Finite Group Databasebeta

← C3×#10591631 #13158633

Subgroups — C3×#10591631 #13158633

Order 1728.

576 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 136 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×Q8×C3⋊Dic3C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic
3C32×#835C212yesmaximal
4C3×#63575C212yesmaximal
5C3×#63572C211yesmaximal, characteristic
6C3×#4793—32nomaximal
7#10591631C311yesmaximal, characteristic, direct product factor
8C2×C33×Q8C2211yescharacteristic
9C2×C3×#627—21no—
10C3×C4⋊(C3⋊Dic3)—21no—
11C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
12C3×C4×C3⋊Dic3—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C2212yes—
15C2×C32×C3⋊C8C2212yes—
16C3×#3724—21no—
17Q8×C3⋊Dic3C2×C311yescharacteristic
18C3×Q8×Dic3—32no—
19C3×Q8×Dic3—61no—
20C3×#888—32no—
21#63919C2×C311yescharacteristic
22C32×#103—32no—
23C3×#835—22no—
24C3×#835D312yes—
25C3×#835C2×C312yes—
26C3×#818—32no—
27C3×Dic3⋊C8—32no—
28C3×#807—32no—
29#63575C2×C312yes—
30#63572C2×C311yescharacteristic
31C33×Q8—21no—
32C33×Q8Q812yescomplemented
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3D411yescharacteristic
38C2×C32×Dic3—22no—
39C3×#627—42no—
40C32×C3⋊C8—22no—
41C3×#477—91no—
42#4793—32no—
43C2×C32×Q8—21no—
44C2×C32×Q8—22no—
45C2×C32×Q8—41no—
46C2×C32×Q8C2×D312yes—
47C2×C32×Q8C22×C311yescharacteristic
48C2×C3×Dic6—62no—
49C2×#627—21no—
50C2×C32×C8—32no—
51C32×C4⋊C4—32no—
52C4⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊Dic3)C22×C311yescharacteristic
54C4×C3⋊Dic3—21no—
55C4×C3⋊Dic3C22×C311yescharacteristic
56C3×C4⋊Dic3—22no—
57C3×C4⋊Dic3—32no—
58C3×C4⋊Dic3—62no—
59C3×C4⋊Dic3—61no—
60C3×C4⋊Dic3—62no—
61C3×C4⋊Dic3C2×D312yes—
62C3×C4⋊Dic3C22×C312yes—
63C3×Dic3⋊C4—62no—
64C3×C4×Dic3—32no—
65C3×C4×Dic3—62no—
66C3×C4×Dic3—61no—
67C3×C4×Dic3—62no—
68C2×C3×C3⋊C8—22no—
69C2×C3×C3⋊C8C2×D312yes—
70C2×C3×C3⋊C8C22×C312yes—
71#3724—21no—
72C22×C33C2×D411yescharacteristic
73C33×C4—22no—
74C33×C4C2×Q811yescharacteristic
75C33×C4#4111yescharacteristic
76C3×C3⋊Dic3—21no—
77C3×C3⋊Dic3—41no—
78C3×C3⋊Dic3Q1612yes—
79C32×Dic3—42no—
80C3×C4⋊Q8—91no—
81C3×C4×Q8—91no—
82Q8×Dic3—32no—
83Q8×Dic3—61no—
84#888—32no—
85C3×#107—92no—
86C3×C4⋊C8—91no—
87C3×#103—32no—
88C3×#103—32no—
89C3×#103—62no—
90#835C3×D312yes—
91#818—32no—
92Dic3⋊C8—32no—
93#807—32no—
94C32×Q8—21no—
95C32×Q8—22no—
96C32×Q8—22no—
97C32×Q8—22no—
98C32×Q8—24no—
99C32×Q8—42no—
100C32×Q8—42no—
101C32×Q8—42no—
102C32×Q8Dic614yescomplemented
103C32×Q8C3×Q812yescomplemented
104C2×C32×C4—21no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—21no—
108C2×C32×C4—22no—
109C2×C32×C4—41no—
110C2×C32×C4—42no—
111C2×C32×C4—42no—
112C2×C32×C4—62no—
113C2×C32×C4C22×D312yes—
114C2×C32×C4C23×C311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3C3×D411yescharacteristic
117C2×C3⋊Dic3C3×D411yescharacteristic
118C2×C3×Dic3—22no—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—42no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—62no—
127C3×Dic6—124no—
128#627—42no—
129C32×C8—62no—
130C3×C3⋊C8—22no—
131C3×C3⋊C8—22no—
132C3×C3⋊C8—42no—
133#477—91noSylow
134C2×C33C22⋊Q811yescharacteristic
135C2×C33C2×Q1611yescharacteristic
136C2×C33AGL(1,ℤ/8)11yescharacteristic
137C2×C3×Q8—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—41no—
140C2×C3×Q8—181no—
141C2×C3×Q8D3211yescharacteristic
142C2×C3×Q8C2×C3×D312yes—
143C2×Dic6—62no—
144C2×C3×C8—32no—
145C2×C3×C8—32no—
146C2×C3×C8—62no—
147C3×C4⋊C4—32no—
148C3×C4⋊C4—32no—
149C3×C4⋊C4—62no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4—181no—
152C3×C4⋊C4—181no—
153C3×C42—91no—
154C3×C42—181no—
155C4⋊Dic3—32no—
156C4⋊Dic3—61no—
157C4⋊Dic3—62no—
158C4⋊Dic3—62no—
159C4⋊Dic3C2×C3×D312yes—
160Dic3⋊C4—62no—
161C4×Dic3—32no—
162C4×Dic3—61no—
163C4×Dic3—62no—
164C4×Dic3—62no—
165C2×C3⋊C8C2×C3×D312yes—
166C22×C32—21no—
167C22×C32—22no—
168C22×C32—41no—
169C22×C32D3×D412yes—
170C22×C32C2×C3×D411yescharacteristic
171C32×C4—22no—
172C32×C4—24no—
173C32×C4—24no—
174C32×C4—21no—
175C32×C4—21no—
176C32×C4—22no—
177C32×C4—22no—
178C32×C4—41no—
179C32×C4—41no—
180C32×C4—44no—
181C32×C4—44no—
182C32×C4—122no—
183C32×C4C2×Dic612yes—
184C32×C4#23712yes—
185C32×C4C2×C3×Q811yescharacteristic
186C32×C4C3×#4111yescharacteristic
187C3⋊Dic3—21no—
188C3⋊Dic3—41no—
189C3⋊Dic3C3×Q1612yes—
190C3×Dic3—34no—
191C3×Dic3—42no—
192C3×Dic3—42no—
193C3×Dic3—62no—
194C3×Dic3—62no—
195C3×Dic3—82no—
196C3×Dic3—121no—
197C3×Dic3—122no—
198C3×Dic3—122no—
199C4⋊Q8—91no—
200C4×Q8—91no—
201#107—92no—
202C4⋊C8—91no—
203#103—32no—
204C33#47711yescharacteristic, complemented, Sylow
205C3×Q8—22no—
206C3×Q8—21no—
207C3×Q8—22no—
208C3×Q8—22no—
209C3×Q8—24no—
210C3×Q8—42no—
211C3×Q8—42no—
212C3×Q8—42no—
213C3×Q8—362no—
214C3×Q8#62712yescomplemented
215C3×Q8C3×Dic614yescomplemented
216C2×C3×C4—22no—
217C2×C3×C4—21no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—21no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4—91no—
227C2×C3×C4—91no—
228C2×C3×C4—122no—
229C2×C3×C4—181no—
230C2×C3×C4C2×D3211yescharacteristic
231C2×C3×C4C22×C3×D312yes—
232C2×Dic3—22no—
233C2×Dic3—32no—
234C2×Dic3—32no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—62no—
238C2×Dic3—62no—
239Dic6—124no—
240C3×C8—62no—
241C3×C8—62no—
242C3×C8—122no—
243C3⋊C8—22no—
244C2×C32—22no—
245C2×C32—22no—
246C2×C32—22no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—41no—
251C2×C32—41no—
252C2×C32—41no—
253C2×C32#87812yes—
254C2×C32#91412yes—
255C2×C32D3×Q1612yes—
256C2×C32C3×C22⋊Q811yescharacteristic
257C2×C32C2×C3×Q1611yescharacteristic
258C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
259C2×Q8—181no—
260C2×Q8C3×D3211yescharacteristic
261C2×C8—32no—
262C4⋊C4—32no—
263C4⋊C4—91no—
264C4⋊C4—181no—
265C4⋊C4—181no—
266C42—91no—
267C42—181no—
268C22×C3—21no—
269C22×C3—22no—
270C22×C3—41no—
271C22×C3#381111yescharacteristic
272C22×C3C3×D3×D412yes—
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—22no—
279C3×C4—21no—
280C3×C4—44no—
281C3×C4—44no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4—92no—
285C3×C4—122no—
286C3×C4—122no—
287C3×C4—181no—
288C3×C4—242no—
289C3×C4—361no—
290C3×C4#380511yescharacteristic
291C3×C4C2×#62711yescharacteristic
292C3×C4C2×C3×Dic612yes—
293C3×C4C3×#23712yes—
294Dic3—34no—
295Dic3—42no—
296Dic3—62no—
297Dic3—62no—
298Dic3—121no—
299Dic3—122no—
300Dic3—122no—
301C32—21no—
302C32—22no—
303C32—41no—
304C32#479312yescomplemented
305C32C3×#47711yescharacteristic, complemented
306Q8—21no—
307Q8—362no—
308Q8C3×#62712yescomplemented
309C2×C4—21no—
310C2×C4—62no—
311C2×C4—91no—
312C2×C4—91no—
313C2×C4—181no—
314C2×C4C2×C3×D3211yescharacteristic
315C8—62no—
316C2×C3—21no—
317C2×C3—22no—
318C2×C3—21no—
319C2×C3—22no—
320C2×C3—21no—
321C2×C3—22no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—41no—
325C2×C3#6394111yescharacteristic
326C2×C3#6394411yescharacteristic
327C2×C3#6398211yescharacteristic
328C2×C3C3×#87812yes—
329C2×C3C3×#91412yes—
330C2×C3C3×D3×Q1612yes—
331C22C3×#381111yescharacteristic
332C4—22no—
333C4—92no—
334C4—122no—
335C4—181no—
336C4—361no—
337C4C3×#380511yescharacteristic
338C4C2×C3×#62711yescharacteristic
339C3—22no—
340C3—21no—
341C3—41no—
342C3#1059163111yescharacteristic, direct product factor
343C3C3×#479312yescomplemented
344C2C3×#6394111yescharacteristic
345C2C3×#6394411yescharacteristic
346C2C3×#6398211yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×Q8D32
hypocenterC32C3×#477
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
7. #10591631C3direct (N × H)
32. C33×Q8Q8semidirect
102. C32×Q8Dic6semidirect
103. C32×Q8C3×Q8semidirect
204. C33#477semidirect
214. C3×Q8#627semidirect
215. C3×Q8C3×Dic6semidirect
304. C32#4793semidirect
305. C32C3×#477semidirect
308. Q8C3×#627semidirect
342. C3#10591631direct (N × H)
343. C3C3×#4793semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊Dic32
C3×#639192
C32×#8352
C3×#635752
C3×#635722
C3×#47933
#105916313