Finite Group Databasebeta

← #10592665⋊C3 #13142605

Subgroups — #10592665⋊C3 #13142605

Order 1728.

659 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic
2C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×#3748⋊C3C211yesmaximal, characteristic, complemented
4#63636⋊C3C212yesmaximal, complemented
5#63637⋊C3C212yesmaximal, complemented
6#10592665C311yesmaximal, characteristic, complemented
7C3×#4966—31nomaximal
8C22×C4×H(3,𝔽3)—21no—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#638⋊C3—21no—
11#3828⋊C3—22no—
12#3828⋊C3C414yescomplemented
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3—21no—
15C2×#634⋊C3C2211yescharacteristic
16C4⋊C4×H(3,𝔽3)—21no—
17C4⋊C4×H(3,𝔽3)C2212yes—
18#3748⋊C3—22no—
19C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
20C2×#3828C2×C311yescharacteristic
21C2×C3×#235—31no—
22C2×C32×C4⋊C4—21no—
23C2×C32×C4⋊C4D311yescharacteristic, complemented
24C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
25C2×#3748C2×C311yescharacteristic, complemented
26C2×C3×#208—31no—
27#63637C2×C312yescomplemented
28#63636C2×C312yescomplemented
29C3×#810—32no—
30C3×#809—32no—
31C23×H(3,𝔽3)D411yescharacteristic
32C22×(C3⋊D3)⋊C3—21no—
33C2×C4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)—42no—
35C2×C4×H(3,𝔽3)C2×C412yes—
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)D412yes—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39#638⋊C3—24no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C4⋊((C3⋊D3)⋊C3)—21no—
42C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
43C4×(C3⋊D3)⋊C3—24no—
44#634⋊C3—21no—
45#634⋊C3C2×C412yescomplemented
46(C3⋊(C3⋊C8))⋊C3—22no—
47C3×#417—91no—
48#4966—31no—
49#4966—91no—
50C22×C32×C4—21no—
51C22×C32×C4—22no—
52C22×C32×C4—21no—
53C22×C32×C4—21no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C22×C311yescharacteristic
56C2×#638—21no—
57#3828—22no—
58#3828C3×C414yescomplemented
59C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
60C2×C4×C3⋊D3—21no—
61C2×#634C22×C311yescharacteristic
62C2×C3×#67—61no—
63C3×#235—34no—
64C3×#235—62no—
65C2×C3×D12—31no—
66C2×C3×C4×D3—61no—
67C2×C3×Dic6—31no—
68C32×C4⋊C4—22no—
69C32×C4⋊C4—22no—
70C32×C4⋊C4—21no—
71C32×C4⋊C4—21no—
72C32×C4⋊C4C2×D312yes—
73C32×C4⋊C4C22×C312yes—
74#3748—22no—
75C2×C3⋊(C3⋊C8)C22×C312yes—
76C3×#208—62no—
77C2×C3×C3⋊C8—32no—
78C22×H(3,𝔽3)—21no—
79C22×H(3,𝔽3)—21no—
80C22×H(3,𝔽3)C22⋊C412yes—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C2×(C3⋊D3)⋊C3—22no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C4×H(3,𝔽3)—42no—
86C4×H(3,𝔽3)C22⋊C412yes—
87C4×H(3,𝔽3)C22×C411yescharacteristic
88C4×H(3,𝔽3)C2×D411yescharacteristic
89(C3⋊Dic3)⋊C3—22no—
90C2×C3×#41—91no—
91C2×#235—31no—
92C2×#235—91no—
93C2×C3×M16—91no—
94C2×C3×C4⋊C4—31no—
95C2×C3×C4⋊C4—31no—
96C2×C3×C4⋊C4—61no—
97C2×C3×C4⋊C4C3×D311yescharacteristic
98C2×#208—31no—
99C2×#208—91no—
100C3×#103—92no—
101C3×#102—92no—
102#810—32no—
103#810—92no—
104#809—32no—
105#809—92no—
106C23×C32—21no—
107C23×C32#6711yescharacteristic
108C23×C32C3×D411yescharacteristic
109C22×C3⋊D3—21no—
110C22×C3×D3—61no—
111C2×C32×C4—24no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—44no—
119C2×C32×C4—42no—
120C2×C32×C4—42no—
121C2×C32×C4C4×D312yes—
122C2×C32×C4D1212yes—
123C2×C32×C4#6711yescharacteristic
124C2×C32×C4C2×C3×C412yes—
125C2×C32×C4C3×D412yes—
126C2×C32×C4C3×D411yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C23×C311yescharacteristic
129#638—24no—
130C2×C3⋊Dic3—21no—
131C4⋊(C3⋊D3)—21no—
132C4⋊(C3⋊D3)C2×C3×C412yescomplemented
133C4×C3⋊D3—24no—
134#634—21no—
135#634C2×C3×C412yescomplemented
136C3×#67—64no—
137C2×C3×Dic3—61no—
138C3×D12—32no—
139C3×D12—61no—
140C3×C4×D3—64no—
141C3×Dic6—32no—
142C3×Dic6—61no—
143C3⋊(C3⋊C8)—22no—
144C3×C3⋊C8—62no—
145#417—91noSylow
146C2×H(3,𝔽3)—22no—
147C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
148C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
149C2×H(3,𝔽3)#13711yescharacteristic
150(C3⋊D3)⋊C3—42no—
151C3×#41—94no—
152C3×#41—182no—
153C2×C3×Q8—91no—
154C2×C3×D4—91no—
155C2×C3×D4—181no—
156C22×C3×C4—21no—
157C22×C3×C4—31no—
158C22×C3×C4—31no—
159C22×C3×C4—61no—
160C22×C3×C4—61no—
161C22×C3×C4—62no—
162C22×C3×C4—61no—
163C22×C3×C4—181no—
164C22×C3×C4C2×C3×D311yescharacteristic
165C2×#67—61no—
166C2×#67—181no—
167#235—34no—
168#235—62no—
169#235—94no—
170#235—182no—
171C2×D12—31no—
172C2×D12—91no—
173C2×C4×D3—61no—
174C2×C4×D3—181no—
175C2×Dic6—31no—
176C2×Dic6—91no—
177C3×M16—182no—
178C2×C3×C8—92no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—32no—
181C3×C4⋊C4—32no—
182C3×C4⋊C4—61no—
183C3×C4⋊C4—61no—
184C3×C4⋊C4—62no—
185C3×C4⋊C4—62no—
186C3×C4⋊C4C2×C3×D312yes—
187#208—62no—
188#208—182no—
189C2×C3⋊C8—32no—
190C2×C3⋊C8—92no—
191C22×C32—22no—
192C22×C32—22no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—22no—
198C22×C32—21no—
199C22×C32(C2×D3)⋊C412yes—
200C22×C32C3×C22⋊C412yes—
201C22×C32C2×#6711yescharacteristic
202C22×C32C2×C3×D411yescharacteristic
203C2×C3⋊D3—22no—
204C2×C3⋊D3—41no—
205C2×C3⋊D3—41no—
206C2×C3×D3—62no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C32×C4—22no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—44no—
213C32×C4—42no—
214C32×C4—42no—
215C32×C4(C2×D3)⋊C412yes—
216C32×C4C3×C22⋊C412yes—
217C32×C4C2×C4×D311yescharacteristic
218C32×C4C2×D1211yescharacteristic
219C32×C4C22×C3×C411yescharacteristic
220C32×C4C2×C3×D411yescharacteristic
221C3⋊Dic3—22no—
222C3×Dic3—62no—
223C2×#41—91no—
224C2×M16—91no—
225C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
226#103—92no—
227#102—92no—
228H(3,𝔽3)#41711yescharacteristic, complemented, Sylow
229C23×C3—31no—
230C23×C3—31no—
231C23×C3—61no—
232C23×C3—181no—
233C23×C3C3×#6711yescharacteristic
234C22×D3—61no—
235C22×D3—181no—
236C3×Q8—92no—
237C3×Q8—181no—
238C3×D4—92no—
239C3×D4—181no—
240C3×D4—184no—
241C2×C3×C4—22no—
242C2×C3×C4—32no—
243C2×C3×C4—32no—
244C2×C3×C4—32no—
245C2×C3×C4—32no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—42no—
251C2×C3×C4—62no—
252C2×C3×C4—62no—
253C2×C3×C4—64no—
254C2×C3×C4—62no—
255C2×C3×C4—62no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—122no—
259C2×C3×C4—122no—
260C2×C3×C4—124no—
261C2×C3×C4—184no—
262C2×C3×C4—181no—
263C2×C3×C4C3×C4×D312yes—
264C2×C3×C4C3×D1212yes—
265C2×C3×C4C3×#6711yescharacteristic
266C2×C3×C4C22×C3×D311yescharacteristic
267#67—64no—
268#67—184no—
269C2×Dic3—61no—
270C2×Dic3—181no—
271D12—32no—
272D12—61no—
273D12—92no—
274D12—181no—
275C4×D3—64no—
276C4×D3—184no—
277Dic6—32no—
278Dic6—61no—
279Dic6—92no—
280Dic6—181no—
281C3×C8—182no—
282C3⋊C8—62no—
283C3⋊C8—182no—
284C2×C32—24no—
285C2×C32—22no—
286C2×C32—22no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32C2×(C2×D3)⋊C411yescharacteristic
291C2×C32#93211yescharacteristic
292C2×C32#94211yescharacteristic
293C2×C32C2×C3×C22⋊C411yescharacteristic
294C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
295C2×C32C3×#13711yescharacteristic
296C3⋊D3—42no—
297C3×D3—122no—
298#41—94no—
299#41—182no—
300C2×Q8—91no—
301C2×D4—91no—
302C2×D4—181no—
303C22×C4—21no—
304C22×C4—181no—
305C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
306M16—182no—
307C2×C8—92no—
308C4⋊C4—21no—
309C4⋊C4C2×(C3⋊D3)⋊C312yes—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—31no—
313C22×C3—31no—
314C22×C3—32no—
315C22×C3—32no—
316C22×C3—61no—
317C22×C3—61no—
318C22×C3—61no—
319C22×C3—61no—
320C22×C3—62no—
321C22×C3—62no—
322C22×C3—61no—
323C22×C3—62no—
324C22×C3—182no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3C3×(C2×D3)⋊C412yes—
328C22×C3C2×C3×#6711yescharacteristic
329C2×D3—62no—
330C2×D3—121no—
331C2×D3—121no—
332C2×D3—182no—
333C2×D3—361no—
334C2×D3—361no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—32no—
340C3×C4—32no—
341C3×C4—42no—
342C3×C4—61no—
343C3×C4—61no—
344C3×C4—62no—
345C3×C4—122no—
346C3×C4—122no—
347C3×C4—124no—
348C3×C4—182no—
349C3×C4C3×(C2×D3)⋊C412yes—
350C3×C4C2×C3×C4×D311yescharacteristic
351C3×C4C2×C3×D1211yescharacteristic
352Dic3—62no—
353Dic3—182no—
354C32—21no—
355C32#496611yescharacteristic, complemented
356C32C3×#41711yescharacteristic, complemented
357C23—181no—
358C23#638⋊C311yescharacteristic
359Q8—92no—
360Q8—181no—
361D4—92no—
362D4—184no—
363D4—181no—
364C2×C4—22no—
365C2×C4—42no—
366C2×C4—181no—
367C2×C4—184no—
368C2×C4C4×(C3⋊D3)⋊C312yes—
369C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
370C2×C4#638⋊C311yescharacteristic
371C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
372C8—182no—
373C2×C3—22no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—62no—
381C2×C3—62no—
382C2×C3—64no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—61no—
386C2×C3—362no—
387C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
388C2×C3C3×#93211yescharacteristic
389C2×C3C3×#94211yescharacteristic
390D3—122no—
391D3—362no—
392C22—21no—
393C22—21no—
394C22—182no—
395C22—361no—
396C22—361no—
397C22#3752⋊C312yes—
398C22C2×#638⋊C311yescharacteristic
399C4—42no—
400C4—182no—
401C4#3752⋊C312yes—
402C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
403C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
404C3—31no—
405C3—31no—
406C3—61no—
407C3C3×#496611yescharacteristic
408C2—22no—
409C2—362no—
410C2C2×#3752⋊C311yescharacteristic
411C2#64141⋊C311yescharacteristic
412C2#64151⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
prime radicalC2×#3828⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#417
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×H(3,𝔽3)C2semidirect
3. C2×#3748⋊C3C2semidirect
4. #63636⋊C3C2semidirect
5. #63637⋊C3C2semidirect
6. #10592665C3semidirect
12. #3828⋊C3C4semidirect
23. C2×C32×C4⋊C4D3semidirect
24. C2×C32×C4⋊C4C2×C3semidirect
25. C2×#3748C2×C3semidirect
27. #63637C2×C3semidirect
28. #63636C2×C3semidirect
42. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
45. #634⋊C3C2×C4semidirect
58. #3828C3×C4semidirect
132. C4⋊(C3⋊D3)C2×C3×C4semidirect
135. #634C2×C3×C4semidirect
225. C2×C4⋊C4(C3⋊D3)⋊C3semidirect
228. H(3,𝔽3)#417semidirect
355. C32#4966semidirect
356. C32C3×#417semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C2×C4⋊C4×H(3,𝔽3)2
C2×#3748⋊C32
#63636⋊C32
#63637⋊C32
#105926653
C3×#49663