Finite Group Databasebeta

← #10592586⋊C3 #13142228

Subgroups — #10592586⋊C3 #13142228

Order 1728.

435 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#3746⋊C3C211yesmaximal, characteristic, complemented
3#64106⋊C3C212yesmaximal
4H(3,𝔽3)×#107C211yesmaximal, characteristic, complemented
5#63637⋊C3C212yesmaximal, complemented
6#10592586C311yesmaximal, characteristic, complemented
7C3×#4887—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C4⋊C4×H(3,𝔽3)C2212yes—
12#3752⋊C3—22no—
13C4⋊((C3⋊Dic3)⋊C3)—22no—
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15C8×(C3⋊D3)⋊C3—22no—
16#3746⋊C3—22no—
17C2×#3746C2×C311yescharacteristic, complemented
18C2×C8×C3⋊D3C2×C311yescharacteristic
19#64106C2×C312yes—
20C2×C3×Dic12—31no—
21C2×C3×D3×C8—31no—
22C3×#897—32no—
23C32×#107—21no—
24C32×#107D311yescharacteristic, complemented
25C32×#107C2×C311yescharacteristic, complemented
26#63637C2×C312yescomplemented
27C3×#810—32no—
28C22×(C3⋊D3)⋊C3D411yescharacteristic
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×(C3⋊Dic3)⋊C3—22no—
32C2×(C3⋊Dic3)⋊C3D411yescharacteristic
33C4×(C3⋊D3)⋊C3—21no—
34C4×(C3⋊D3)⋊C3—21no—
35#634⋊C3—22no—
36#634⋊C3—22no—
37C8×H(3,𝔽3)D412yes—
38(C3⋊(C3⋊C8))⋊C3—21no—
39C3×#467—91no—
40#4887—31no—
41#4887—91no—
42C2×C4×C3⋊D3C22×C311yescharacteristic
43C2×#634C22×C312yes—
44C2×C3×C4×D3—31no—
45C2×C3×Dic6—32no—
46C2×C32×C8—21no—
47C2×C32×C8C2×D311yescharacteristic
48C2×C32×C8C22×C311yescharacteristic
49C32×C4⋊C4—22no—
50C32×C4⋊C4C2×D312yes—
51C32×C4⋊C4C22×C312yes—
52#3752—22no—
53C4⋊(C3⋊Dic3)—22no—
54C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
55#3746—22no—
56C8×C3⋊D3—22no—
57C3×(C2×D3)⋊C4—62no—
58C3×C4⋊Dic3—62no—
59C2×C3×C3⋊C8—31no—
60C3×Dic12—62no—
61C3×D3×C8—62no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C2×(C3⋊D3)⋊C3—21no—
64C2×(C3⋊D3)⋊C3—21no—
65C2×(C3⋊D3)⋊C3Q1612yescomplemented
66C4×H(3,𝔽3)—42no—
67C4×H(3,𝔽3)C2×D411yescharacteristic
68C4×H(3,𝔽3)#4111yescharacteristic
69(C3⋊Dic3)⋊C3—21no—
70(C3⋊Dic3)⋊C3—42no—
71C2×C3×Q16—91no—
72C22×C3×C8—91no—
73C3×C22⋊Q8—92no—
74C2×Dic12—31no—
75C2×Dic12—91no—
76C2×D3×C8—31no—
77C2×D3×C8—91no—
78#897—32no—
79#897—92no—
80C3×#107—31no—
81C3×#107—31no—
82C3×#107—61no—
83C3×#107C3×D311yescharacteristic
84C3×#103—92no—
85#810—32no—
86#810—92no—
87C22×C3⋊D3C3×D411yescharacteristic
88C22×C3×D3—31no—
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—21no—
92C2×C32×C4—42no—
93C2×C32×C4C22×D311yescharacteristic
94C2×C32×C4C23×C311yescharacteristic
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3C3×D411yescharacteristic
97C4×C3⋊D3—21no—
98C4×C3⋊D3—21no—
99#634—22no—
100#634—22no—
101C2×C3×Dic3—31no—
102C2×C3×Dic3—62no—
103C3×C4×D3—61no—
104C3×C4×D3—61no—
105C3×Dic6—62no—
106C3×Dic6—62no—
107C32×C8—22no—
108C32×C8D1212yes—
109C32×C8C3×D412yes—
110C3⋊(C3⋊C8)—21no—
111C3×C3⋊C8—61no—
112#467—91noSylow
113C2×H(3,𝔽3)#12111yescharacteristic
114C2×H(3,𝔽3)C2×Q1611yescharacteristic
115C2×H(3,𝔽3)#13511yescharacteristic
116(C3⋊D3)⋊C3—22no—
117C2×C3×Q8—92no—
118C22×C3×C4—91no—
119C2×C4×D3—31no—
120C2×C4×D3—91no—
121C2×Dic6—32no—
122C2×Dic6—92no—
123C3×Q16—182no—
124C2×C3×C8—31no—
125C2×C3×C8—31no—
126C2×C3×C8—61no—
127C2×C3×C8—91no—
128C2×C3×C8—182no—
129C2×C3×C8C2×C3×D311yescharacteristic
130C3×C4⋊C4—32no—
131C3×C4⋊C4—32no—
132C3×C4⋊C4—62no—
133C3×C4⋊C4—182no—
134C3×C4⋊C4C2×C3×D312yes—
135C3×C22⋊C4—182no—
136(C2×D3)⋊C4—62no—
137(C2×D3)⋊C4—182no—
138C4⋊Dic3—62no—
139C4⋊Dic3—182no—
140C2×C3⋊C8—31no—
141C2×C3⋊C8—91no—
142Dic12—62no—
143Dic12—182no—
144D3×C8—62no—
145D3×C8—182no—
146C22×C32—21no—
147C22×C32D3×D411yescharacteristic
148C22×C32C2×C3×D411yescharacteristic
149C2×C3⋊D3—21no—
150C2×C3⋊D3—21no—
151C2×C3⋊D3C3×Q1612yescomplemented
152C2×C3×D3—32no—
153C2×C3×D3—61no—
154C2×C3×D3—61no—
155C32×C4—21no—
156C32×C4—21no—
157C32×C4—42no—
158C32×C4—42no—
159C32×C4—82no—
160C32×C4C2×D1211yescharacteristic
161C32×C4#23911yescharacteristic
162C32×C4C2×C3×D411yescharacteristic
163C32×C4C3×#4111yescharacteristic
164C3⋊Dic3—21no—
165C3⋊Dic3—42no—
166C3×Dic3—61no—
167C3×Dic3—122no—
168C2×Q16—91no—
169C22×C8—91no—
170C22⋊Q8—92no—
171#107(C3⋊D3)⋊C311yescharacteristic, complemented
172#103—92no—
173H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
174C23×C3—91no—
175C22×D3—31no—
176C22×D3—91no—
177C3×Q8—182no—
178C3×Q8—182no—
179C2×C3×C4—22no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—62no—
183C2×C3×C4—62no—
184C2×C3×C4—61no—
185C2×C3×C4—91no—
186C2×C3×C4—122no—
187C2×C3×C4—181no—
188C2×C3×C4—181no—
189C2×C3×C4—182no—
190C2×C3×C4C22×C3×D311yescharacteristic
191C2×Dic3—31no—
192C2×Dic3—62no—
193C2×Dic3—91no—
194C2×Dic3—182no—
195C4×D3—61no—
196C4×D3—61no—
197C4×D3—181no—
198C4×D3—181no—
199Dic6—62no—
200Dic6—62no—
201Dic6—182no—
202Dic6—182no—
203C3×C8—32no—
204C3×C8—32no—
205C3×C8—62no—
206C3×C8—181no—
207C3×C8C3×D1212yes—
208C3⋊C8—61no—
209C3⋊C8—181no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—21no—
213C2×C32#89511yescharacteristic
214C2×C32#91211yescharacteristic
215C2×C32D3×Q1611yescharacteristic
216C2×C32C3×#12111yescharacteristic
217C2×C32C2×C3×Q1611yescharacteristic
218C2×C32C3×#13511yescharacteristic
219C3⋊D3—22no—
220C3×D3—62no—
221C2×Q8—92no—
222C22×C4—91no—
223Q16—182no—
224C2×C8—91no—
225C2×C8—182no—
226C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
227C4⋊C4—182no—
228C4⋊C4C2×(C3⋊D3)⋊C312yes—
229C22⋊C4—182no—
230C22×C3—31no—
231C22×C3—31no—
232C22×C3—61no—
233C22×C3—92no—
234C22×C3—181no—
235C22×C3—181no—
236C22×C3C3×D3×D411yescharacteristic
237C2×D3—32no—
238C2×D3—61no—
239C2×D3—61no—
240C2×D3—92no—
241C2×D3—181no—
242C2×D3—181no—
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—31no—
247C3×C4—42no—
248C3×C4—61no—
249C3×C4—61no—
250C3×C4—122no—
251C3×C4—122no—
252C3×C4—181no—
253C3×C4—242no—
254C3×C4—362no—
255C3×C4C2×C3×D1211yescharacteristic
256C3×C4C3×#23911yescharacteristic
257Dic3—61no—
258Dic3—122no—
259Dic3—181no—
260Dic3—362no—
261C32—21no—
262C32#488711yescharacteristic, complemented
263C32C3×#46711yescharacteristic, complemented
264C23—91no—
265Q8—182no—
266Q8—182no—
267C2×C4—22no—
268C2×C4—91no—
269C2×C4—181no—
270C2×C4—181no—
271C2×C4—182no—
272C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
273C8—181no—
274C8C4⋊((C3⋊D3)⋊C3)12yes—
275C2×C3—31no—
276C2×C3—31no—
277C2×C3—31no—
278C2×C3—31no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—61no—
282C2×C3—61no—
283C2×C3—61no—
284C2×C3—182no—
285C2×C3C3×#89511yescharacteristic
286C2×C3C3×#91211yescharacteristic
287C2×C3C3×D3×Q1611yescharacteristic
288D3—62no—
289D3—182no—
290C22—92no—
291C22—181no—
292C22—181no—
293C22D4×(C3⋊D3)⋊C311yescharacteristic
294C4—42no—
295C4—181no—
296C4—362no—
297C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
298C4#8872211yescharacteristic
299C3—31no—
300C3—31no—
301C3—61no—
302C3C3×#488711yescharacteristic
303C2—182no—
304C2#64104⋊C311yescharacteristic
305C2#64121⋊C311yescharacteristic
306C2Q16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#467
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
2. C2×#3746⋊C3C2semidirect
4. H(3,𝔽3)×#107C2semidirect
5. #63637⋊C3C2semidirect
6. #10592586C3semidirect
17. C2×#3746C2×C3semidirect
24. C32×#107D3semidirect
25. C32×#107C2×C3semidirect
26. #63637C2×C3semidirect
65. C2×(C3⋊D3)⋊C3Q16semidirect
151. C2×C3⋊D3C3×Q16semidirect
171. #107(C3⋊D3)⋊C3semidirect
173. H(3,𝔽3)#467semidirect
262. C32#4887semidirect
263. C32C3×#467semidirect

Maximal subgroups

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

subgroupindex
C2×C8×(C3⋊D3)⋊C32
C2×#3746⋊C32
#64106⋊C32
H(3,𝔽3)×#1072
#63637⋊C32
#105925863
C3×#48873