(C3⋊Dic3)⋊C3semidirect
419. C22C4⋊((C3⋊Dic3)⋊C3)semidirect

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×C4⋊((C3⋊Dic3)⋊C3)2
#63658⋊C32
#105926493
C3×#49503