Finite Group Databasebeta

← #10592654⋊C3 #13142505

Subgroups — #10592654⋊C3 #13142505

Order 1728.

1903 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×#3757⋊C3C211yesmaximal, characteristic, complemented
4C2×#3752⋊C3C214yesmaximal, complemented
5#10592654C311yesmaximal, characteristic, complemented
6C3×#4955—31nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C23×(C3⋊D3)⋊C3—22no—
9C23×(C3⋊D3)⋊C3—22no—
10C23×(C3⋊D3)⋊C3—28no—
11C23×(C3⋊D3)⋊C3C414yescomplemented
12C23×(C3⋊D3)⋊C3C2212yes—
13C22×C4×H(3,𝔽3)C2212yescomplemented
14C22×(C3⋊Dic3)⋊C3C2212yescomplemented
15C22⋊C4×H(3,𝔽3)—22no—
16#3757⋊C3—22no—
17#3752⋊C3—28no—
18C24×C3⋊D3C2×C311yescharacteristic
19C24×C3×D3—31no—
20C2×C32×C22⋊C4—21no—
21C2×C32×C22⋊C4D311yescharacteristic, complemented
22C2×C32×C22⋊C4C2×C311yescharacteristic, complemented
23C2×#3757C2×C311yescharacteristic, complemented
24C2×#3752C2×C314yescomplemented
25C2×C3×#217—31no—
26C2×C3×(C2×D3)⋊C4—34no—
27C23×H(3,𝔽3)—24no—
28C23×H(3,𝔽3)C2×C412yes—
29C23×H(3,𝔽3)D412yes—
30C23×H(3,𝔽3)D412yes—
31C23×H(3,𝔽3)C2311yescharacteristic
32C22×(C3⋊D3)⋊C3—216no—
33C22×(C3⋊D3)⋊C3—216no—
34C22×(C3⋊D3)⋊C3—28no—
35C22×(C3⋊D3)⋊C3—22no—
36C22×(C3⋊D3)⋊C3—24no—
37C22×(C3⋊D3)⋊C3—22no—
38C22×(C3⋊D3)⋊C3—216no—
39C22×(C3⋊D3)⋊C3C2×C414yes—
40C22×(C3⋊D3)⋊C3D418yes—
41C2×C4×H(3,𝔽3)—22no—
42C2×C4×H(3,𝔽3)—24no—
43C2×(C3⋊Dic3)⋊C3—24no—
44C2×(C3⋊Dic3)⋊C3—22no—
45C3×#379—91no—
46#4955—31no—
47#4955—91no—
48C24×C32—21no—
49C24×C32C2×D311yescharacteristic
50C24×C32C22×C311yescharacteristic
51C23×C3⋊D3—22no—
52C23×C3⋊D3—22no—
53C23×C3⋊D3—28no—
54C23×C3⋊D3C3×C414yescomplemented
55C23×C3⋊D3C22×C312yes—
56C23×C3×D3—32no—
57C23×C3×D3—34no—
58C23×C3×D3—68no—
59C23×C3×D3—62no—
60C23×C3×D3—62no—
61C22×C32×C4—22no—
62C22×C32×C4C2×D312yescomplemented
63C22×C32×C4C22×C312yescomplemented
64C22×C3⋊Dic3C22×C312yescomplemented
65C22×C3×Dic3—32no—
66C32×C22⋊C4—24no—
67C32×C22⋊C4—22no—
68C32×C22⋊C4—22no—
69#3757—22no—
70#3752—28no—
71C3×#217—62no—
72C3×(C2×D3)⋊C4—68no—
73C22×H(3,𝔽3)—24no—
74C22×H(3,𝔽3)—24no—
75C22×H(3,𝔽3)—24no—
76C22×H(3,𝔽3)C22⋊C414yescomplemented
77C22×H(3,𝔽3)C22×C411yescharacteristic
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C22×H(3,𝔽3)C2×D414yes—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C2×(C3⋊D3)⋊C3—232no—
82C2×(C3⋊D3)⋊C3—24no—
83C2×(C3⋊D3)⋊C3—24no—
84C2×(C3⋊D3)⋊C3—28no—
85C2×(C3⋊D3)⋊C3—28no—
86C2×(C3⋊D3)⋊C3C22⋊C418yescomplemented
87C4×H(3,𝔽3)—42no—
88(C3⋊Dic3)⋊C3—42no—
89C25×C3—91no—
90C24×D3—31no—
91C24×D3—91no—
92C2×C3×C22⋊C4—31no—
93C2×C3×C22⋊C4—31no—
94C2×C3×C22⋊C4—61no—
95C2×C3×C22⋊C4—94no—
96C2×C3×C22⋊C4—91no—
97C2×C3×C22⋊C4C3×D311yescharacteristic
98C2×#217—31no—
99C2×#217—91no—
100C2×(C2×D3)⋊C4—34no—
101C2×(C2×D3)⋊C4—94no—
102C23×C32—24no—
103C23×C32—24no—
104C23×C32—22no—
105C23×C32—22no—
106C23×C32—22no—
107C23×C32—21no—
108C23×C32—44no—
109C23×C32C4×D312yes—
110C23×C32D1212yes—
111C23×C32#6712yes—
112C23×C32C2×C3×C412yes—
113C23×C32C3×D412yes—
114C23×C32C3×D412yes—
115C23×C32C22×D311yescharacteristic
116C23×C32C23×C311yescharacteristic
117C22×C3⋊D3—216no—
118C22×C3⋊D3—216no—
119C22×C3⋊D3—28no—
120C22×C3⋊D3—22no—
121C22×C3⋊D3—24no—
122C22×C3⋊D3—22no—
123C22×C3⋊D3—216no—
124C22×C3⋊D3C2×C3×C414yes—
125C22×C3⋊D3C3×D418yes—
126C22×C3×D3—34no—
127C22×C3×D3—38no—
128C22×C3×D3—616no—
129C22×C3×D3—616no—
130C22×C3×D3—616no—
131C22×C3×D3—68no—
132C22×C3×D3—62no—
133C22×C3×D3—64no—
134C22×C3×D3—62no—
135C2×C32×C4—24no—
136C2×C32×C4—24no—
137C2×C32×C4—22no—
138C2×C32×C4—24no—
139C2×C32×C4—22no—
140C2×C32×C4—44no—
141C2×C3⋊Dic3—24no—
142C2×C3⋊Dic3—22no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—64no—
145#379—91noSylow
146C2×H(3,𝔽3)—24no—
147C2×H(3,𝔽3)C2×C22⋊C412yes—
148C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
149C2×H(3,𝔽3)#12012yes—
150C2×H(3,𝔽3)#12012yes—
151(C3⋊D3)⋊C3—28no—
152C24×C3—31no—
153C24×C3—31no—
154C24×C3—61no—
155C24×C3—92no—
156C24×C3—94no—
157C24×C3—182no—
158C24×C3—182no—
159C24×C3—188no—
160C24×C3C2×C3×D311yescharacteristic
161C23×D3—32no—
162C23×D3—34no—
163C23×D3—68no—
164C23×D3—62no—
165C23×D3—62no—
166C23×D3—92no—
167C23×D3—94no—
168C23×D3—188no—
169C23×D3—182no—
170C23×D3—182no—
171C22×C3×C4—32no—
172C22×C3×C4—32no—
173C22×C3×C4—62no—
174C22×C3×C4—92no—
175C22×C3×C4C2×C3×D312yes—
176C22×Dic3—32no—
177C22×Dic3—92no—
178C3×C22⋊C4—22no—
179C3×C22⋊C4—62no—
180C3×C22⋊C4—62no—
181C3×C22⋊C4—64no—
182C3×C22⋊C4—188no—
183C3×C22⋊C4—182no—
184#217—62no—
185#217—182no—
186(C2×D3)⋊C4—68no—
187(C2×D3)⋊C4—188no—
188C22×C32—24no—
189C22×C32—24no—
190C22×C32—24no—
191C22×C32—24no—
192C22×C32—24no—
193C22×C32—24no—
194C22×C32—24no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—24no—
198C22×C32—21no—
199C22×C32—44no—
200C22×C32—44no—
201C22×C32—44no—
202C22×C32(C2×D3)⋊C414yescomplemented
203C22×C32C3×C22⋊C414yescomplemented
204C22×C32C2×C4×D311yescharacteristic
205C22×C32C2×D1211yescharacteristic
206C22×C32D3×D414yes—
207C22×C32C2×#6711yescharacteristic
208C22×C32C22×C3×C411yescharacteristic
209C22×C32C2×C3×D411yescharacteristic
210C22×C32C2×C3×D414yes—
211C22×C32C2×C3×D411yescharacteristic
212C2×C3⋊D3—232no—
213C2×C3⋊D3—24no—
214C2×C3⋊D3—24no—
215C2×C3⋊D3—28no—
216C2×C3⋊D3—28no—
217C2×C3⋊D3C3×C22⋊C418yescomplemented
218C2×C3×D3—38no—
219C2×C3×D3—632no—
220C2×C3×D3—64no—
221C2×C3×D3—68no—
222C2×C3×D3—64no—
223C2×C3×D3—68no—
224C32×C4—44no—
225C32×C4—42no—
226C32×C4—42no—
227C3⋊Dic3—42no—
228C3×Dic3—122no—
229C25—91no—
230C2×C22⋊C4—91no—
231C2×C22⋊C4—94no—
232C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
233H(3,𝔽3)#37911yescharacteristic, complemented, Sylow
234C23×C3—24no—
235C23×C3—32no—
236C23×C3—32no—
237C23×C3—32no—
238C23×C3—32no—
239C23×C3—32no—
240C23×C3—32no—
241C23×C3—31no—
242C23×C3—31no—
243C23×C3—64no—
244C23×C3—64no—
245C23×C3—62no—
246C23×C3—62no—
247C23×C3—62no—
248C23×C3—61no—
249C23×C3—98no—
250C23×C3—94no—
251C23×C3—124no—
252C23×C3—182no—
253C23×C3—184no—
254C23×C3—182no—
255C23×C3—188no—
256C23×C3—1816no—
257C23×C3—1816no—
258C23×C3—1816no—
259C23×C3C3×C4×D312yes—
260C23×C3C3×D1212yes—
261C23×C3C3×#6712yes—
262C23×C3C22×C3×D311yescharacteristic
263C22×D3—34no—
264C22×D3—38no—
265C22×D3—616no—
266C22×D3—616no—
267C22×D3—616no—
268C22×D3—68no—
269C22×D3—62no—
270C22×D3—64no—
271C22×D3—62no—
272C22×D3—94no—
273C22×D3—98no—
274C22×D3—1816no—
275C22×D3—182no—
276C22×D3—184no—
277C22×D3—182no—
278C22×D3—188no—
279C22×D3—1816no—
280C22×D3—1816no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—64no—
284C2×C3×C4—62no—
285C2×C3×C4—62no—
286C2×C3×C4—64no—
287C2×C3×C4—64no—
288C2×C3×C4—124no—
289C2×C3×C4—182no—
290C2×C3×C4—184no—
291C2×Dic3—62no—
292C2×Dic3—64no—
293C2×Dic3—184no—
294C2×Dic3—182no—
295C2×C32—24no—
296C2×C32—24no—
297C2×C32—22no—
298C2×C32—22no—
299C2×C32—21no—
300C2×C32—22no—
301C2×C32—44no—
302C2×C32D3×C22⋊C412yes—
303C2×C32#88212yes—
304C2×C32C2×(C2×D3)⋊C411yescharacteristic
305C2×C32#93712yes—
306C2×C32C2×C3×C22⋊C412yes—
307C2×C32C2×C3×C22⋊C411yescharacteristic
308C2×C32C3×#12012yes—
309C2×C32C3×#12012yes—
310C3⋊D3—28no—
311C3×D3—68no—
312C24—92no—
313C24—94no—
314C24—188no—
315C24—182no—
316C24—182no—
317C24C2×(C3⋊D3)⋊C311yescharacteristic
318C22×C4—92no—
319C22×C4C2×(C3⋊D3)⋊C312yescomplemented
320C22⋊C4—22no—
321C22⋊C4—188no—
322C22⋊C4—182no—
323C22×C3—24no—
324C22×C3—24no—
325C22×C3—24no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—31no—
329C22×C3—31no—
330C22×C3—34no—
331C22×C3—34no—
332C22×C3—31no—
333C22×C3—31no—
334C22×C3—34no—
335C22×C3—34no—
336C22×C3—64no—
337C22×C3—64no—
338C22×C3—64no—
339C22×C3—64no—
340C22×C3—64no—
341C22×C3—64no—
342C22×C3—61no—
343C22×C3—61no—
344C22×C3—64no—
345C22×C3—61no—
346C22×C3—64no—
347C22×C3—98no—
348C22×C3—124no—
349C22×C3—124no—
350C22×C3—124no—
351C22×C3—184no—
352C22×C3—188no—
353C22×C3—184no—
354C22×C3—188no—
355C22×C3—1832no—
356C22×C3C3×(C2×D3)⋊C414yes—
357C22×C3C2×C3×C4×D311yescharacteristic
358C22×C3C2×C3×D1211yescharacteristic
359C22×C3C3×D3×D414yes—
360C22×C3C2×C3×#6711yescharacteristic
361C2×D3—38no—
362C2×D3—632no—
363C2×D3—64no—
364C2×D3—68no—
365C2×D3—64no—
366C2×D3—68no—
367C2×D3—98no—
368C2×D3—184no—
369C2×D3—188no—
370C2×D3—184no—
371C2×D3—188no—
372C2×D3—1832no—
373C3×C4—42no—
374C3×C4—122no—
375C3×C4—122no—
376C3×C4—124no—
377C3×C4—362no—
378Dic3—122no—
379Dic3—362no—
380C32—21no—
381C32#495511yescharacteristic, complemented
382C32C3×#37911yescharacteristic, complemented
383C23—24no—
384C23—94no—
385C23—98no—
386C23—1816no—
387C23—1816no—
388C23—1816no—
389C23—182no—
390C23—184no—
391C23—182no—
392C23—188no—
393C23C4×(C3⋊D3)⋊C312yes—
394C23C4⋊((C3⋊D3)⋊C3)12yes—
395C23#638⋊C312yes—
396C23C22×(C3⋊D3)⋊C311yescharacteristic
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—182no—
400C2×C4—184no—
401C2×C3—24no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—32no—
405C2×C3—32no—
406C2×C3—32no—
407C2×C3—32no—
408C2×C3—32no—
409C2×C3—32no—
410C2×C3—64no—
411C2×C3—64no—
412C2×C3—61no—
413C2×C3—62no—
414C2×C3—62no—
415C2×C3—62no—
416C2×C3—124no—
417C2×C3—188no—
418C2×C3C3×D3×C22⋊C412yes—
419C2×C3C3×#88212yes—
420C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
421C2×C3C3×#93712yes—
422D3—68no—
423D3—188no—
424C22—24no—
425C22—24no—
426C22—24no—
427C22—98no—
428C22—1832no—
429C22—184no—
430C22—188no—
431C22—184no—
432C22—188no—
433C22#3752⋊C314yescomplemented
434C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
435C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
436C22D4×(C3⋊D3)⋊C314yes—
437C22C2×#638⋊C311yescharacteristic
438C4—42no—
439C4—362no—
440C3—31no—
441C3—31no—
442C3—61no—
443C3C3×#495511yescharacteristic
444C2—24no—
445C2—188no—
446C2C22⋊C4×(C3⋊D3)⋊C312yes—
447C2#64091⋊C312yes—
448C2C2×#3752⋊C311yescharacteristic
449C2#64146⋊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×(C3⋊D3)⋊C3C2
cosocleC23×C32C23×C3
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#379
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×H(3,𝔽3)C2semidirect
3. C2×#3757⋊C3C2semidirect
4. C2×#3752⋊C3C2semidirect
5. #10592654C3semidirect
11. C23×(C3⋊D3)⋊C3C4semidirect
13. C22×C4×H(3,𝔽3)C22semidirect
14. C22×(C3⋊Dic3)⋊C3C22semidirect
21. C2×C32×C22⋊C4D3semidirect
22. C2×C32×C22⋊C4C2×C3semidirect
23. C2×#3757C2×C3semidirect
24. C2×#3752C2×C3semidirect
54. C23×C3⋊D3C3×C4semidirect
62. C22×C32×C4C2×D3semidirect
63. C22×C32×C4C22×C3semidirect
64. C22×C3⋊Dic3C22×C3semidirect
76. C22×H(3,𝔽3)C22⋊C4semidirect
86. C2×(C3⋊D3)⋊C3C22⋊C4semidirect
202. C22×C32(C2×D3)⋊C4semidirect
203. C22×C32C3×C22⋊C4semidirect
217. C2×C3⋊D3C3×C22⋊C4semidirect
232. C2×C22⋊C4(C3⋊D3)⋊C3semidirect
233. H(3,𝔽3)#379semidirect
319. C22×C4C2×(C3⋊D3)⋊C3semidirect
381. C32#4955semidirect
382. C32C3×#379semidirect
433. C22#3752⋊C3semidirect

Maximal subgroups

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

subgroupindex
C24×(C3⋊D3)⋊C32
C2×C22⋊C4×H(3,𝔽3)2
C2×#3757⋊C32
C2×#3752⋊C32
#105926543
C3×#49553