Finite Group Databasebeta

← #10592478⋊C3 #13141796

Subgroups — #10592478⋊C3 #13141796

Order 1728.

421 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64147⋊C3C211yesmaximal, characteristic
2#64101⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
4#63659⋊C3C211yesmaximal, characteristic, complemented
5#63647⋊C3C211yesmaximal, characteristic, complemented
6#63644⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592478C311yesmaximal, characteristic, complemented
9C3×#4779—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#3757⋊C3—21no—
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4×(C3⋊Dic3)⋊C3—21no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64147C2×C311yescharacteristic
22#64101C2×C311yescharacteristic, complemented
23C3×#938—31no—
24C3×#892—31no—
25C32×#102—21no—
26C32×#102D311yescharacteristic, complemented
27C32×#102C2×C311yescharacteristic, complemented
28#63659C2×C311yescharacteristic, complemented
29#63647C2×C311yescharacteristic, complemented
30#63644C2×C311yescharacteristic, complemented
31#63634C2×C311yescharacteristic, complemented
32C3×#832—31no—
33C3×(C2×D3)⋊C8—31no—
34C3×#817—31no—
35C3×#807—31no—
36C23×H(3,𝔽3)—21no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42#638⋊C3—42no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C4×(C3⋊D3)⋊C3—21no—
47C4×(C3⋊D3)⋊C3—21no—
48C8×H(3,𝔽3)—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50C3×#482—91no—
51#4779—31no—
52#4779—91no—
53C2×C32×D4—21no—
54C2×C32×D4C2×D311yescharacteristic
55C2×C32×D4C22×C311yescharacteristic
56C2×#638—21no—
57C2×C4×C3⋊D3C22×C311yescharacteristic
58C2×C3×#67—61no—
59C2×C3×C4×D3—31no—
60C2×C32×C8—21no—
61C2×C32×C8C2×D311yescharacteristic
62C2×C32×C8C22×C311yescharacteristic
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C4⋊C4C22×C311yescharacteristic
66#3757—21no—
67#3752—21no—
68C4⋊(C3⋊Dic3)C22×C311yescharacteristic
69C4⋊(C3⋊Dic3)C22×C311yescharacteristic
70C4×C3⋊Dic3—21no—
71C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
72C3×#217—61no—
73C3×(C2×D3)⋊C4—61no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—31no—
76C3×C4×Dic3—61no—
77C2×C3×C3⋊C8—31no—
78C22×H(3,𝔽3)—41no—
79C22×H(3,𝔽3)—41no—
80C22×H(3,𝔽3)—41no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C2×(C3⋊D3)⋊C3—21no—
83C2×(C3⋊D3)⋊C3—21no—
84C2×(C3⋊D3)⋊C3—21no—
85C4×H(3,𝔽3)—41no—
86C4×H(3,𝔽3)#4111yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88(C3⋊Dic3)⋊C3—21no—
89(C3⋊Dic3)⋊C3—41no—
90(C3⋊Dic3)⋊C3—41no—
91C3×#121—91no—
92C3×#117—91no—
93#938—31no—
94#938—91no—
95#892—31no—
96#892—91no—
97C3×#107—91no—
98C3×#106—91no—
99C3×#102—31no—
100C3×#102—31no—
101C3×#102—61no—
102C3×#102—91no—
103C3×#102C3×D311yescharacteristic
104C3×C22⋊C8—91no—
105#832—31no—
106#832—91no—
107(C2×D3)⋊C8—31no—
108(C2×D3)⋊C8—91no—
109#817—31no—
110#817—91no—
111#807—31no—
112#807—91no—
113C23×C32—21no—
114C23×C32—21no—
115C23×C32—41no—
116C22×C3⋊D3C3×D411yescharacteristic
117C22×C3×D3—31no—
118C32×D4—22no—
119C32×D4—21no—
120C32×D4—21no—
121C32×D4—21no—
122C32×D4—21no—
123C32×D4—41no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—41no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C23×C311yescharacteristic
130#638—42no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3C3×D411yescharacteristic
134C4×C3⋊D3—21no—
135C4×C3⋊D3—21no—
136C3×#67—122no—
137C2×C3×Dic3—31no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C32×C8—21no—
143C32×C8—21no—
144C32×C8—41no—
145C3⋊(C3⋊C8)—21no—
146C3×C3⋊C8—61no—
147#482—91noSylow
148C2×H(3,𝔽3)—81no—
149C2×H(3,𝔽3)#12311yescharacteristic
150C2×H(3,𝔽3)#13511yescharacteristic
151C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
152(C3⋊D3)⋊C3—41no—
153C2×C3×D4—31no—
154C2×C3×D4—31no—
155C2×C3×D4—61no—
156C2×C3×D4—181no—
157C2×C3×D4C2×C3×D311yescharacteristic
158C22×C3×C4—91no—
159C2×#67—61no—
160C2×#67—181no—
161C2×C4×D3—31no—
162C2×C4×D3—91no—
163C2×C3×C8—31no—
164C2×C3×C8—31no—
165C2×C3×C8—61no—
166C2×C3×C8—91no—
167C2×C3×C8C2×C3×D311yescharacteristic
168C3×C4⋊C4—31no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—61no—
171C3×C4⋊C4—91no—
172C3×C4⋊C4—91no—
173C3×C4⋊C4C2×C3×D311yescharacteristic
174C3×C22⋊C4—181no—
175C3×C22⋊C4—181no—
176C3×C42—181no—
177#217—61no—
178#217—181no—
179(C2×D3)⋊C4—61no—
180(C2×D3)⋊C4—181no—
181C4⋊Dic3—31no—
182C4⋊Dic3—31no—
183C4⋊Dic3—91no—
184C4⋊Dic3—91no—
185C4×Dic3—61no—
186C4×Dic3—181no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—91no—
189C22×C32—21no—
190C22×C32—42no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32—81no—
198C22×C32—81no—
199C22×C32D3×D411yescharacteristic
200C22×C32C2×C3×D411yescharacteristic
201C2×C3⋊D3—21no—
202C2×C3⋊D3—21no—
203C2×C3⋊D3—21no—
204C2×C3×D3—61no—
205C2×C3×D3—61no—
206C2×C3×D3—61no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4—41no—
210C32×C4—41no—
211C32×C4—81no—
212C32×C4#23511yescharacteristic
213C32×C4#23711yescharacteristic
214C32×C4C3×#4111yescharacteristic
215C32×C4C3×#4111yescharacteristic
216C3⋊Dic3—21no—
217C3⋊Dic3—41no—
218C3⋊Dic3—41no—
219C3×Dic3—61no—
220C3×Dic3—121no—
221C3×Dic3—121no—
222#121—91no—
223#117—91no—
224#107—91no—
225#106—91no—
226#102—91no—
227#102(C3⋊D3)⋊C311yescharacteristic, complemented
228C22⋊C8—91no—
229H(3,𝔽3)#48211yescharacteristic, complemented, Sylow
230C23×C3—21no—
231C23×C3—61no—
232C23×C3—61no—
233C23×C3—91no—
234C23×C3—121no—
235C22×D3—31no—
236C22×D3—91no—
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—61no—
240C3×D4—61no—
241C3×D4—61no—
242C3×D4—61no—
243C3×D4—62no—
244C3×D4—121no—
245C3×D4—362no—
246C2×C3×C4—21no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—91no—
253C2×C3×C4—121no—
254C2×C3×C4—181no—
255C2×C3×C4—181no—
256C2×C3×C4—181no—
257C2×C3×C4—181no—
258C2×C3×C4C22×C3×D311yescharacteristic
259#67—122no—
260#67—362no—
261C2×Dic3—31no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—91no—
265C2×Dic3—181no—
266C2×Dic3—181no—
267C4×D3—61no—
268C4×D3—61no—
269C4×D3—181no—
270C4×D3—181no—
271C3×C8—21no—
272C3×C8—61no—
273C3×C8—61no—
274C3×C8—121no—
275C3×C8—181no—
276C3⋊C8—61no—
277C3⋊C8—181no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—82no—
282C2×C32—81no—
283C2×C32—81no—
284C2×C32#88311yescharacteristic
285C2×C32#91111yescharacteristic
286C2×C32#91611yescharacteristic
287C2×C32C3×#12311yescharacteristic
288C2×C32C3×#13511yescharacteristic
289C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
290C3⋊D3—41no—
291C3×D3—121no—
292C2×D4—181no—
293C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
294C22×C4—91no—
295C2×C8—91no—
296C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
297C4⋊C4—91no—
298C4⋊C4—91no—
299C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
300C22⋊C4—181no—
301C22⋊C4—181no—
302C42—181no—
303C22×C3—31no—
304C22×C3—31no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—61no—
309C22×C3—121no—
310C22×C3—121no—
311C22×C3—121no—
312C22×C3—121no—
313C22×C3—121no—
314C22×C3—121no—
315C22×C3—122no—
316C22×C3—181no—
317C22×C3—181no—
318C22×C3—181no—
319C22×C3—241no—
320C22×C3—241no—
321C22×C3C3×D3×D411yescharacteristic
322C2×D3—61no—
323C2×D3—61no—
324C2×D3—61no—
325C2×D3—181no—
326C2×D3—181no—
327C2×D3—181no—
328C3×C4—31no—
329C3×C4—31no—
330C3×C4—31no—
331C3×C4—31no—
332C3×C4—41no—
333C3×C4—61no—
334C3×C4—61no—
335C3×C4—121no—
336C3×C4—121no—
337C3×C4—181no—
338C3×C4—241no—
339C3×C4—361no—
340C3×C4—361no—
341C3×C4C3×#23511yescharacteristic
342C3×C4C3×#23711yescharacteristic
343Dic3—61no—
344Dic3—121no—
345Dic3—121no—
346Dic3—181no—
347Dic3—361no—
348Dic3—361no—
349C32—21no—
350C32#477911yescharacteristic, complemented
351C32C3×#48211yescharacteristic, complemented
352C23—21no—
353C23—91no—
354D4—21no—
355D4—21no—
356D4—362no—
357C2×C4—21no—
358C2×C4—91no—
359C2×C4—181no—
360C2×C4—181no—
361C2×C4—181no—
362C2×C4—181no—
363C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
364C8—21no—
365C8—181no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—61no—
373C2×C3—61no—
374C2×C3—61no—
375C2×C3—81no—
376C2×C3—241no—
377C2×C3—241no—
378C2×C3—242no—
379C2×C3—361no—
380C2×C3C3×#88311yescharacteristic
381C2×C3C3×#91111yescharacteristic
382C2×C3C3×#91611yescharacteristic
383D3—121no—
384D3—361no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—181no—
389C22—181no—
390C22—181no—
391C22D4×(C3⋊D3)⋊C311yescharacteristic
392C4—41no—
393C4—181no—
394C4—361no—
395C4—361no—
396C4#3828⋊C311yescharacteristic
397C4#3830⋊C311yescharacteristic
398C3—31no—
399C3—31no—
400C3—61no—
401C3C3×#477911yescharacteristic
402C2—81no—
403C2—361no—
404C2#64092⋊C311yescharacteristic
405C2#64120⋊C311yescharacteristic
406C2#64125⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
prime radical#64147⋊C3C2
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#482
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. #64101⋊C3C2semidirect
3. H(3,𝔽3)×#102C2semidirect
4. #63659⋊C3C2semidirect
5. #63647⋊C3C2semidirect
6. #63644⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592478C3semidirect
22. #64101C2×C3semidirect
26. C32×#102D3semidirect
27. C32×#102C2×C3semidirect
28. #63659C2×C3semidirect
29. #63647C2×C3semidirect
30. #63644C2×C3semidirect
31. #63634C2×C3semidirect
227. #102(C3⋊D3)⋊C3semidirect
229. H(3,𝔽3)#482semidirect
350. C32#4779semidirect
351. C32C3×#482semidirect

Maximal subgroups

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

subgroupindex
#64147⋊C32
#64101⋊C32
H(3,𝔽3)×#1022
#63659⋊C32
#63647⋊C32
#63644⋊C32
#63634⋊C32
#105924783
C3×#47793