Finite Group Databasebeta

← C3×#10591628 #13158657

Subgroups — C3×#10591628 #13158657

Order 1728.

714 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 140 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×#64153C211yesmaximal, characteristic
2C2×C32×#216C212yesmaximal
3C3×#63873C211yesmaximal, characteristic
4C2×C3×#3719C212yesmaximal
5C3×#63553C211yesmaximal, characteristic
6C3×#5183—32nomaximal
7#10591628C311yesmaximal, characteristic, direct product factor
8C2×C33×Q8C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C2×C32×Dic6C2212yes—
11C3×#3750—22no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×#216—24no—
14C2×C32×C3⋊C8C2212yes—
15C3×#3724—22no—
16C3×#3719—24no—
17C2×C32×Q16—32no—
18#64153C2×C311yescharacteristic
19C3×#944—32no—
20C3×#944—32no—
21C3×#944—61no—
22C2×C3×#216—22no—
23C2×C3×#216—32no—
24C2×C3×#216D312yes—
25C2×C3×#216C2×C312yes—
26C2×C3×Dic12—32no—
27#63873C2×C311yescharacteristic
28C2×#3719C2×C312yes—
29C3×C8×Dic3—32no—
30#63553C2×C311yescharacteristic
31C33×Q8—22no—
32C2×C33×C4—21no—
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3D412yes—
36C3×#634—22no—
37C2×C32×Dic3—22no—
38C32×Dic6—22no—
39C32×Dic6—22no—
40C32×C3⋊C8D414yes—
41C3×#494—91no—
42#5183—32no—
43C2×C32×Q8—21no—
44C2×C32×Q8—22no—
45C2×C32×Q8—32no—
46C2×C32×Q8—41no—
47C2×C32×Q8C2×D312yes—
48C2×C32×Q8C22×C311yescharacteristic
49C2×#634C22×C311yescharacteristic
50C2×C3×Dic6—22no—
51C2×C3×Dic6—32no—
52C2×C3×Dic6—61no—
53C2×C3×Dic6C2×D312yes—
54C2×C3×Dic6C22×C312yes—
55C32×Q16—64no—
56C2×C32×C8—32no—
57#3750—22no—
58C4×C3⋊Dic3C22×C311yescharacteristic
59C3×#216—28no—
60C3×#216—24no—
61C3×#216—24no—
62C3×#216—64no—
63C3×Dic3⋊C4—64no—
64C3×Dic3⋊C4—64no—
65C3×Dic3⋊C4—64no—
66C3×C4×Dic3—32no—
67C3×C4×Dic3—61no—
68C2×C3×C3⋊C8—22no—
69C2×C3×C3⋊C8C2×D312yes—
70C2×C3×C3⋊C8C22×C312yes—
71C3×Dic12—64no—
72#3724—22no—
73#3719—24no—
74C22×C33C2×D411yescharacteristic
75C33×C4—41no—
76C33×C4C2×D412yes—
77C3×C3⋊Dic3—41no—
78C3×C3⋊Dic3Q1614yescomplemented
79C32×Dic3—42no—
80C2×C3×Q16—32no—
81C2×C3×Q16—32no—
82C2×C3×Q16—62no—
83C2×C3×Q16—92no—
84C3×C4⋊Q8—91no—
85C3×C4⋊Q8—91no—
86#944—32no—
87#944—32no—
88#944—61no—
89C2×#216—32no—
90C2×#216C3×D312yes—
91C2×Dic12—32no—
92C3×C4×C8—91no—
93C8×Dic3—32no—
94C32×Q8—22no—
95C32×Q8—24no—
96C32×Q8—22no—
97C32×Q8—22no—
98C32×Q8—42no—
99C32×Q8—42no—
100C32×Q8—44no—
101C32×Q8—62no—
102C32×Q8—62no—
103C2×C32×C4—21no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—21no—
107C2×C32×C4—22no—
108C2×C32×C4—41no—
109C2×C32×C4—42no—
110C2×C32×C4—42no—
111C2×C32×C4—62no—
112C2×C32×C4C22×D312yes—
113C2×C32×C4C23×C311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3C3×D412yes—
116#634—22no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—34no—
120C2×C3×Dic3—42no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C3×Dic6—24no—
125C3×Dic6—22no—
126C3×Dic6—22no—
127C3×Dic6—22no—
128C3×Dic6—22no—
129C3×Dic6—42no—
130C3×Dic6—62no—
131C3×Dic6—62no—
132C3×Dic6—122no—
133C32×C8—34no—
134C3×C3⋊C8—24no—
135C3×C3⋊C8#6714yes—
136C3×C3⋊C8C3×D414yes—
137#494—91noSylow
138C2×C33#12711yescharacteristic
139C2×C33C2×Q1612yes—
140C2×C3×Q8—21no—
141C2×C3×Q8—22no—
142C2×C3×Q8—32no—
143C2×C3×Q8—32no—
144C2×C3×Q8—41no—
145C2×C3×Q8—62no—
146C2×C3×Q8—91no—
147C2×C3×Q8D3211yescharacteristic
148C2×C3×Q8C2×C3×D312yes—
149C2×Dic6—32no—
150C2×Dic6—61no—
151C2×Dic6C2×C3×D312yes—
152C3×Q16—64no—
153C3×Q16—64no—
154C3×Q16—68no—
155C3×Q16—184no—
156C2×C3×C8—32no—
157C2×C3×C8—32no—
158C2×C3×C8—62no—
159C3×C4⋊C4—182no—
160C3×C4⋊C4—182no—
161C3×C42—91no—
162#216—24no—
163#216—64no—
164Dic3⋊C4—64no—
165Dic3⋊C4—64no—
166Dic3⋊C4—64no—
167C4×Dic3—32no—
168C4×Dic3—61no—
169C2×C3⋊C8C2×C3×D312yes—
170Dic12—64no—
171C22×C32—21no—
172C22×C32—22no—
173C22×C32—41no—
174C22×C32D3×D412yes—
175C22×C32C2×C3×D411yescharacteristic
176C32×C4—22no—
177C32×C4—22no—
178C32×C4—22no—
179C32×C4—42no—
180C32×C4—42no—
181C32×C4—44no—
182C32×C4—44no—
183C32×C4—42no—
184C32×C4—41no—
185C32×C4—122no—
186C32×C4D3×D412yes—
187C32×C4C2×#6712yes—
188C32×C4C2×C3×D412yes—
189C3⋊Dic3—41no—
190C3⋊Dic3C3×Q1614yescomplemented
191C3×Dic3—38no—
192C3×Dic3—44no—
193C3×Dic3—42no—
194C3×Dic3—42no—
195C3×Dic3—64no—
196C3×Dic3—122no—
197C3×Dic3—122no—
198C2×Q16—32no—
199C2×Q16—92no—
200C4⋊Q8—91no—
201C4⋊Q8—91no—
202C4×C8—91no—
203C33#49411yescharacteristic, complemented, Sylow
204C3×Q8—24no—
205C3×Q8—22no—
206C3×Q8—22no—
207C3×Q8—22no—
208C3×Q8—42no—
209C3×Q8—42no—
210C3×Q8—44no—
211C3×Q8—64no—
212C3×Q8—62no—
213C3×Q8—62no—
214C3×Q8—62no—
215C3×Q8—62no—
216C3×Q8—122no—
217C3×Q8—182no—
218C2×C3×C4—22no—
219C2×C3×C4—21no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—21no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—92no—
229C2×C3×C4—122no—
230C2×C3×C4—181no—
231C2×C3×C4C2×D3211yescharacteristic
232C2×C3×C4C22×C3×D312yes—
233C2×Dic3—22no—
234C2×Dic3—34no—
235C2×Dic3—62no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238Dic6—22no—
239Dic6—22no—
240Dic6—62no—
241Dic6—62no—
242Dic6—122no—
243C3×C8—34no—
244C3×C8—34no—
245C3×C8—64no—
246C3⋊C8C3×#6714yes—
247C2×C32—24no—
248C2×C32—22no—
249C2×C32—22no—
250C2×C32—21no—
251C2×C32—41no—
252C2×C32—42no—
253C2×C32D3×Q1614yes—
254C2×C32#94012yes—
255C2×C32C3×#12711yescharacteristic
256C2×C32C2×C3×Q1612yes—
257C2×Q8—32no—
258C2×Q8—91no—
259C2×Q8C3×D3211yescharacteristic
260Q16—64no—
261Q16—184no—
262C2×C8—32no—
263C4⋊C4—182no—
264C4⋊C4—182no—
265C42—91no—
266C22×C3—21no—
267C22×C3—22no—
268C22×C3—41no—
269C22×C3#381111yescharacteristic
270C22×C3C3×D3×D412yes—
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—22no—
274C3×C4—42no—
275C3×C4—44no—
276C3×C4—44no—
277C3×C4—42no—
278C3×C4—41no—
279C3×C4—42no—
280C3×C4—94no—
281C3×C4—124no—
282C3×C4—122no—
283C3×C4—122no—
284C3×C4—361no—
285C3×C4D3×#6712yes—
286C3×C4C3×D3×D412yes—
287C3×C4C2×C3×#6712yes—
288Dic3—38no—
289Dic3—42no—
290Dic3—64no—
291Dic3—122no—
292Dic3—122no—
293C32—22no—
294C32—21no—
295C32—41no—
296C32#518312yescomplemented
297C32C3×#49411yescharacteristic, complemented
298Q8—22no—
299Q8—62no—
300Q8—62no—
301Q8—182no—
302C2×C4—21no—
303C2×C4—62no—
304C2×C4—92no—
305C2×C4—181no—
306C2×C4C2×C3×D3211yescharacteristic
307C8—34no—
308C2×C3—22no—
309C2×C3—24no—
310C2×C3—21no—
311C2×C3—22no—
312C2×C3—42no—
313C2×C3—41no—
314C2×C3#6394412yes—
315C2×C3#6397911yescharacteristic
316C2×C3C3×D3×Q1614yes—
317C2×C3C3×#94012yes—
318C22C3×#381111yescharacteristic
319C4—41no—
320C4—94no—
321C4—122no—
322C4—361no—
323C4C3×D3×#6712yes—
324C3—22no—
325C3—21no—
326C3—41no—
327C3#1059162811yescharacteristic, direct product factor
328C3C3×#518312yescomplemented
329C2C3×#6394412yes—
330C2C3×#6397911yescharacteristic
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×#494
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. #10591628C3direct (N × H)
78. C3×C3⋊Dic3Q16semidirect
190. C3⋊Dic3C3×Q16semidirect
203. C33#494semidirect
296. C32#5183semidirect
297. C32C3×#494semidirect
327. C3#10591628direct (N × H)
328. C3C3×#5183semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
C2×C32×#2162
C3×#638732
C2×C3×#37192
C3×#635532
C3×#51833
#105916283