Finite Group Databasebeta

← #10592539⋊C3 #13142061

Subgroups — #10592539⋊C3 #13142061

Order 1728.

374 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64081⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic, complemented
3#63647⋊C3C211yesmaximal, characteristic
4#63647⋊C3C211yesmaximal, characteristic
5#63642⋊C3C211yesmaximal, characteristic, complemented
6C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
7C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
8#10592539C311yesmaximal, characteristic, complemented
9C3×#4840—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C42×H(3,𝔽3)C2211yescharacteristic
14#3752⋊C3C411yescharacteristic
15#3752⋊C3C411yescharacteristic
16#3750⋊C3C411yescharacteristic
17#3750⋊C3C411yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64081C2×C311yescharacteristic
22C3×#872—31no—
23C32×C4⋊C8—21no—
24C32×C4⋊C8D311yescharacteristic, complemented
25C32×C4⋊C8C2×C311yescharacteristic, complemented
26#63647C2×C311yescharacteristic
27#63647C2×C311yescharacteristic
28#63642C2×C311yescharacteristic, complemented
29C8×C3⋊Dic3C2×C311yescharacteristic, complemented
30C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
31C3×(C2×D3)⋊C8—31no—
32C3×(C2×D3)⋊C8—31no—
33C3×#815—31no—
34C3×C8×Dic3—31no—
35C3×C4⋊(C3⋊C8)—31no—
36C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)C2×C411yescharacteristic
38C2×C4×H(3,𝔽3)C2×C411yescharacteristic
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
41C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
42C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
43C4×(C3⋊D3)⋊C3—21no—
44C4×(C3⋊D3)⋊C3—21no—
45C8×H(3,𝔽3)—21no—
46C8×H(3,𝔽3)—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49C3×#433—91no—
50#4840—31no—
51#4840—91no—
52C2×C4×C3⋊D3C22×C311yescharacteristic
53C2×C3×C4×D3—31no—
54C2×C32×C8—21no—
55C2×C32×C8—21no—
56C2×C32×C8C2×D311yescharacteristic
57C2×C32×C8C2×D311yescharacteristic
58C2×C32×C8C22×C311yescharacteristic
59C2×C32×C8C22×C311yescharacteristic
60C32×C42—21no—
61C32×C42C2×D311yescharacteristic
62C32×C42C22×C311yescharacteristic
63#3752C3×C411yescharacteristic
64#3752C3×C411yescharacteristic
65#3750C3×C411yescharacteristic
66#3750C3×C411yescharacteristic
67C4×C3⋊Dic3C22×C311yescharacteristic
68C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
69C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
70C3×(C2×D3)⋊C4—31no—
71C3×(C2×D3)⋊C4—31no—
72C3×Dic3⋊C4—31no—
73C3×Dic3⋊C4—31no—
74C3×C4×Dic3—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C22×H(3,𝔽3)C22×C411yescharacteristic
78C2×(C3⋊D3)⋊C3—21no—
79C2×(C3⋊D3)⋊C3—21no—
80C2×(C3⋊D3)⋊C3—21no—
81C4×H(3,𝔽3)—21no—
82C4×H(3,𝔽3)—21no—
83C4×H(3,𝔽3)#4111yescharacteristic
84C4×H(3,𝔽3)#4111yescharacteristic
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—21no—
88C3×#117—91no—
89#872—31no—
90#872—91no—
91C3×C4⋊C8—31no—
92C3×C4⋊C8—31no—
93C3×C4⋊C8—61no—
94C3×C4⋊C8—91no—
95C3×C4⋊C8C3×D311yescharacteristic
96C3×C22⋊C8—91no—
97C3×C22⋊C8—91no—
98C3×#97—91no—
99C3×C4×C8—91no—
100(C2×D3)⋊C8—31no—
101(C2×D3)⋊C8—31no—
102(C2×D3)⋊C8—91no—
103(C2×D3)⋊C8—91no—
104#815—31no—
105#815—91no—
106C8×Dic3—31no—
107C8×Dic3—91no—
108C4⋊(C3⋊C8)—31no—
109C4⋊(C3⋊C8)—91no—
110C22×C3⋊D3C2×C3×C411yescharacteristic
111C22×C3×D3—31no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4C4×D311yescharacteristic
116C2×C32×C4C4×D311yescharacteristic
117C2×C32×C4C2×C3×C411yescharacteristic
118C2×C32×C4C2×C3×C411yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C23×C311yescharacteristic
121C2×C3⋊Dic3C2×C3×C411yescharacteristic
122C2×C3⋊Dic3C2×C3×C411yescharacteristic
123C2×C3⋊Dic3C2×C3×C411yescharacteristic
124C4×C3⋊D3—21no—
125C4×C3⋊D3—21no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—31no—
129C3×C4×D3—61no—
130C3×C4×D3—61no—
131C32×C8—21no—
132C32×C8—21no—
133C32×C8—21no—
134C32×C8—21no—
135C32×C8—41no—
136C32×C8—41no—
137C3⋊(C3⋊C8)—21no—
138C3⋊(C3⋊C8)—21no—
139C3×C3⋊C8—61no—
140C3×C3⋊C8—61no—
141#433—91noSylow
142C2×H(3,𝔽3)#11711yescharacteristic
143C2×H(3,𝔽3)#13111yescharacteristic
144C2×H(3,𝔽3)#13111yescharacteristic
145(C3⋊D3)⋊C3—41no—
146C22×C3×C4—91no—
147C2×C4×D3—31no—
148C2×C4×D3—91no—
149C2×C3×C8—31no—
150C2×C3×C8—31no—
151C2×C3×C8—31no—
152C2×C3×C8—31no—
153C2×C3×C8—61no—
154C2×C3×C8—61no—
155C2×C3×C8—91no—
156C2×C3×C8—91no—
157C2×C3×C8C2×C3×D311yescharacteristic
158C2×C3×C8C2×C3×D311yescharacteristic
159C3×C4⋊C4—91no—
160C3×C4⋊C4—91no—
161C3×C22⋊C4—91no—
162C3×C22⋊C4—91no—
163C3×C42—31no—
164C3×C42—31no—
165C3×C42—61no—
166C3×C42—91no—
167C3×C42C2×C3×D311yescharacteristic
168(C2×D3)⋊C4—31no—
169(C2×D3)⋊C4—31no—
170(C2×D3)⋊C4—91no—
171(C2×D3)⋊C4—91no—
172Dic3⋊C4—31no—
173Dic3⋊C4—31no—
174Dic3⋊C4—91no—
175Dic3⋊C4—91no—
176C4×Dic3—31no—
177C4×Dic3—91no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—91no—
182C22×C32—21no—
183C22×C32C2×C4×D311yescharacteristic
184C22×C32C22×C3×C411yescharacteristic
185C2×C3⋊D3—21no—
186C2×C3⋊D3—21no—
187C2×C3⋊D3—21no—
188C2×C3×D3—61no—
189C2×C3×D3—61no—
190C2×C3×D3—61no—
191C32×C4—22no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4—21no—
196C32×C4—21no—
197C32×C4—21no—
198C32×C4—41no—
199C32×C4#23711yescharacteristic
200C32×C4#23911yescharacteristic
201C32×C4C3×#4111yescharacteristic
202C32×C4C3×#4111yescharacteristic
203C3⋊Dic3—21no—
204C3⋊Dic3—21no—
205C3⋊Dic3—21no—
206C3×Dic3—61no—
207C3×Dic3—61no—
208C3×Dic3—61no—
209#117—91no—
210C4⋊C8—91no—
211C4⋊C8(C3⋊D3)⋊C311yescharacteristic, complemented
212C22⋊C8—91no—
213C22⋊C8—91no—
214#97—91no—
215C4×C8—91no—
216H(3,𝔽3)#43311yescharacteristic, complemented, Sylow
217C23×C3—91no—
218C22×D3—31no—
219C22×D3—91no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—31no—
224C2×C3×C4—31no—
225C2×C3×C4—31no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—91no—
232C2×C3×C4—181no—
233C2×C3×C4—181no—
234C2×C3×C4C3×C4×D311yescharacteristic
235C2×C3×C4C3×C4×D311yescharacteristic
236C2×C3×C4C22×C3×D311yescharacteristic
237C2×Dic3—31no—
238C2×Dic3—31no—
239C2×Dic3—31no—
240C2×Dic3—91no—
241C2×Dic3—91no—
242C2×Dic3—91no—
243C4×D3—61no—
244C4×D3—61no—
245C4×D3—181no—
246C4×D3—181no—
247C3×C8—21no—
248C3×C8—21no—
249C3×C8—61no—
250C3×C8—61no—
251C3×C8—61no—
252C3×C8—61no—
253C3×C8—121no—
254C3×C8—121no—
255C3×C8—181no—
256C3×C8—181no—
257C3⋊C8—61no—
258C3⋊C8—61no—
259C3⋊C8—181no—
260C3⋊C8—181no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32#89211yescharacteristic
265C2×C32#90111yescharacteristic
266C2×C32#90711yescharacteristic
267C2×C32C3×#11711yescharacteristic
268C2×C32C3×#13111yescharacteristic
269C2×C32C3×#13111yescharacteristic
270C3⋊D3—41no—
271C3×D3—121no—
272C22×C4—91no—
273C2×C8—91no—
274C2×C8—91no—
275C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
276C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
277C4⋊C4—91no—
278C4⋊C4—91no—
279C22⋊C4—91no—
280C22⋊C4—91no—
281C42—91no—
282C42C2×(C3⋊D3)⋊C311yescharacteristic
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—61no—
286C22×C3—181no—
287C22×C3—181no—
288C22×C3—181no—
289C22×C3C2×C3×C4×D311yescharacteristic
290C2×D3—61no—
291C2×D3—61no—
292C2×D3—61no—
293C2×D3—181no—
294C2×D3—181no—
295C2×D3—181no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—61no—
303C3×C4—61no—
304C3×C4—61no—
305C3×C4—61no—
306C3×C4—62no—
307C3×C4—61no—
308C3×C4—61no—
309C3×C4—121no—
310C3×C4—181no—
311C3×C4—181no—
312C3×C4—181no—
313C3×C4C3×#23711yescharacteristic
314C3×C4C3×#23911yescharacteristic
315Dic3—61no—
316Dic3—61no—
317Dic3—61no—
318Dic3—181no—
319Dic3—181no—
320Dic3—181no—
321C32—21no—
322C32#484011yescharacteristic, complemented
323C32C3×#43311yescharacteristic, complemented
324C23—91no—
325C2×C4—91no—
326C2×C4—91no—
327C2×C4—91no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
331C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
332C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
333C8—21no—
334C8—21no—
335C8—181no—
336C8—181no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—61no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—361no—
347C2×C3C3×#89211yescharacteristic
348C2×C3C3×#90111yescharacteristic
349C2×C3C3×#90711yescharacteristic
350D3—121no—
351D3—361no—
352C22—181no—
353C22—181no—
354C22—181no—
355C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
356C4—21no—
357C4—21no—
358C4—181no—
359C4—181no—
360C4—181no—
361C4#3830⋊C311yescharacteristic
362C4#8872211yescharacteristic
363C3—31no—
364C3—31no—
365C3—61no—
366C3C3×#484011yescharacteristic
367C2—361no—
368C2#64101⋊C311yescharacteristic
369C2#64110⋊C311yescharacteristic
370C2#64116⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C4⋊C8C2
socleC22×C3C2×C3×C4×D3
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC4⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#433
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×C4⋊C8C2semidirect
5. #63642⋊C3C2semidirect
6. C8×(C3⋊Dic3)⋊C3C2semidirect
7. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
8. #10592539C3semidirect
24. C32×C4⋊C8D3semidirect
25. C32×C4⋊C8C2×C3semidirect
28. #63642C2×C3semidirect
29. C8×C3⋊Dic3C2×C3semidirect
30. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
211. C4⋊C8(C3⋊D3)⋊C3semidirect
216. H(3,𝔽3)#433semidirect
322. C32#4840semidirect
323. C32C3×#433semidirect

Maximal subgroups

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

subgroupindex
#64081⋊C32
H(3,𝔽3)×C4⋊C82
#63647⋊C32
#63642⋊C32
C8×(C3⋊Dic3)⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105925393
C3×#48403