Finite Group Databasebeta

← #10592419⋊C3 #13141603

Subgroups — #10592419⋊C3 #13141603

Order 1728.

386 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64135⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic
4#63642⋊C3C211yesmaximal, characteristic, complemented
5#63641⋊C3C211yesmaximal, characteristic, complemented
6#63641⋊C3C211yesmaximal, characteristic, complemented
7C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
8#10592419C311yesmaximal, characteristic, complemented
9C3×#4720—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13#3757⋊C3C411yescharacteristic
14#3757⋊C3C411yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64135C2×C311yescharacteristic
22C3×#926—31no—
23C32×C22⋊C8—21no—
24C32×C22⋊C8D311yescharacteristic
25C32×C22⋊C8C2×C311yescharacteristic
26#63657C2×C311yescharacteristic
27#63642C2×C311yescharacteristic, complemented
28#63641C2×C311yescharacteristic, complemented
29#63641C2×C311yescharacteristic, complemented
30C8×C3⋊Dic3C2×C311yescharacteristic, complemented
31C3×#830—31no—
32C3×#815—31no—
33C3×Dic3⋊C8—31no—
34C3×Dic3⋊C8—31no—
35C3×C8×Dic3—31no—
36C23×H(3,𝔽3)C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2×C411yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
42C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
43C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
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#4720—31no—
51#4720—91no—
52C22×C32×C4—21no—
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C22×C311yescharacteristic
55C2×C32×C8—21no—
56C2×C32×C8—21no—
57C2×C32×C8C2×D311yescharacteristic
58C2×C32×C8C2×D311yescharacteristic
59C2×C32×C8C22×C311yescharacteristic
60C2×C32×C8C22×C311yescharacteristic
61#3757C3×C411yescharacteristic
62#3757C3×C411yescharacteristic
63C4⋊(C3⋊Dic3)C3×C411yescharacteristic
64C4⋊(C3⋊Dic3)C3×C411yescharacteristic
65C4×C3⋊Dic3C22×C311yescharacteristic
66C4×C3⋊Dic3C22×C311yescharacteristic
67C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
68C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
69C3×#217—31no—
70C3×#217—31no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C3×C4×Dic3—31no—
74C3×C4×Dic3—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C22×H(3,𝔽3)—21no—
78C22×H(3,𝔽3)—21no—
79C22×H(3,𝔽3)—21no—
80C22×H(3,𝔽3)C22×C411yescharacteristic
81C4×H(3,𝔽3)—21no—
82C4×H(3,𝔽3)#4111yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84(C3⋊Dic3)⋊C3—21no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—21no—
88C3×#117—91no—
89#926—31no—
90#926—91no—
91C3×C4⋊C8—91no—
92C3×C4⋊C8—91no—
93C3×C22⋊C8—31no—
94C3×C22⋊C8—31no—
95C3×C22⋊C8—61no—
96C3×C22⋊C8—91no—
97C3×C22⋊C8C3×D311yescharacteristic
98C3×#97—91no—
99C3×C4×C8—91no—
100#830—31no—
101#830—91no—
102#815—31no—
103#815—91no—
104Dic3⋊C8—31no—
105Dic3⋊C8—31no—
106Dic3⋊C8—91no—
107Dic3⋊C8—91no—
108C8×Dic3—31no—
109C8×Dic3—91no—
110C23×C32—21no—
111C23×C32C4×D311yescharacteristic
112C23×C32C2×C3×C411yescharacteristic
113C2×C32×C4—21no—
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×C4C4×D311yescharacteristic
122C2×C32×C4C2×C3×C411yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C23×C311yescharacteristic
125C2×C3⋊Dic3C2×C3×C411yescharacteristic
126C2×C3⋊Dic3C2×C3×C411yescharacteristic
127C2×C3⋊Dic3C2×C3×C411yescharacteristic
128C2×C3⋊Dic3C2×C3×C411yescharacteristic
129C2×C3×Dic3—31no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—31no—
133C32×C8—21no—
134C32×C8—21no—
135C32×C8—21no—
136C32×C8—21no—
137C32×C8—41no—
138C32×C8—41no—
139C3⋊(C3⋊C8)—21no—
140C3⋊(C3⋊C8)—21no—
141C3×C3⋊C8—61no—
142C3×C3⋊C8—61no—
143#433—91noSylow
144C2×H(3,𝔽3)—41no—
145C2×H(3,𝔽3)#11711yescharacteristic
146C2×H(3,𝔽3)#13111yescharacteristic
147C2×H(3,𝔽3)#13111yescharacteristic
148C22×C3×C4—31no—
149C22×C3×C4—31no—
150C22×C3×C4—61no—
151C22×C3×C4C2×C3×D311yescharacteristic
152C2×C3×C8—31no—
153C2×C3×C8—31no—
154C2×C3×C8—31no—
155C2×C3×C8—31no—
156C2×C3×C8—61no—
157C2×C3×C8—61no—
158C2×C3×C8—91no—
159C2×C3×C8—91no—
160C2×C3×C8C2×C3×D311yescharacteristic
161C2×C3×C8C2×C3×D311yescharacteristic
162C3×C4⋊C4—91no—
163C3×C4⋊C4—91no—
164C3×C22⋊C4—91no—
165C3×C22⋊C4—91no—
166C3×C42—91no—
167C3×C42—91no—
168#217—31no—
169#217—31no—
170#217—91no—
171#217—91no—
172C4⋊Dic3—31no—
173C4⋊Dic3—31no—
174C4⋊Dic3—91no—
175C4⋊Dic3—91no—
176C4×Dic3—31no—
177C4×Dic3—31no—
178C4×Dic3—91no—
179C4×Dic3—91no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—91no—
184C22×C32—22no—
185C22×C32—21no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32C2×C4×D311yescharacteristic
195C22×C32C22×C3×C411yescharacteristic
196C32×C4—21no—
197C32×C4—21no—
198C32×C4—21no—
199C32×C4—21no—
200C32×C4—41no—
201C32×C4#23711yescharacteristic
202C32×C4#23711yescharacteristic
203C32×C4C3×#4111yescharacteristic
204C32×C4C3×#4111yescharacteristic
205C3⋊Dic3—21no—
206C3⋊Dic3—21no—
207C3⋊Dic3—21no—
208C3⋊Dic3—21no—
209C3×Dic3—61no—
210C3×Dic3—61no—
211C3×Dic3—61no—
212C3×Dic3—61no—
213#117—91no—
214C4⋊C8—91no—
215C4⋊C8—91no—
216C22⋊C8—91no—
217C22⋊C8(C3⋊D3)⋊C311yescharacteristic
218#97—91no—
219C4×C8—91no—
220H(3,𝔽3)#43311yescharacteristic, complemented, Sylow
221C23×C3—31no—
222C23×C3—31no—
223C23×C3—61no—
224C23×C3C3×C4×D311yescharacteristic
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—31no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—91no—
238C2×C3×C4—91no—
239C2×C3×C4—91no—
240C2×C3×C4—91no—
241C2×C3×C4—121no—
242C2×C3×C4—121no—
243C2×C3×C4C3×C4×D311yescharacteristic
244C2×C3×C4C22×C3×D311yescharacteristic
245C2×Dic3—31no—
246C2×Dic3—31no—
247C2×Dic3—31no—
248C2×Dic3—31no—
249C2×Dic3—91no—
250C2×Dic3—91no—
251C2×Dic3—91no—
252C2×Dic3—91no—
253C3×C8—21no—
254C3×C8—21no—
255C3×C8—61no—
256C3×C8—61no—
257C3×C8—61no—
258C3×C8—61no—
259C3×C8—121no—
260C3×C8—121no—
261C3×C8—181no—
262C3×C8—181no—
263C3⋊C8—61no—
264C3⋊C8—61no—
265C3⋊C8—181no—
266C3⋊C8—181no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—42no—
271C2×C32—41no—
272C2×C32—41no—
273C2×C32#87711yescharacteristic
274C2×C32#90111yescharacteristic
275C2×C32#90711yescharacteristic
276C2×C32C3×#11711yescharacteristic
277C2×C32C3×#13111yescharacteristic
278C2×C32C3×#13111yescharacteristic
279C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
280C2×C8—91no—
281C2×C8—91no—
282C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
283C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
284C4⋊C4—91no—
285C4⋊C4—91no—
286C22⋊C4—91no—
287C22⋊C4—91no—
288C42—91no—
289C42—91no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—61no—
296C22×C3—61no—
297C22×C3—61no—
298C22×C3—61no—
299C22×C3—61no—
300C22×C3—61no—
301C22×C3—62no—
302C22×C3—61no—
303C22×C3—121no—
304C22×C3—121no—
305C22×C3C2×C3×C4×D311yescharacteristic
306C3×C4—21no—
307C3×C4—31no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—61no—
312C3×C4—61no—
313C3×C4—61no—
314C3×C4—61no—
315C3×C4—121no—
316C3×C4—181no—
317C3×C4—181no—
318C3×C4—181no—
319C3×C4—181no—
320C3×C4C3×#23711yescharacteristic
321C3×C4C3×#23711yescharacteristic
322Dic3—61no—
323Dic3—61no—
324Dic3—61no—
325Dic3—61no—
326Dic3—181no—
327Dic3—181no—
328Dic3—181no—
329Dic3—181no—
330C32—21no—
331C32#472011yescharacteristic, complemented
332C32C3×#43311yescharacteristic, complemented
333C23C4×(C3⋊D3)⋊C311yescharacteristic
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—91no—
337C2×C4—91no—
338C2×C4—91no—
339C2×C4—91no—
340C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
341C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
342C8—21no—
343C8—21no—
344C8—181no—
345C8—181no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—41no—
353C2×C3—61no—
354C2×C3—61no—
355C2×C3—61no—
356C2×C3—121no—
357C2×C3—121no—
358C2×C3—122no—
359C2×C3C3×#87711yescharacteristic
360C2×C3C3×#90111yescharacteristic
361C2×C3C3×#90711yescharacteristic
362C22—21no—
363C22—21no—
364C22—21no—
365C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
366C4—21no—
367C4—181no—
368C4—181no—
369C4—181no—
370C4—181no—
371C4#3830⋊C311yescharacteristic
372C4#3830⋊C311yescharacteristic
373C3—31no—
374C3—31no—
375C3—61no—
376C3C3×#472011yescharacteristic
377C2—41no—
378C2#64086⋊C311yescharacteristic
379C2#64110⋊C311yescharacteristic
380C2#64116⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊C8C2
socleC22×C3C2×C3×C4×D3
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC22⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#433
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #63642⋊C3C2semidirect
5. #63641⋊C3C2semidirect
6. #63641⋊C3C2semidirect
7. C8×(C3⋊Dic3)⋊C3C2semidirect
8. #10592419C3semidirect
27. #63642C2×C3semidirect
28. #63641C2×C3semidirect
29. #63641C2×C3semidirect
30. C8×C3⋊Dic3C2×C3semidirect
220. H(3,𝔽3)#433semidirect
331. C32#4720semidirect
332. C32C3×#433semidirect

Maximal subgroups

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

subgroupindex
#64135⋊C32
H(3,𝔽3)×C22⋊C82
#63657⋊C32
#63642⋊C32
#63641⋊C32
C8×(C3⋊Dic3)⋊C32
#105924193
C3×#47203