Finite Group Databasebeta

← C3×#10593300 #13166669

Subgroups — C3×#10593300 #13166669

Order 1728.

380 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 58 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
1C3×#64226C211yesmaximal, characteristic, complemented
2C3×#64224C211yesmaximal, characteristic
3C3×#63803C211yesmaximal, characteristic, complemented
4C3×#63796C211yesmaximal, characteristic, complemented
5C3×#63770C211yesmaximal, characteristic, complemented
6C3×#63728C211yesmaximal, characteristic, complemented
7C3×#63727C211yesmaximal, characteristic, complemented
8#10593300C311yesmaximal, characteristic, direct product factor
9C3×#3802C2211yescharacteristic
10C3×#3798C2211yescharacteristic
11C3×#3797C2211yescharacteristic
12C3×#3788C2211yescharacteristic
13C3×#3788C2211yescharacteristic
14C3×#3776—21no—
15C3×#3775—21no—
16C3×#3775—21no—
17C3×#3774—21no—
18C3×C8×C3⋊D3C2211yescharacteristic
19C32×#204—21no—
20C32×D3×C8—21no—
21C3×#3715—21no—
22C3×#3712C2211yescharacteristic
23C3×#3711—21no—
24#64226C2×C311yescharacteristic, complemented
25#64224C2×C311yescharacteristic
26C3×D3×SD16—61no—
27C3×#901—61no—
28#63803C2×C311yescharacteristic, complemented
29#63796C2×C311yescharacteristic, complemented
30#63770C2×C311yescharacteristic, complemented
31#63728C2×C311yescharacteristic, complemented
32#63727C2×C311yescharacteristic, complemented
33C2×C3×D32—21no—
34C3×C4×C3⋊D3C2311yescharacteristic
35C32×D12—21no—
36C32×C4×D3—21no—
37C32×Dic6—21no—
38C3×#627—21no—
39C3×#627D411yescharacteristic
40C3×#626—21no—
41C3×#625—21no—
42C3×#625D411yescharacteristic
43C3×#624—21no—
44C3×#622—21no—
45C3×#622—21no—
46C33×C8D411yescharacteristic
47C3×C3⋊(C3⋊C8)D411yescharacteristic
48C32×C3⋊C8—21no—
49C3×#471—91nomaximal
50C3×D3×Q8—61no—
51C3×D3×D4—61no—
52C3×#235—61no—
53#3802C22×C311yescharacteristic
54#3798C22×C311yescharacteristic
55#3797C22×C311yescharacteristic
56#3788C22×C311yescharacteristic
57#3788C22×C311yescharacteristic
58#3776—21no—
59#3775—21no—
60#3775—21no—
61#3774—21no—
62C32×SD16—61no—
63C2×C32×C8—61no—
64C8×C3⋊D3C22×C311yescharacteristic
65C3×#215—61no—
66C3×#214—61no—
67C3×#208—61no—
68C3×#204—21no—
69C3×#204—21no—
70C3×#204—41no—
71C3×ΓL(1,𝔽25)—61no—
72C3×ΓL(1,𝔽25)—61no—
73C3×D3×C8—21no—
74C3×D3×C8—21no—
75C3×D3×C8—41no—
76C3×D3×C8—61no—
77C3×D3×C8—61no—
78#3715—21no—
79#3712C22×C311yescharacteristic
80#3711—21no—
81C2×C3×C3⋊D3#4111yescharacteristic
82C2×C32×D3—21no—
83C2×C32×D3—41no—
84C3×D32—81no—
85C33×C4C2×D411yescharacteristic
86C3×C3⋊Dic3C2×D411yescharacteristic
87C32×Dic3—21no—
88C32×Dic3—41no—
89C3×#137—91no—
90C3×AGL(1,ℤ/8)—91no—
91C2×C3×SD16—91no—
92C3×#131—91no—
93D3×SD16—61no—
94#901—61no—
95C3×#108—91no—
96C3×#101—91no—
97C3×#100—91no—
98C22×C3×D3—121no—
99C2×D32—21no—
100C32×Q8—61no—
101C32×D4—61no—
102C2×C32×C4—61no—
103C4×C3⋊D3C23×C311yescharacteristic
104C3×#67—61no—
105C3×#67—61no—
106C3×#67—121no—
107C3×D12—21no—
108C3×D12—21no—
109C3×D12—41no—
110C3×D12—61no—
111C3×C4×D3—21no—
112C3×C4×D3—21no—
113C3×C4×D3—41no—
114C3×C4×D3—61no—
115C3×C4×D3—61no—
116C3×C4×D3—121no—
117C3×Dic6—21no—
118C3×Dic6—21no—
119C3×Dic6—41no—
120C3×Dic6—61no—
121C3×Dic6—121no—
122#627—21no—
123#627C3×D411yescharacteristic
124#626—21no—
125#625—21no—
126#625C3×D411yescharacteristic
127#624—21no—
128#622—21no—
129#622—21no—
130C32×C8—21no—
131C32×C8—21no—
132C32×C8—41no—
133C32×C8—41no—
134C32×C8—61no—
135C32×C8C3×D411yescharacteristic
136C3⋊(C3⋊C8)C3×D411yescharacteristic
137C3×C3⋊C8—21no—
138C3×C3⋊C8—21no—
139C3×C3⋊C8—41no—
140C3×C3⋊C8—61no—
141C3×C3⋊C8—61no—
142#471—91noSylow
143C2×C33#12111yescharacteristic
144C3×C3⋊D3—21no—
145C32×D3—41no—
146C32×D3—81no—
147C3×#41—91no—
148C2×C3×Q8—91no—
149C2×C3×D4—91no—
150D3×Q8—61no—
151D3×D4—61no—
152#235—61no—
153C3×Q16—181no—
154C3×SD16—61no—
155C3×SD16—61no—
156C3×SD16—121no—
157C3×SD16—181no—
158C3×SD16—181no—
159C3×SD16—181no—
160C3×D8—181no—
161C3×M16—91no—
162C3×M16—91no—
163C3×M16—91no—
164C3×M16—181no—
165C2×C3×C8—61no—
166C2×C3×C8—61no—
167C2×C3×C8—91no—
168C2×C3×C8—121no—
169#215—61no—
170#214—61no—
171#208—61no—
172#204—21no—
173ΓL(1,𝔽25)—61no—
174ΓL(1,𝔽25)—61no—
175D3×C8—21no—
176D3×C8—61no—
177D3×C8—61no—
178C22×C32—61no—
179C22×C32—121no—
180C2×C3⋊D3C3×#4111yescharacteristic
181C2×C3×D3—21no—
182C2×C3×D3—21no—
183C2×C3×D3—41no—
184C2×C3×D3—41no—
185C2×C3×D3—41no—
186C2×C3×D3—61no—
187C2×C3×D3—61no—
188C2×C3×D3—81no—
189C2×C3×D3—242no—
190D32—81no—
191C32×C4—21no—
192C32×C4—21no—
193C32×C4—41no—
194C32×C4—41no—
195C32×C4—61no—
196C32×C4—121no—
197C32×C4C2×C3×D411yescharacteristic
198C3⋊Dic3C2×C3×D411yescharacteristic
199C3×Dic3—21no—
200C3×Dic3—21no—
201C3×Dic3—41no—
202C3×Dic3—41no—
203C3×Dic3—41no—
204C3×Dic3—61no—
205C3×Dic3—61no—
206C3×Dic3—81no—
207#137—91no—
208AGL(1,ℤ/8)—91no—
209C2×SD16—91no—
210#131—91no—
211#108—91no—
212#101—91no—
213#100—91no—
214C33#47111yescharacteristic, complemented, Sylow
215C23×C3—181no—
216C22×D3—121no—
217C3×Q8—61no—
218C3×Q8—61no—
219C3×Q8—91no—
220C3×Q8—121no—
221C3×Q8—181no—
222C3×D4—61no—
223C3×D4—61no—
224C3×D4—91no—
225C3×D4—121no—
226C3×D4—181no—
227C3×D4—181no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—91no—
231C2×C3×C4—121no—
232C2×C3×C4—181no—
233#67—61no—
234#67—61no—
235#67—121no—
236D12—21no—
237D12—61no—
238C4×D3—21no—
239C4×D3—61no—
240C4×D3—61no—
241C4×D3—121no—
242Dic6—21no—
243Dic6—61no—
244Dic6—121no—
245C3×C8—21no—
246C3×C8—21no—
247C3×C8—41no—
248C3×C8—41no—
249C3×C8—61no—
250C3×C8—61no—
251C3×C8—91no—
252C3×C8—121no—
253C3×C8—181no—
254C3×C8—181no—
255C3×C8GO+(4,𝔽2)11yescharacteristic
256C3⋊C8—21no—
257C3⋊C8—61no—
258C3⋊C8—61no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—121no—
264C2×C32—241no—
265C2×C32C3×#12111yescharacteristic
266C3⋊D3—21no—
267C3×D3—41no—
268C3×D3—42no—
269C3×D3—41no—
270C3×D3—62no—
271C3×D3—81no—
272C3×D3—81no—
273C3×D3—121no—
274C3×D3—161no—
275#41—91no—
276C2×Q8—91no—
277C2×D4—91no—
278Q16—181no—
279SD16—61no—
280SD16—181no—
281SD16—181no—
282SD16—181no—
283D8—181no—
284M16—91no—
285M16—91no—
286M16—91no—
287M16—181no—
288C2×C8—61no—
289C2×C8—91no—
290C22×C3—61no—
291C22×C3—61no—
292C22×C3—91no—
293C22×C3—121no—
294C22×C3—121no—
295C22×C3—121no—
296C22×C3—241no—
297C22×C3—721no—
298C2×D3—21no—
299C2×D3—41no—
300C2×D3—61no—
301C2×D3—61no—
302C2×D3—242no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—61no—
308C3×C4—61no—
309C3×C4—91no—
310C3×C4—121no—
311C3×C4—121no—
312C3×C4—121no—
313C3×C4—241no—
314C3×C4C2×GO+(4,𝔽2)11yescharacteristic
315Dic3—21no—
316Dic3—41no—
317Dic3—61no—
318Dic3—61no—
319C32—21no—
320C32—21no—
321C32—41no—
322C32—41no—
323C32C3×#47111yescharacteristic, complemented
324C23—181no—
325Q8—61no—
326Q8—91no—
327Q8—181no—
328D4—61no—
329D4—91no—
330D4—181no—
331D4—181no—
332C2×C4—61no—
333C2×C4—91no—
334C2×C4—181no—
335C8—61no—
336C8—91no—
337C8—181no—
338C8—181no—
339C8C3×GO+(4,𝔽2)11yescharacteristic
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—121no—
345C2×C3—122no—
346C2×C3—121no—
347C2×C3—181no—
348C2×C3—241no—
349C2×C3—241no—
350C2×C3—481no—
351C2×C3#6423011yescharacteristic
352D3—41no—
353D3—62no—
354D3—81no—
355D3—121no—
356C22—61no—
357C22—91no—
358C22—121no—
359C22—721no—
360C4—61no—
361C4—91no—
362C4—121no—
363C4C2×C3×GO+(4,𝔽2)11yescharacteristic
364C3—21no—
365C3—21no—
366C3—41no—
367C3—41no—
368C3#1059330011yescharacteristic, direct product factor
369C2—121no—
370C2—181no—
371C2—241no—
372C2C3×#6423011yescharacteristic
subgroupG/H
centerC2×C3#64230
commutator, cosocleC4×C3⋊D3C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33#121
prime radicalC3×#64224C2
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#471
LCSG ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
1. C3×#64226C2semidirect
3. C3×#63803C2semidirect
4. C3×#63796C2semidirect
5. C3×#63770C2semidirect
6. C3×#63728C2semidirect
7. C3×#63727C2semidirect
8. #10593300C3direct (N × H)
24. #64226C2×C3semidirect
28. #63803C2×C3semidirect
29. #63796C2×C3semidirect
30. #63770C2×C3semidirect
31. #63728C2×C3semidirect
32. #63727C2×C3semidirect
214. C33#471semidirect
323. C32C3×#471semidirect
368. C3#10593300direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642262
C3×#642242
C3×#638032
C3×#637962
C3×#637702
C3×#637282
C3×#637272
#105933003
C3×#4719