Finite Group Databasebeta

← C3×#10593366 #13166708

Subgroups — C3×#10593366 #13166708

Order 1728.

492 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 74 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×#63919C211yesmaximal, characteristic, complemented
2C3×#63914C211yesmaximal, characteristic
3C3×C4×#622C211yesmaximal, characteristic, complemented
4C3×#63739C214yesmaximal, complemented
5#10593366C311yesmaximal, characteristic, direct product factor
6C2×C32×D12—21no—
7C2×C3×#627—21no—
8C2×C3×#625—21no—
9C2×C3×#625C2212yes—
10C2×C3×#622C2212yes—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×C4⋊Dic3—21no—
13C3×#3724C2212yes—
14C3×#940—61no—
15C3×#888—61no—
16#63919C2×C311yescharacteristic, complemented
17#63914C2×C311yescharacteristic
18C4×#622C2×C311yescharacteristic, complemented
19#63739C2×C314yescomplemented
20C22×C32×D3—22no—
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3D411yescharacteristic
23C2×C3×C3⋊Dic3C2311yescharacteristic
24C2×C32×Dic3—22no—
25C32×D12—42no—
26C3×#627—42no—
27C3×#625—24no—
28C3×#625—42no—
29C3×#622—22no—
30C3×#486—91nomaximal
31C2×C32×D4—61no—
32C2×C3×#67—62no—
33C2×C3×#67—62no—
34C2×C3×D12—21no—
35C2×C3×D12—21no—
36C2×C3×D12—41no—
37C2×C3×Dic6—62no—
38C2×#627—21no—
39C2×#625—21no—
40C2×#625C22×C312yes—
41C2×#622C22×C312yes—
42C32×C4⋊C4—61no—
43C4×C3⋊Dic3C22×C311yescharacteristic
44C3×C4⋊Dic3—21no—
45C3×C4⋊Dic3—21no—
46C3×C4⋊Dic3—41no—
47C3×Dic3⋊C4—62no—
48C3×C4×Dic3—61no—
49C3×C4×Dic3—61no—
50#3724C22×C312yes—
51C22×C33C2×D411yescharacteristic
52C2×C32×D3—42no—
53C2×C32×D3—42no—
54C2×C32×D3—42no—
55C33×C4SD1612yes—
56C3×C3⋊Dic3D812yes—
57C3×C3⋊Dic3#4112yes—
58C32×Dic3—42no—
59C3×C4⋊Q8—91no—
60C3×#127—91no—
61#940—61no—
62#888—61no—
63C3×#102—94no—
64C3×C4×C8—91no—
65C23×C32—62no—
66C22×C3×D3—22no—
67C22×C3×D3—22no—
68C22×C3×D3—42no—
69C32×D4—122no—
70C2×C32×C4—21no—
71C2×C32×C4—21no—
72C2×C32×C4—41no—
73C2×C32×C4—41no—
74C2×C32×C4—62no—
75C2×C32×C4C3×D411yescharacteristic
76C2×C3⋊Dic3C3×D411yescharacteristic
77C2×C3⋊Dic3C23×C311yescharacteristic
78C3×#67—124no—
79C3×#67—124no—
80C2×C3×Dic3—22no—
81C2×C3×Dic3—22no—
82C2×C3×Dic3—42no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C3×D12—42no—
88C3×D12—44no—
89C3×D12—42no—
90C3×Dic6—124no—
91#627—42no—
92#625—24no—
93#625—42no—
94#622—22no—
95#486—91noSylow
96C2×C33#12411yescharacteristic
97C2×C33C2×D811yescharacteristic
98C2×C33C2×SD1611yescharacteristic
99C32×D3—82no—
100C2×C3×Q8—181no—
101C2×C3×D4—61no—
102C2×C3×D4—61no—
103C2×C3×D4—92no—
104C2×C3×D4—121no—
105C2×C3×D4—181no—
106C2×#67—62no—
107C2×#67—62no—
108C2×D12—21no—
109C2×Dic6—62no—
110C2×C3×C8—92no—
111C3×C4⋊C4—61no—
112C3×C4⋊C4—61no—
113C3×C4⋊C4—92no—
114C3×C4⋊C4—121no—
115C3×C42—91no—
116C4⋊Dic3—21no—
117Dic3⋊C4—62no—
118C4×Dic3—61no—
119C4×Dic3—61no—
120C22×C32—21no—
121C22×C32—21no—
122C22×C32—41no—
123C22×C32—41no—
124C22×C32—122no—
125C22×C32—122no—
126C22×C32—122no—
127C22×C32C2×C3×D411yescharacteristic
128C2×C3×D3—42no—
129C2×C3×D3—42no—
130C2×C3×D3—42no—
131C2×C3×D3—44no—
132C2×C3×D3—42no—
133C2×C3×D3—42no—
134C2×C3×D3—42no—
135C2×C3×D3—82no—
136C2×C3×D3—82no—
137C32×C4—22no—
138C32×C4—22no—
139C32×C4—42no—
140C32×C4—42no—
141C32×C4—122no—
142C32×C4C3×SD1612yes—
143C3⋊Dic3C3×D812yes—
144C3⋊Dic3C3×#4112yes—
145C3×Dic3—42no—
146C3×Dic3—42no—
147C3×Dic3—62no—
148C3×Dic3—62no—
149C3×Dic3—62no—
150C3×Dic3—62no—
151C3×Dic3—82no—
152C4⋊Q8—91no—
153#127—91no—
154#102—94no—
155C4×C8—91no—
156C33#48611yescharacteristic, complemented, Sylow
157C23×C3—62no—
158C23×C3—62no—
159C23×C3—122no—
160C22×D3—22no—
161C3×Q8—362no—
162C3×D4—122no—
163C3×D4—124no—
164C3×D4—122no—
165C3×D4—184no—
166C3×D4—362no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—62no—
172C2×C3×C4—62no—
173C2×C3×C4—91no—
174C2×C3×C4—91no—
175C2×C3×C4—122no—
176C2×C3×C4GO+(4,𝔽2)11yescharacteristic
177#67—124no—
178#67—124no—
179C2×Dic3—22no—
180C2×Dic3—61no—
181C2×Dic3—61no—
182C2×Dic3—61no—
183C2×Dic3—61no—
184D12—42no—
185Dic6—124no—
186C3×C8—182no—
187C2×C32—21no—
188C2×C32—21no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32—41no—
194C2×C32—41no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32—41no—
199C2×C32—242no—
200C2×C32C3×#12411yescharacteristic
201C2×C32C2×C3×D811yescharacteristic
202C2×C32C2×C3×SD1611yescharacteristic
203C3×D3—82no—
204C3×D3—84no—
205C3×D3—82no—
206C2×Q8—181no—
207C2×D4—61no—
208C2×D4—92no—
209C2×D4—181no—
210C2×C8—92no—
211C4⋊C4—61no—
212C4⋊C4—92no—
213C42—91no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—41no—
217C22×C3—41no—
218C22×C3—122no—
219C22×C3—122no—
220C22×C3—122no—
221C22×C3—124no—
222C22×C3—122no—
223C22×C3—122no—
224C22×C3—122no—
225C22×C3—242no—
226C22×C3—242no—
227C22×C3C2×GO+(4,𝔽2)11yescharacteristic
228C2×D3—42no—
229C2×D3—42no—
230C2×D3—42no—
231C3×C4—22no—
232C3×C4—22no—
233C3×C4—42no—
234C3×C4—42no—
235C3×C4—92no—
236C3×C4—92no—
237C3×C4—122no—
238C3×C4—122no—
239C3×C4—242no—
240C3×C4#377512yes—
241Dic3—42no—
242Dic3—62no—
243Dic3—62no—
244Dic3—62no—
245Dic3—62no—
246C32—21no—
247C32—21no—
248C32—41no—
249C32—41no—
250C32C3×#48611yescharacteristic, complemented
251C23—62no—
252Q8—362no—
253D4—122no—
254D4—184no—
255D4—362no—
256C2×C4—62no—
257C2×C4—91no—
258C2×C4—91no—
259C2×C4C3×GO+(4,𝔽2)11yescharacteristic
260C8—182no—
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—21no—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—41no—
268C2×C3—41no—
269C2×C3—41no—
270C2×C3—41no—
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—242no—
274C2×C3—244no—
275C2×C3—242no—
276C2×C3#6422111yescharacteristic
277C2×C3#6422511yescharacteristic
278C2×C3C2×#377511yescharacteristic
279D3—82no—
280C22—122no—
281C22—122no—
282C22—122no—
283C22C2×C3×GO+(4,𝔽2)11yescharacteristic
284C4—92no—
285C4—92no—
286C4—122no—
287C4C3×#377512yes—
288C3—21no—
289C3—21no—
290C3—41no—
291C3—41no—
292C3#1059336611yescharacteristic, direct product factor
293C2—242no—
294C2C3×#6422111yescharacteristic
295C2C3×#6422511yescharacteristic
296C2C2×C3×#377511yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC3×#63914C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#486
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#63919C2semidirect
3. C3×C4×#622C2semidirect
4. C3×#63739C2semidirect
5. #10593366C3direct (N × H)
16. #63919C2×C3semidirect
18. C4×#622C2×C3semidirect
19. #63739C2×C3semidirect
156. C33#486semidirect
250. C32C3×#486semidirect
292. C3#10593366direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#639192
C3×#639142
C3×C4×#6222
C3×#637392
#105933663
C3×#4869