Finite Group Databasebeta

← #10592405⋊C3 #13141530

Subgroups — #10592405⋊C3 #13141530

Order 1728.

474 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3743⋊C3C211yesmaximal, characteristic, complemented
2C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
4H(3,𝔽3)×#97C211yesmaximal, characteristic, complemented
5#63647⋊C3C211yesmaximal, characteristic, complemented
6#63647⋊C3C211yesmaximal, characteristic, complemented
7C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
8#10592405C311yesmaximal, characteristic, complemented
9C3×#4706—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C42×H(3,𝔽3)C2211yescharacteristic
16#3752⋊C3C411yescharacteristic
17#3752⋊C3C411yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#3743⋊C3—22no—
22C8×(C3⋊D3)⋊C3—22no—
23C2×#3743C2×C311yescharacteristic, complemented
24C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
25C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
26C2×C3×ΓL(1,𝔽25)—31no—
27C2×C3×D3×C8—31no—
28C3×C4×D12—31no—
29C32×#97—21no—
30C32×#97D311yescharacteristic, complemented
31C32×#97C2×C311yescharacteristic, complemented
32#63647C2×C311yescharacteristic, complemented
33#63647C2×C311yescharacteristic, complemented
34C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
35C3×(C2×D3)⋊C8—31no—
36C3×(C2×D3)⋊C8—31no—
37C3×C4⋊(C3⋊C8)—31no—
38C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
39C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
40C2×C4×H(3,𝔽3)C2×C411yescharacteristic
41C2×C4×H(3,𝔽3)C2×C411yescharacteristic
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
45C4⋊((C3⋊D3)⋊C3)—22no—
46C4×(C3⋊D3)⋊C3—21no—
47C4×(C3⋊D3)⋊C3—21no—
48C4×(C3⋊D3)⋊C3—21no—
49C4×(C3⋊D3)⋊C3—21no—
50C8×H(3,𝔽3)—21no—
51C8×H(3,𝔽3)D412yescomplemented
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54C3×#435—91no—
55#4706—31no—
56#4706—91no—
57C2×C4⋊(C3⋊D3)C3×C411yescharacteristic
58C2×C4×C3⋊D3C22×C311yescharacteristic
59C2×C4×C3⋊D3C22×C311yescharacteristic
60C2×C3×D12—31no—
61C2×C3×C4×D3—31no—
62C2×C3×C4×D3—31no—
63C2×C32×C8—21no—
64C2×C32×C8—21no—
65C2×C32×C8C2×D311yescharacteristic
66C2×C32×C8C2×D311yescharacteristic
67C2×C32×C8C22×C311yescharacteristic
68C2×C32×C8C22×C311yescharacteristic
69C32×C42—21no—
70C32×C42C2×D311yescharacteristic
71C32×C42C22×C311yescharacteristic
72#3752C3×C411yescharacteristic
73#3752C3×C411yescharacteristic
74C4⋊(C3⋊Dic3)C3×C411yescharacteristic
75C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
76C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
77#3743—22no—
78C8×C3⋊D3—22no—
79C3×(C2×D3)⋊C4—31no—
80C3×(C2×D3)⋊C4—31no—
81C3×C4⋊Dic3—31no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—31no—
84C3×ΓL(1,𝔽25)—62no—
85C3×D3×C8—62no—
86C22×H(3,𝔽3)C22×C411yescharacteristic
87C2×(C3⋊D3)⋊C3—21no—
88C2×(C3⋊D3)⋊C3—21no—
89C2×(C3⋊D3)⋊C3—21no—
90C2×(C3⋊D3)⋊C3—21no—
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3M1612yescomplemented
93C4×H(3,𝔽3)—21no—
94C4×H(3,𝔽3)—21no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—21no—
99C2×C3×M16—91no—
100C22×C3×C8—91no—
101C3×C4×D4—91no—
102C2×ΓL(1,𝔽25)—31no—
103C2×ΓL(1,𝔽25)—91no—
104C2×D3×C8—31no—
105C2×D3×C8—91no—
106C4×D12—31no—
107C4×D12—91no—
108C3×C4⋊C8—91no—
109C3×C22⋊C8—91no—
110C3×C22⋊C8—91no—
111C3×#97—31no—
112C3×#97—31no—
113C3×#97—61no—
114C3×#97C3×D311yescharacteristic
115(C2×D3)⋊C8—31no—
116(C2×D3)⋊C8—31no—
117(C2×D3)⋊C8—91no—
118(C2×D3)⋊C8—91no—
119C4⋊(C3⋊C8)—31no—
120C4⋊(C3⋊C8)—91no—
121C22×C3⋊D3C2×C3×C411yescharacteristic
122C22×C3⋊D3C2×C3×C411yescharacteristic
123C22×C3×D3—31no—
124C22×C3×D3—31no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4C4×D311yescharacteristic
129C2×C32×C4C4×D311yescharacteristic
130C2×C32×C4C2×C3×C411yescharacteristic
131C2×C32×C4C2×C3×C411yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C23×C311yescharacteristic
134C2×C3⋊Dic3C2×C3×C411yescharacteristic
135C2×C3⋊Dic3C2×C3×C411yescharacteristic
136C4⋊(C3⋊D3)—22no—
137C4×C3⋊D3—21no—
138C4×C3⋊D3—21no—
139C4×C3⋊D3—21no—
140C4×C3⋊D3—21no—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—31no—
143C3×D12—62no—
144C3×C4×D3—61no—
145C3×C4×D3—61no—
146C3×C4×D3—61no—
147C3×C4×D3—61no—
148C32×C8—22no—
149C32×C8—21no—
150C32×C8—21no—
151C32×C8—22no—
152C32×C8D1212yescomplemented
153C32×C8C3×D412yescomplemented
154C3⋊(C3⋊C8)—21no—
155C3⋊(C3⋊C8)—21no—
156C3×C3⋊C8—61no—
157C3×C3⋊C8—61no—
158#435—91noSylow
159C2×H(3,𝔽3)C4×D411yescharacteristic
160C2×H(3,𝔽3)C2×M1611yescharacteristic
161C2×H(3,𝔽3)#13111yescharacteristic
162(C3⋊D3)⋊C3—22no—
163(C3⋊D3)⋊C3—41no—
164C2×C3×D4—91no—
165C22×C3×C4—91no—
166C22×C3×C4—91no—
167C2×D12—31no—
168C2×D12—91no—
169C2×C4×D3—31no—
170C2×C4×D3—31no—
171C2×C4×D3—91no—
172C2×C4×D3—91no—
173C3×M16—182no—
174C2×C3×C8—31no—
175C2×C3×C8—31no—
176C2×C3×C8—31no—
177C2×C3×C8—31no—
178C2×C3×C8—61no—
179C2×C3×C8—61no—
180C2×C3×C8—91no—
181C2×C3×C8—91no—
182C2×C3×C8—182no—
183C2×C3×C8C2×C3×D311yescharacteristic
184C2×C3×C8C2×C3×D311yescharacteristic
185C3×C4⋊C4—91no—
186C3×C22⋊C4—91no—
187C3×C22⋊C4—91no—
188C3×C42—31no—
189C3×C42—31no—
190C3×C42—61no—
191C3×C42C2×C3×D311yescharacteristic
192(C2×D3)⋊C4—31no—
193(C2×D3)⋊C4—31no—
194(C2×D3)⋊C4—91no—
195(C2×D3)⋊C4—91no—
196C4⋊Dic3—31no—
197C4⋊Dic3—91no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—91no—
201C2×C3⋊C8—91no—
202ΓL(1,𝔽25)—62no—
203ΓL(1,𝔽25)—182no—
204D3×C8—62no—
205D3×C8—182no—
206C22×C32—21no—
207C22×C32C2×C4×D311yescharacteristic
208C22×C32C22×C3×C411yescharacteristic
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3—21no—
212C2×C3⋊D3—21no—
213C2×C3⋊D3—21no—
214C2×C3⋊D3C3×M1612yescomplemented
215C2×C3×D3—32no—
216C2×C3×D3—61no—
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220C2×C3×D3—61no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—41no—
228C32×C4—41no—
229C32×C4C2×D1211yescharacteristic
230C32×C4#23511yescharacteristic
231C32×C4C2×C3×D411yescharacteristic
232C32×C4C3×#4111yescharacteristic
233C3⋊Dic3—21no—
234C3⋊Dic3—21no—
235C3×Dic3—61no—
236C3×Dic3—61no—
237C2×M16—91no—
238C22×C8—91no—
239C4×D4—91no—
240C4⋊C8—91no—
241C22⋊C8—91no—
242C22⋊C8—91no—
243#97(C3⋊D3)⋊C311yescharacteristic, complemented
244H(3,𝔽3)#43511yescharacteristic, complemented, Sylow
245C23×C3—91no—
246C23×C3—91no—
247C22×D3—31no—
248C22×D3—31no—
249C22×D3—91no—
250C22×D3—91no—
251C3×D4—182no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—91no—
262C2×C3×C4—91no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4—181no—
266C2×C3×C4—181no—
267C2×C3×C4C3×C4×D311yescharacteristic
268C2×C3×C4C3×C4×D311yescharacteristic
269C2×C3×C4C22×C3×D311yescharacteristic
270C2×Dic3—31no—
271C2×Dic3—31no—
272C2×Dic3—91no—
273C2×Dic3—91no—
274D12—62no—
275D12—182no—
276C4×D3—61no—
277C4×D3—61no—
278C4×D3—61no—
279C4×D3—61no—
280C4×D3—181no—
281C4×D3—181no—
282C4×D3—181no—
283C4×D3—181no—
284C3×C8—21no—
285C3×C8—32no—
286C3×C8—32no—
287C3×C8—61no—
288C3×C8—61no—
289C3×C8—62no—
290C3×C8—62no—
291C3×C8—181no—
292C3×C8—181no—
293C3×C8C3×D1212yes—
294C3⋊C8—61no—
295C3⋊C8—61no—
296C3⋊C8—181no—
297C3⋊C8—181no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32C4×D1211yescharacteristic
302C2×C32D3×M1611yescharacteristic
303C2×C32#90711yescharacteristic
304C2×C32C3×C4×D411yescharacteristic
305C2×C32C2×C3×M1611yescharacteristic
306C2×C32C3×#13111yescharacteristic
307C3⋊D3—22no—
308C3⋊D3—41no—
309C3×D3—62no—
310C3×D3—121no—
311C2×D4—91no—
312C22×C4—91no—
313C22×C4—91no—
314M16—182no—
315C2×C8—91no—
316C2×C8—91no—
317C2×C8—182no—
318C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
319C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
320C4⋊C4—91no—
321C22⋊C4—91no—
322C22⋊C4—91no—
323C42C2×(C3⋊D3)⋊C311yescharacteristic
324C22×C3—31no—
325C22×C3—31no—
326C22×C3—61no—
327C22×C3—92no—
328C22×C3—181no—
329C22×C3—181no—
330C22×C3—181no—
331C22×C3—181no—
332C22×C3—181no—
333C22×C3C2×C3×C4×D311yescharacteristic
334C2×D3—32no—
335C2×D3—61no—
336C2×D3—61no—
337C2×D3—61no—
338C2×D3—61no—
339C2×D3—61no—
340C2×D3—92no—
341C2×D3—181no—
342C2×D3—181no—
343C2×D3—181no—
344C2×D3—181no—
345C2×D3—181no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—61no—
353C3×C4—61no—
354C3×C4—61no—
355C3×C4—61no—
356C3×C4—61no—
357C3×C4—61no—
358C3×C4—121no—
359C3×C4—121no—
360C3×C4—181no—
361C3×C4—181no—
362C3×C4C2×C3×D1211yescharacteristic
363C3×C4C3×#23511yescharacteristic
364Dic3—61no—
365Dic3—61no—
366Dic3—181no—
367Dic3—181no—
368C32—21no—
369C32#470611yescharacteristic, complemented
370C32C3×#43511yescharacteristic, complemented
371C23—91no—
372C23—91no—
373D4—182no—
374C2×C4—91no—
375C2×C4—91no—
376C2×C4—181no—
377C2×C4—181no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
381C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
382C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
383C8—21no—
384C8—181no—
385C8—181no—
386C8C4⋊((C3⋊D3)⋊C3)12yescomplemented
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—61no—
394C2×C3—61no—
395C2×C3—61no—
396C2×C3—182no—
397C2×C3—361no—
398C2×C3C3×C4×D1211yescharacteristic
399C2×C3C3×D3×M1611yescharacteristic
400C2×C3C3×#90711yescharacteristic
401D3—62no—
402D3—121no—
403D3—182no—
404D3—361no—
405C22—92no—
406C22—181no—
407C22—181no—
408C22—181no—
409C22—181no—
410C22—181no—
411C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
412C4—21no—
413C4—21no—
414C4—181no—
415C4—181no—
416C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
417C4#3828⋊C311yescharacteristic
418C3—31no—
419C3—31no—
420C3—61no—
421C3C3×#470611yescharacteristic
422C2—182no—
423C2—361no—
424C2C4×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
425C2M16×(C3⋊D3)⋊C311yescharacteristic
426C2#64116⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#97C2
socleC22×C3C2×C3×C4×D3
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C4
cosocleC2×C32×C4C23×C3
hypercenter#97(C3⋊D3)⋊C3
hypocenterC32C3×#435
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ #97

Complements

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

NH ≅ G/Ntype
1. C2×#3743⋊C3C2semidirect
2. C2×C8×(C3⋊D3)⋊C3C2semidirect
4. H(3,𝔽3)×#97C2semidirect
5. #63647⋊C3C2semidirect
6. #63647⋊C3C2semidirect
7. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
8. #10592405C3semidirect
23. C2×#3743C2×C3semidirect
24. C2×C8×C3⋊D3C2×C3semidirect
30. C32×#97D3semidirect
31. C32×#97C2×C3semidirect
32. #63647C2×C3semidirect
33. #63647C2×C3semidirect
34. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
51. C8×H(3,𝔽3)D4semidirect
92. C2×(C3⋊D3)⋊C3M16semidirect
152. C32×C8D12semidirect
153. C32×C8C3×D4semidirect
214. C2×C3⋊D3C3×M16semidirect
243. #97(C3⋊D3)⋊C3semidirect
244. H(3,𝔽3)#435semidirect
369. C32#4706semidirect
370. C32C3×#435semidirect
386. C8C4⋊((C3⋊D3)⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×#3743⋊C32
C2×C8×(C3⋊D3)⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
H(3,𝔽3)×#972
#63647⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105924053
C3×#47063