Finite Group Databasebeta

← #10592580⋊C3 #13142222

Subgroups — #10592580⋊C3 #13142222

Order 1728.

449 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3743⋊C3C211yesmaximal, characteristic
2#64101⋊C3C211yesmaximal, characteristic
3C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
4H(3,𝔽3)×#107C211yesmaximal, characteristic, complemented
5#63644⋊C3C211yesmaximal, characteristic, complemented
6#63635⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592580C311yesmaximal, characteristic, complemented
9C3×#4881—31nomaximal
10C2×C4×(C3⋊D3)⋊C3—21no—
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#3752⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3—21no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#3743⋊C3C414yescomplemented
22C2×#3743C2×C311yescharacteristic
23#64101C2×C311yescharacteristic
24C4⋊C4×C3⋊D3C2×C311yescharacteristic
25C2×C3×ΓL(1,𝔽25)—31no—
26C3×#892—31no—
27C3×D3×C4⋊C4—31no—
28C32×#107—21no—
29C32×#107D311yescharacteristic, complemented
30C32×#107C2×C311yescharacteristic, complemented
31#63644C2×C311yescharacteristic, complemented
32#63635C2×C311yescharacteristic, complemented
33#63634C2×C311yescharacteristic, complemented
34C3×#817—31no—
35C3×#808—31no—
36C3×#807—31no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C4×(C3⋊D3)⋊C3—42no—
45C4×(C3⋊D3)⋊C3C2×C412yes—
46C4×(C3⋊D3)⋊C3Q812yes—
47C8×H(3,𝔽3)C2×C412yescomplemented
48(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
49C3×#428—91no—
50#4881—31no—
51#4881—91no—
52C2×C4×C3⋊D3—21no—
53C2×C4×C3⋊D3C22×C311yescharacteristic
54C2×C3×C4×D3—31no—
55C2×C3×C4×D3—61no—
56C2×C32×C8—21no—
57C2×C32×C8C2×D311yescharacteristic
58C2×C32×C8C22×C311yescharacteristic
59C32×C4⋊C4—21no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C4⋊C4C2×D311yescharacteristic
63C32×C4⋊C4C22×C311yescharacteristic
64C32×C4⋊C4C22×C311yescharacteristic
65#3752—21no—
66C4⋊(C3⋊Dic3)C22×C311yescharacteristic
67C4⋊(C3⋊Dic3)C22×C311yescharacteristic
68#3750—21no—
69C4×C3⋊Dic3—21no—
70C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
71#3743C3×C414yescomplemented
72C3×(C2×D3)⋊C4—61no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—31no—
75C3×Dic3⋊C4—61no—
76C3×C4×Dic3—61no—
77C2×C3×C3⋊C8—31no—
78C3×ΓL(1,𝔽25)—34no—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3—21no—
82C2×(C3⋊D3)⋊C3C4⋊C412yes—
83C4×H(3,𝔽3)—41no—
84C4×H(3,𝔽3)—41no—
85C4×H(3,𝔽3)C22×C411yescharacteristic
86C4×H(3,𝔽3)C2×Q811yescharacteristic
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89(C3⋊Dic3)⋊C3C4⋊C412yes—
90C2×C3×M16—91no—
91C3×#117—91no—
92C2×C3×C4⋊C4—91no—
93C2×ΓL(1,𝔽25)—31no—
94C2×ΓL(1,𝔽25)—91no—
95#892—31no—
96#892—91no—
97D3×C4⋊C4—31no—
98D3×C4⋊C4—91no—
99C3×#107—31no—
100C3×#107—31no—
101C3×#107—61no—
102C3×#107—91no—
103C3×#107C3×D311yescharacteristic
104C3×#106—91no—
105C3×#106—91no—
106#817—31no—
107#817—91no—
108#808—31no—
109#808—91no—
110#807—31no—
111#807—91no—
112C22×C3⋊D3C3×D411yescharacteristic
113C22×C3×D3—31no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—41no—
120C2×C32×C4—41no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C23×C311yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3C3×D411yescharacteristic
126C4×C3⋊D3—42no—
127C4×C3⋊D3C2×C3×C412yes—
128C4×C3⋊D3C3×Q812yes—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C3×C4×D3—32no—
133C3×C4×D3—32no—
134C3×C4×D3—122no—
135C32×C8—22no—
136C32×C8C4×D312yescomplemented
137C32×C8C2×C3×C412yescomplemented
138C3⋊(C3⋊C8)C2×C3×C412yescomplemented
139C3×C3⋊C8—32no—
140#428—91noSylow
141C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
142C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
143C2×H(3,𝔽3)#13711yescharacteristic
144(C3⋊D3)⋊C3—22no—
145C22×C3×C4—91no—
146C22×C3×C4—181no—
147C2×C4×D3—31no—
148C2×C4×D3—61no—
149C2×C4×D3—91no—
150C2×C4×D3—181no—
151C3×M16—94no—
152C2×C3×C8—31no—
153C2×C3×C8—31no—
154C2×C3×C8—61no—
155C2×C3×C8—91no—
156C2×C3×C8C2×C3×D311yescharacteristic
157C3×C4⋊C4—31no—
158C3×C4⋊C4—31no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4—61no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4—181no—
166C3×C4⋊C4C2×C3×D311yescharacteristic
167C3×C4⋊C4C2×C3×D311yescharacteristic
168C3×C22⋊C4—181no—
169C3×C42—181no—
170(C2×D3)⋊C4—61no—
171(C2×D3)⋊C4—181no—
172C4⋊Dic3—31no—
173C4⋊Dic3—31no—
174C4⋊Dic3—91no—
175C4⋊Dic3—91no—
176Dic3⋊C4—61no—
177Dic3⋊C4—181no—
178C4×Dic3—61no—
179C4×Dic3—181no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—91no—
182ΓL(1,𝔽25)—34no—
183ΓL(1,𝔽25)—94no—
184C22×C32—21no—
185C22×C32D3×D411yescharacteristic
186C22×C32C2×C3×D411yescharacteristic
187C2×C3⋊D3—21no—
188C2×C3⋊D3—21no—
189C2×C3⋊D3C3×C4⋊C412yes—
190C2×C3×D3—32no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4—41no—
196C32×C4—41no—
197C32×C4—41no—
198C32×C4—41no—
199C32×C4—81no—
200C32×C4—81no—
201C32×C4C2×C4×D311yescharacteristic
202C32×C4D3×Q811yescharacteristic
203C32×C4C22×C3×C411yescharacteristic
204C32×C4C2×C3×Q811yescharacteristic
205C3⋊Dic3—41no—
206C3⋊Dic3—41no—
207C3⋊Dic3C3×C4⋊C412yes—
208C3×Dic3—32no—
209C3×Dic3—121no—
210C3×Dic3—121no—
211C2×M16—91no—
212#117—91no—
213C2×C4⋊C4—91no—
214#107—91no—
215#107(C3⋊D3)⋊C311yescharacteristic, complemented
216#106—91no—
217#106—91no—
218H(3,𝔽3)#42811yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C22×D3—31no—
221C22×D3—91no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—31no—
225C2×C3×C4—31no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—61no—
231C2×C3×C4—92no—
232C2×C3×C4—92no—
233C2×C3×C4—91no—
234C2×C3×C4—121no—
235C2×C3×C4—121no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4—362no—
239C2×C3×C4C22×C3×D311yescharacteristic
240C2×Dic3—31no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—91no—
244C2×Dic3—181no—
245C2×Dic3—181no—
246C4×D3—32no—
247C4×D3—32no—
248C4×D3—92no—
249C4×D3—92no—
250C4×D3—122no—
251C4×D3—362no—
252C3×C8—32no—
253C3×C8—32no—
254C3×C8—62no—
255C3×C8—92no—
256C3×C8C3×C4×D312yes—
257C3⋊C8—32no—
258C3⋊C8—92no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32D3×C4⋊C411yescharacteristic
263C2×C32#91111yescharacteristic
264C2×C32#91811yescharacteristic
265C2×C32C2×C3×C4⋊C411yescharacteristic
266C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
267C2×C32C3×#13711yescharacteristic
268C3⋊D3—22no—
269C3×D3—62no—
270C22×C4—91no—
271C22×C4—181no—
272M16—94no—
273C2×C8—91no—
274C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
275C4⋊C4—91no—
276C4⋊C4—91no—
277C4⋊C4—181no—
278C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
279C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
280C22⋊C4—181no—
281C42—181no—
282C22×C3—31no—
283C22×C3—31no—
284C22×C3—61no—
285C22×C3—92no—
286C22×C3—181no—
287C22×C3—181no—
288C22×C3C3×D3×D411yescharacteristic
289C2×D3—32no—
290C2×D3—61no—
291C2×D3—61no—
292C2×D3—92no—
293C2×D3—181no—
294C2×D3—181no—
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—31no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—61no—
302C3×C4—61no—
303C3×C4—92no—
304C3×C4—121no—
305C3×C4—121no—
306C3×C4—121no—
307C3×C4—121no—
308C3×C4—241no—
309C3×C4—241no—
310C3×C4—361no—
311C3×C4—361no—
312C3×C4C2×C3×C4×D311yescharacteristic
313C3×C4C3×D3×Q811yescharacteristic
314Dic3—32no—
315Dic3—92no—
316Dic3—121no—
317Dic3—121no—
318Dic3—361no—
319Dic3—361no—
320C32—21no—
321C32#488111yescharacteristic, complemented
322C32C3×#42811yescharacteristic, complemented
323C23—91no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—91no—
327C2×C4—92no—
328C2×C4—92no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—362no—
332C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
333C8—92no—
334C8C4×(C3⋊D3)⋊C312yescomplemented
335C2×C3—31no—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—61no—
342C2×C3—61no—
343C2×C3—61no—
344C2×C3—182no—
345C2×C3C3×D3×C4⋊C411yescharacteristic
346C2×C3C3×#91111yescharacteristic
347C2×C3C3×#91811yescharacteristic
348D3—62no—
349D3—182no—
350C22—92no—
351C22—181no—
352C22—181no—
353C22D4×(C3⋊D3)⋊C311yescharacteristic
354C4—41no—
355C4—41no—
356C4—92no—
357C4—361no—
358C4—361no—
359C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
360C4Q8×(C3⋊D3)⋊C311yescharacteristic
361C3—31no—
362C3—31no—
363C3—61no—
364C3C3×#488111yescharacteristic
365C2—182no—
366C2C4⋊C4×(C3⋊D3)⋊C311yescharacteristic
367C2#64120⋊C311yescharacteristic
368C2#64127⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×(C3⋊D3)⋊C3D4
cosocleC2×C32×C4C23×C3
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#428
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
4. H(3,𝔽3)×#107C2semidirect
5. #63644⋊C3C2semidirect
6. #63635⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592580C3semidirect
21. #3743⋊C3C4semidirect
29. C32×#107D3semidirect
30. C32×#107C2×C3semidirect
31. #63644C2×C3semidirect
32. #63635C2×C3semidirect
33. #63634C2×C3semidirect
47. C8×H(3,𝔽3)C2×C4semidirect
48. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
71. #3743C3×C4semidirect
136. C32×C8C4×D3semidirect
137. C32×C8C2×C3×C4semidirect
138. C3⋊(C3⋊C8)C2×C3×C4semidirect
215. #107(C3⋊D3)⋊C3semidirect
218. H(3,𝔽3)#428semidirect
321. C32#4881semidirect
322. C32C3×#428semidirect
334. C8C4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3743⋊C32
#64101⋊C32
C4⋊C4×(C3⋊D3)⋊C32
H(3,𝔽3)×#1072
#63644⋊C32
#63635⋊C32
#63634⋊C32
#105925803
C3×#48813