Finite Group Databasebeta

← #10592702⋊C3 #13142697

Subgroups — #10592702⋊C3 #13142697

Order 1728.

447 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 92 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
1H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic, complemented
4#63636⋊C3C211yesmaximal, characteristic, complemented
5#63636⋊C3C211yesmaximal, characteristic, complemented
6#63635⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592702C311yesmaximal, characteristic, complemented
9C3×#5003—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C22⋊C4×H(3,𝔽3)—21no—
16C42×H(3,𝔽3)—21no—
17#3752⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C32×#117—21no—
22C32×#117D311yescharacteristic, complemented
23C32×#117C2×C311yescharacteristic, complemented
24#64139C2×C311yescharacteristic
25C3×#930—31no—
26#63657C2×C311yescharacteristic, complemented
27#63636C2×C311yescharacteristic, complemented
28#63636C2×C311yescharacteristic, complemented
29#63635C2×C311yescharacteristic, complemented
30#63634C2×C311yescharacteristic, complemented
31C3×#830—31no—
32C3×#809—31no—
33C3×#809—31no—
34C3×#808—31no—
35C3×#807—31no—
36C23×H(3,𝔽3)D411yescharacteristic
37C22×(C3⋊D3)⋊C3—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)D411yescharacteristic
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#638⋊C3—42no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50C3×#482—91no—
51#5003—31no—
52#5003—91no—
53C22×C32×C4—21no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C22×C311yescharacteristic
56C2×#638—21no—
57C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
58C2×C3×#67—61no—
59C2×C3×D12—31no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4—21no—
62C32×C4⋊C4C2×D311yescharacteristic
63C32×C4⋊C4C2×D311yescharacteristic
64C32×C4⋊C4C22×C311yescharacteristic
65C32×C4⋊C4C22×C311yescharacteristic
66C32×C22⋊C4—22no—
67C32×C22⋊C4—21no—
68C32×C22⋊C4—21no—
69C32×C42—22no—
70C32×C42—21no—
71C32×C42—21no—
72#3752—21no—
73C4⋊(C3⋊Dic3)C22×C311yescharacteristic
74C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
75C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
76C3×(C2×D3)⋊C4—61no—
77C3×C4⋊Dic3—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C22×H(3,𝔽3)—21no—
81C22×H(3,𝔽3)—21no—
82C22×H(3,𝔽3)—21no—
83C22×H(3,𝔽3)C2×D411yescharacteristic
84C2×(C3⋊D3)⋊C3—41no—
85C2×(C3⋊D3)⋊C3—41no—
86C2×(C3⋊D3)⋊C3—41no—
87C4×H(3,𝔽3)—21no—
88C4×H(3,𝔽3)—41no—
89C4×H(3,𝔽3)—41no—
90C4×H(3,𝔽3)#4111yescharacteristic
91C4×H(3,𝔽3)#4111yescharacteristic
92(C3⋊Dic3)⋊C3—41no—
93C3×#121—91no—
94C3×#117—31no—
95C3×#117—31no—
96C3×#117—61no—
97C3×#117C3×D311yescharacteristic
98#930—31no—
99#930—91no—
100C3×#107—91no—
101C3×#106—91no—
102C3×#102—91no—
103C3×#102—91no—
104C3×C22⋊C8—91no—
105#830—31no—
106#830—91no—
107#809—31no—
108#809—31no—
109#809—91no—
110#809—91no—
111#808—31no—
112#808—91no—
113#807—31no—
114#807—91no—
115C23×C32—21no—
116C23×C32#6711yescharacteristic
117C23×C32C3×D411yescharacteristic
118C22×C3⋊D3—21no—
119C22×C3×D3—61no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—22no—
127C2×C32×C4—22no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4#6711yescharacteristic
135C2×C32×C4C3×D411yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C23×C311yescharacteristic
138#638—42no—
139C2×C3⋊Dic3—21no—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)—21no—
142C3×#67—122no—
143C2×C3×Dic3—61no—
144C3×D12—61no—
145C3×D12—61no—
146C3⋊(C3⋊C8)—21no—
147C3⋊(C3⋊C8)—21no—
148C3×C3⋊C8—61no—
149C3×C3⋊C8—61no—
150#482—91noSylow
151C2×H(3,𝔽3)—41no—
152C2×H(3,𝔽3)#12311yescharacteristic
153C2×H(3,𝔽3)#13511yescharacteristic
154C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
155(C3⋊D3)⋊C3—81no—
156C2×C3×D4—91no—
157C2×C3×D4—181no—
158C22×C3×C4—31no—
159C22×C3×C4—31no—
160C22×C3×C4—61no—
161C22×C3×C4C2×C3×D311yescharacteristic
162C2×#67—61no—
163C2×#67—181no—
164C2×D12—31no—
165C2×D12—91no—
166C2×C3×C8—91no—
167C2×C3×C8—91no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—31no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—61no—
173C3×C4⋊C4—61no—
174C3×C4⋊C4—91no—
175C3×C4⋊C4C2×C3×D311yescharacteristic
176C3×C4⋊C4C2×C3×D311yescharacteristic
177C3×C22⋊C4—21no—
178C3×C22⋊C4—61no—
179C3×C22⋊C4—61no—
180C3×C22⋊C4—62no—
181C3×C22⋊C4—181no—
182C3×C42—21no—
183C3×C42—61no—
184C3×C42—61no—
185C3×C42—62no—
186(C2×D3)⋊C4—61no—
187(C2×D3)⋊C4—181no—
188C4⋊Dic3—31no—
189C4⋊Dic3—91no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—31no—
192C2×C3⋊C8—91no—
193C2×C3⋊C8—91no—
194C22×C32—22no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—21no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32C2×#6711yescharacteristic
205C22×C32C2×C3×D411yescharacteristic
206C2×C3⋊D3—41no—
207C2×C3⋊D3—41no—
208C2×C3⋊D3—41no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C2×C3×D3—121no—
212C32×C4—22no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—21no—
216C32×C4—21no—
217C32×C4—42no—
218C32×C4—42no—
219C32×C4—41no—
220C32×C4—41no—
221C32×C4—41no—
222C32×C4—41no—
223C32×C4#23511yescharacteristic
224C32×C4#23511yescharacteristic
225C32×C4C3×#4111yescharacteristic
226C32×C4C3×#4111yescharacteristic
227C3⋊Dic3—41no—
228C3×Dic3—121no—
229#121—91no—
230#117(C3⋊D3)⋊C311yescharacteristic, complemented
231#107—91no—
232#106—91no—
233#102—91no—
234#102—91no—
235C22⋊C8—91no—
236H(3,𝔽3)#48211yescharacteristic, complemented, Sylow
237C23×C3—31no—
238C23×C3—31no—
239C23×C3—61no—
240C23×C3—181no—
241C23×C3C3×#6711yescharacteristic
242C22×D3—61no—
243C22×D3—181no—
244C3×D4—181no—
245C3×D4—181no—
246C3×D4—362no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—62no—
266C2×C3×C4—62no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—181no—
270C2×C3×C4C3×#6711yescharacteristic
271C2×C3×C4C22×C3×D311yescharacteristic
272#67—122no—
273#67—362no—
274C2×Dic3—61no—
275C2×Dic3—181no—
276D12—61no—
277D12—61no—
278D12—181no—
279D12—181no—
280C3×C8—181no—
281C3×C8—181no—
282C3⋊C8—61no—
283C3⋊C8—61no—
284C3⋊C8—181no—
285C3⋊C8—181no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—42no—
290C2×C32—41no—
291C2×C32—41no—
292C2×C32#92911yescharacteristic
293C2×C32#94911yescharacteristic
294C2×C32#95011yescharacteristic
295C2×C32C3×#12311yescharacteristic
296C2×C32C3×#13511yescharacteristic
297C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
298C3⋊D3—81no—
299C3×D3—241no—
300C2×D4—91no—
301C2×D4—181no—
302C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
303C2×C8—91no—
304C2×C8—91no—
305C4⋊C4—91no—
306C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
307C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—181no—
310C42—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—31no—
315C22×C3—31no—
316C22×C3—61no—
317C22×C3—61no—
318C22×C3—61no—
319C22×C3—61no—
320C22×C3—61no—
321C22×C3—61no—
322C22×C3—62no—
323C22×C3—61no—
324C22×C3—121no—
325C22×C3—121no—
326C22×C3—361no—
327C22×C3—361no—
328C22×C3—361no—
329C22×C3C2×C3×#6711yescharacteristic
330C2×D3—121no—
331C2×D3—121no—
332C2×D3—121no—
333C2×D3—361no—
334C2×D3—361no—
335C2×D3—361no—
336C3×C4—21no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—31no—
341C3×C4—41no—
342C3×C4—41no—
343C3×C4—61no—
344C3×C4—61no—
345C3×C4—62no—
346C3×C4—61no—
347C3×C4—61no—
348C3×C4—121no—
349C3×C4—121no—
350C3×C4—121no—
351C3×C4—121no—
352C3×C4—122no—
353C3×C4—122no—
354C3×C4—361no—
355C3×C4C3×#23511yescharacteristic
356C3×C4C3×#23511yescharacteristic
357Dic3—121no—
358Dic3—361no—
359C32—21no—
360C32#500311yescharacteristic, complemented
361C32C3×#48211yescharacteristic, complemented
362C23—181no—
363C23#638⋊C311yescharacteristic
364D4—181no—
365D4—181no—
366D4—362no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—181no—
372C2×C4#638⋊C311yescharacteristic
373C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
374C8—181no—
375C8—181no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—41no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—61no—
386C2×C3—121no—
387C2×C3—121no—
388C2×C3—122no—
389C2×C3—721no—
390C2×C3C3×#92911yescharacteristic
391C2×C3C3×#94911yescharacteristic
392C2×C3C3×#95011yescharacteristic
393D3—241no—
394D3—721no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—361no—
399C22—361no—
400C22—361no—
401C22C2×#638⋊C311yescharacteristic
402C4—21no—
403C4—41no—
404C4—41no—
405C4—361no—
406C4#3828⋊C311yescharacteristic
407C4#3828⋊C311yescharacteristic
408C3—31no—
409C3—31no—
410C3—61no—
411C3C3×#500311yescharacteristic
412C2—41no—
413C2—721no—
414C2#64138⋊C311yescharacteristic
415C2#64158⋊C311yescharacteristic
416C2#64159⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#117C2
socleC22×C3C2×C3×#67
prime radical#64139⋊C3C2
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#482
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#117C2semidirect
3. #63657⋊C3C2semidirect
4. #63636⋊C3C2semidirect
5. #63636⋊C3C2semidirect
6. #63635⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592702C3semidirect
22. C32×#117D3semidirect
23. C32×#117C2×C3semidirect
26. #63657C2×C3semidirect
27. #63636C2×C3semidirect
28. #63636C2×C3semidirect
29. #63635C2×C3semidirect
30. #63634C2×C3semidirect
230. #117(C3⋊D3)⋊C3semidirect
236. H(3,𝔽3)#482semidirect
360. C32#5003semidirect
361. C32C3×#482semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1172
#64139⋊C32
#63657⋊C32
#63636⋊C32
#63635⋊C32
#63634⋊C32
#105927023
C3×#50033