Finite Group Databasebeta

← #10592812⋊C3 #13143128

Subgroups — #10592812⋊C3 #13143128

Order 1728.

629 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic
2C22×C8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#64135⋊C3C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal, complemented
5#63643⋊C3C212yesmaximal, complemented
6#10592812C311yesmaximal, characteristic, complemented
7C3×#5113—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#638⋊C3—21no—
10#3828⋊C3—22no—
11#3828⋊C3C414yescomplemented
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3—21no—
14C2×#634⋊C3C2211yescharacteristic
15C2×C8×H(3,𝔽3)—22no—
16C2×C8×H(3,𝔽3)C2212yes—
17#3757⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2212yes—
19C4×(C3⋊Dic3)⋊C3—21no—
20C2×#3828C2×C311yescharacteristic
21C2×C3×#235—31no—
22C22×C32×C8—21no—
23C22×C32×C8D311yescharacteristic, complemented
24C22×C32×C8C2×C311yescharacteristic, complemented
25#64135C2×C311yescharacteristic, complemented
26C3×#926—31no—
27#63648C2×C312yescomplemented
28#63643C2×C312yescomplemented
29C3×#821—32no—
30C3×#816—32no—
31C23×H(3,𝔽3)D411yescharacteristic
32C22×(C3⋊D3)⋊C3—21no—
33C2×C4×H(3,𝔽3)C2×C412yes—
34C2×C4×H(3,𝔽3)D411yescharacteristic
35C2×C4×H(3,𝔽3)D412yes—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37#638⋊C3—24no—
38C2×(C3⋊Dic3)⋊C3—21no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C4⋊((C3⋊D3)⋊C3)—21no—
41C4⋊((C3⋊D3)⋊C3)C2×C412yes—
42C4×(C3⋊D3)⋊C3—24no—
43#634⋊C3—21no—
44#634⋊C3C2×C412yes—
45C8×H(3,𝔽3)—22no—
46C3×#416—91no—
47#5113—31no—
48#5113—91no—
49C22×C32×C4—21no—
50C22×C32×C4C2×D311yescharacteristic
51C22×C32×C4C22×C311yescharacteristic
52C2×#638—21no—
53#3828—22no—
54#3828C3×C414yescomplemented
55C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
56C2×C4×C3⋊D3—21no—
57C2×#634C22×C311yescharacteristic
58C2×C3×#67—61no—
59C3×#235—34no—
60C3×#235—62no—
61C2×C3×D12—31no—
62C2×C3×C4×D3—61no—
63C2×C3×Dic6—31no—
64C2×C32×C8—24no—
65C2×C32×C8—22no—
66C2×C32×C8—22no—
67C2×C32×C8—22no—
68C2×C32×C8C2×D312yes—
69C2×C32×C8C22×C312yes—
70#3757—21no—
71C4⋊(C3⋊Dic3)C22×C312yes—
72C4×C3⋊Dic3—21no—
73C3×#217—61no—
74C3×C4⋊Dic3—32no—
75C3×C4×Dic3—61no—
76C22×H(3,𝔽3)—22no—
77C22×H(3,𝔽3)C22⋊C412yes—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—22no—
80C2×(C3⋊D3)⋊C3—42no—
81C4×H(3,𝔽3)C22⋊C412yes—
82C4×H(3,𝔽3)C22×C411yescharacteristic
83C4×H(3,𝔽3)C2×D411yescharacteristic
84(C3⋊Dic3)⋊C3—22no—
85(C3⋊Dic3)⋊C3—42no—
86C2×C3×#41—91no—
87C2×#235—31no—
88C2×#235—91no—
89C22×C3×C8—31no—
90C22×C3×C8—31no—
91C22×C3×C8—61no—
92C22×C3×C8C3×D311yescharacteristic
93C3×#117—91no—
94#926—31no—
95#926—91no—
96C3×#103—92no—
97C3×#102—92no—
98#821—32no—
99#821—92no—
100#816—32no—
101#816—92no—
102C23×C32—21no—
103C23×C32D1211yescharacteristic
104C23×C32C3×D411yescharacteristic
105C22×C3⋊D3—21no—
106C22×C3×D3—61no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—22no—
110C2×C32×C4—21no—
111C2×C32×C4C4×D312yes—
112C2×C32×C4D1211yescharacteristic
113C2×C32×C4#6712yes—
114C2×C32×C4C2×C3×C412yes—
115C2×C32×C4C3×D411yescharacteristic
116C2×C32×C4C3×D412yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C32×C4C23×C311yescharacteristic
119#638—24no—
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—22no—
122C4⋊(C3⋊D3)—21no—
123C4⋊(C3⋊D3)C2×C3×C412yes—
124C4×C3⋊D3—24no—
125#634—21no—
126#634C2×C3×C412yes—
127C3×#67—64no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—62no—
130C3×D12—32no—
131C3×D12—61no—
132C3×C4×D3—64no—
133C3×Dic6—32no—
134C3×Dic6—61no—
135C32×C8—24no—
136C32×C8—22no—
137C32×C8—22no—
138#416—91noSylow
139C2×H(3,𝔽3)—22no—
140C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
141C2×H(3,𝔽3)#13512yes—
142(C3⋊D3)⋊C3—42no—
143C3×#41—94no—
144C3×#41—182no—
145C2×C3×Q8—91no—
146C2×C3×D4—91no—
147C2×C3×D4—181no—
148C22×C3×C4—31no—
149C22×C3×C4—31no—
150C22×C3×C4—61no—
151C22×C3×C4—181no—
152C22×C3×C4C2×C3×D311yescharacteristic
153C2×#67—61no—
154C2×#67—181no—
155#235—34no—
156#235—62no—
157#235—94no—
158#235—182no—
159C2×D12—31no—
160C2×D12—91no—
161C2×C4×D3—61no—
162C2×C4×D3—181no—
163C2×Dic6—31no—
164C2×Dic6—91no—
165C2×C3×C8—22no—
166C2×C3×C8—32no—
167C2×C3×C8—32no—
168C2×C3×C8—62no—
169C2×C3×C8—62no—
170C2×C3×C8—64no—
171C2×C3×C8—62no—
172C2×C3×C8C2×C3×D312yes—
173C3×C4⋊C4—92no—
174C3×C22⋊C4—181no—
175C3×C42—181no—
176#217—61no—
177#217—181no—
178C4⋊Dic3—32no—
179C4⋊Dic3—92no—
180C4×Dic3—61no—
181C4×Dic3—181no—
182C22×C32—24no—
183C22×C32—22no—
184C22×C32—22no—
185C22×C32—22no—
186C22×C32—21no—
187C22×C32(C2×D3)⋊C412yes—
188C22×C32C3×C22⋊C412yes—
189C22×C32C2×D1211yescharacteristic
190C22×C32C2×C3×D411yescharacteristic
191C2×C3⋊D3—22no—
192C2×C3⋊D3—42no—
193C2×C3×D3—62no—
194C2×C3×D3—122no—
195C32×C4—22no—
196C32×C4—21no—
197C32×C4—21no—
198C32×C4(C2×D3)⋊C412yes—
199C32×C4C3×C22⋊C412yes—
200C32×C4C2×C4×D311yescharacteristic
201C32×C4C2×#6711yescharacteristic
202C32×C4C22×C3×C411yescharacteristic
203C32×C4C2×C3×D411yescharacteristic
204C3⋊Dic3—22no—
205C3⋊Dic3—42no—
206C3×Dic3—62no—
207C3×Dic3—122no—
208C2×#41—91no—
209C22×C8(C3⋊D3)⋊C311yescharacteristic, complemented
210#117—91no—
211#103—92no—
212#102—92no—
213H(3,𝔽3)#41611yescharacteristic, complemented, Sylow
214C23×C3—31no—
215C23×C3—31no—
216C23×C3—61no—
217C23×C3—181no—
218C23×C3C3×D1211yescharacteristic
219C22×D3—61no—
220C22×D3—181no—
221C3×Q8—92no—
222C3×Q8—181no—
223C3×D4—92no—
224C3×D4—181no—
225C3×D4—184no—
226C2×C3×C4—32no—
227C2×C3×C4—32no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—32no—
231C2×C3×C4—32no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—62no—
235C2×C3×C4—61no—
236C2×C3×C4—62no—
237C2×C3×C4—61no—
238C2×C3×C4—184no—
239C2×C3×C4—181no—
240C2×C3×C4—182no—
241C2×C3×C4C3×C4×D312yes—
242C2×C3×C4C3×D1211yescharacteristic
243C2×C3×C4C3×#6712yes—
244C2×C3×C4C22×C3×D311yescharacteristic
245#67—64no—
246#67—184no—
247C2×Dic3—61no—
248C2×Dic3—62no—
249C2×Dic3—181no—
250C2×Dic3—182no—
251D12—32no—
252D12—61no—
253D12—92no—
254D12—181no—
255C4×D3—64no—
256C4×D3—184no—
257Dic6—32no—
258Dic6—61no—
259Dic6—92no—
260Dic6—181no—
261C3×C8—22no—
262C3×C8—62no—
263C3×C8—62no—
264C3×C8—64no—
265C2×C32—24no—
266C2×C32—22no—
267C2×C32—22no—
268C2×C32—22no—
269C2×C32—21no—
270C2×C32#90412yes—
271C2×C32C2×(C2×D3)⋊C411yescharacteristic
272C2×C32C2×C3×C22⋊C411yescharacteristic
273C2×C32C3×#13512yes—
274C3⋊D3—42no—
275C3×D3—122no—
276#41—94no—
277#41—182no—
278C2×Q8—91no—
279C2×D4—91no—
280C2×D4—181no—
281C22×C4—181no—
282C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
283C2×C8—22no—
284C2×C8C2×(C3⋊D3)⋊C312yes—
285C4⋊C4—92no—
286C22⋊C4—181no—
287C42—181no—
288C22×C3—22no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—32no—
292C22×C3—32no—
293C22×C3—62no—
294C22×C3—62no—
295C22×C3—64no—
296C22×C3—61no—
297C22×C3—62no—
298C22×C3—182no—
299C22×C3—362no—
300C22×C3C3×(C2×D3)⋊C412yes—
301C22×C3C2×C3×D1211yescharacteristic
302C2×D3—62no—
303C2×D3—122no—
304C2×D3—182no—
305C2×D3—362no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—32no—
311C3×C4—32no—
312C3×C4—61no—
313C3×C4—61no—
314C3×C4—62no—
315C3×C4—182no—
316C3×C4—362no—
317C3×C4C3×(C2×D3)⋊C412yes—
318C3×C4C2×C3×C4×D311yescharacteristic
319C3×C4C2×C3×#6711yescharacteristic
320Dic3—62no—
321Dic3—122no—
322Dic3—182no—
323Dic3—362no—
324C32—21no—
325C32#511311yescharacteristic, complemented
326C32C3×#41611yescharacteristic, complemented
327C23—181no—
328C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
329Q8—92no—
330Q8—181no—
331D4—92no—
332D4—184no—
333D4—181no—
334C2×C4—181no—
335C2×C4—184no—
336C2×C4—182no—
337C2×C4C4×(C3⋊D3)⋊C312yes—
338C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
339C2×C4#638⋊C312yes—
340C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
341C8—22no—
342C2×C3—22no—
343C2×C3—32no—
344C2×C3—32no—
345C2×C3—31no—
346C2×C3—31no—
347C2×C3—62no—
348C2×C3—62no—
349C2×C3—64no—
350C2×C3—62no—
351C2×C3—61no—
352C2×C3—362no—
353C2×C3C3×#90412yes—
354C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
355D3—122no—
356D3—362no—
357C22—22no—
358C22—182no—
359C22—362no—
360C22#3752⋊C312yes—
361C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
362C4—182no—
363C4—362no—
364C4#3752⋊C312yes—
365C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
366C4C2×#638⋊C311yescharacteristic
367C3—31no—
368C3—31no—
369C3—61no—
370C3C3×#511311yescharacteristic
371C2—22no—
372C2—362no—
373C2#64113⋊C312yes—
374C2C2×#3752⋊C311yescharacteristic
subgroupG/H
centerC2×C4C4⋊((C3⋊D3)⋊C3)
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC22×C8×H(3,𝔽3)C2
socleC22×C3C2×C3×D12
prime radicalC2×#3828⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenterC22×C8(C3⋊D3)⋊C3
hypocenterC32C3×#416
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. C22×C8×H(3,𝔽3)C2semidirect
3. #64135⋊C3C2semidirect
4. #63648⋊C3C2semidirect
5. #63643⋊C3C2semidirect
6. #10592812C3semidirect
11. #3828⋊C3C4semidirect
23. C22×C32×C8D3semidirect
24. C22×C32×C8C2×C3semidirect
25. #64135C2×C3semidirect
27. #63648C2×C3semidirect
28. #63643C2×C3semidirect
54. #3828C3×C4semidirect
209. C22×C8(C3⋊D3)⋊C3semidirect
213. H(3,𝔽3)#416semidirect
325. C32#5113semidirect
326. C32C3×#416semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C22×C8×H(3,𝔽3)2
#64135⋊C32
#63648⋊C32
#63643⋊C32
#105928123
C3×#51133