Finite Group Databasebeta

← #10592418⋊C3 #13141601

Subgroups — #10592418⋊C3 #13141601

Order 1728.

435 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 122 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×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic
4#63642⋊C3C212yesmaximal, complemented
5#63641⋊C3C212yesmaximal, complemented
6#10592418C311yesmaximal, characteristic, complemented
7C3×#4719—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3C411yescharacteristic
10C22×(C3⋊Dic3)⋊C3C411yescharacteristic
11C2×C8×H(3,𝔽3)C2212yes—
12C4×(C3⋊Dic3)⋊C3C411yescharacteristic
13C4×(C3⋊Dic3)⋊C3C411yescharacteristic
14C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C2×C4×C3⋊Dic3C2×C311yescharacteristic
18C2×C3×C4×Dic3—31no—
19C32×C22⋊C8—21no—
20C32×C22⋊C8D311yescharacteristic
21C32×C22⋊C8C2×C311yescharacteristic
22#63657C2×C311yescharacteristic
23#63642C2×C312yescomplemented
24#63641C2×C312yescomplemented
25C3×#830—31no—
26C3×#815—32no—
27C3×Dic3⋊C8—32no—
28C23×H(3,𝔽3)C2×C411yescharacteristic
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)C2×C411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3—22no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
35C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
36C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
37C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
38C8×H(3,𝔽3)—22no—
39(C3⋊(C3⋊C8))⋊C3—22no—
40C3×#432—91no—
41#4719—31no—
42#4719—91no—
43C22×C32×C4—21no—
44C22×C32×C4C2×D311yescharacteristic
45C22×C32×C4C22×C311yescharacteristic
46C22×C3⋊Dic3C3×C411yescharacteristic
47C22×C3⋊Dic3C3×C411yescharacteristic
48C22×C3×Dic3—31no—
49C22×C3×Dic3—31no—
50C2×C32×C8—22no—
51C2×C32×C8C2×D312yes—
52C2×C32×C8C22×C312yes—
53C4×C3⋊Dic3C3×C411yescharacteristic
54C4×C3⋊Dic3C3×C411yescharacteristic
55C4×C3⋊Dic3C22×C311yescharacteristic
56C4×C3⋊Dic3C22×C311yescharacteristic
57C2×C3⋊(C3⋊C8)C22×C312yes—
58C3×C4×Dic3—31no—
59C3×C4×Dic3—31no—
60C3×C4×Dic3—31no—
61C3×C4×Dic3—31no—
62C2×C3×C3⋊C8—32no—
63C22×H(3,𝔽3)—21no—
64C22×H(3,𝔽3)—21no—
65C22×H(3,𝔽3)M1612yes—
66C22×H(3,𝔽3)C22×C411yescharacteristic
67C4×H(3,𝔽3)—21no—
68C4×H(3,𝔽3)#4112yes—
69(C3⋊Dic3)⋊C3—21no—
70(C3⋊Dic3)⋊C3—21no—
71(C3⋊Dic3)⋊C3—21no—
72(C3⋊Dic3)⋊C3M1612yes—
73C2×C3×C42—91no—
74C2×C4×Dic3—31no—
75C2×C4×Dic3—91no—
76C3×C4⋊C8—92no—
77C3×C22⋊C8—31no—
78C3×C22⋊C8—31no—
79C3×C22⋊C8—61no—
80C3×C22⋊C8—91no—
81C3×C22⋊C8C3×D311yescharacteristic
82C3×#97—92no—
83#830—31no—
84#830—91no—
85#815—32no—
86#815—92no—
87Dic3⋊C8—32no—
88Dic3⋊C8—92no—
89C23×C32—21no—
90C23×C32C4×D311yescharacteristic
91C23×C32C2×C3×C411yescharacteristic
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—42no—
97C2×C32×C4C4×D311yescharacteristic
98C2×C32×C4C2×C3×C411yescharacteristic
99C2×C32×C4C22×D311yescharacteristic
100C2×C32×C4C23×C311yescharacteristic
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3C2×C3×C411yescharacteristic
104C2×C3⋊Dic3C2×C3×C411yescharacteristic
105C2×C3⋊Dic3C2×C3×C411yescharacteristic
106C2×C3⋊Dic3C2×C3×C411yescharacteristic
107C2×C3×Dic3—31no—
108C2×C3×Dic3—31no—
109C2×C3×Dic3—31no—
110C2×C3×Dic3—31no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C32×C8—22no—
114C32×C8—22no—
115C32×C8—42no—
116C3⋊(C3⋊C8)—22no—
117C3×C3⋊C8—62no—
118#432—91noSylow
119C2×H(3,𝔽3)—22no—
120C2×H(3,𝔽3)#11711yescharacteristic
121C2×H(3,𝔽3)C2×M1611yescharacteristic
122C2×H(3,𝔽3)C2×M1611yescharacteristic
123C22×C3×C4—31no—
124C22×C3×C4—31no—
125C22×C3×C4—61no—
126C22×C3×C4—91no—
127C22×C3×C4—91no—
128C22×C3×C4C2×C3×D311yescharacteristic
129C22×Dic3—31no—
130C22×Dic3—31no—
131C22×Dic3—91no—
132C22×Dic3—91no—
133C2×C3×C8—32no—
134C2×C3×C8—32no—
135C2×C3×C8—62no—
136C2×C3×C8—92no—
137C2×C3×C8C2×C3×D312yes—
138C3×C42—91no—
139C3×C42—91no—
140C3×C42—91no—
141C3×C42—91no—
142C4×Dic3—31no—
143C4×Dic3—31no—
144C4×Dic3—31no—
145C4×Dic3—31no—
146C4×Dic3—91no—
147C4×Dic3—91no—
148C4×Dic3—91no—
149C4×Dic3—91no—
150C2×C3⋊C8—32no—
151C2×C3⋊C8—92no—
152C22×C32—21no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—22no—
157C22×C32—21no—
158C22×C32—41no—
159C22×C32—41no—
160C22×C32ΓL(1,𝔽25)12yes—
161C22×C32C3×M1612yes—
162C22×C32C2×C4×D311yescharacteristic
163C22×C32C22×C3×C411yescharacteristic
164C32×C4—21no—
165C32×C4—21no—
166C32×C4—22no—
167C32×C4—41no—
168C32×C4#23712yes—
169C32×C4C3×#4112yes—
170C3⋊Dic3—21no—
171C3⋊Dic3—21no—
172C3⋊Dic3—21no—
173C3⋊Dic3C3×M1612yes—
174C3×Dic3—32no—
175C3×Dic3—61no—
176C3×Dic3—61no—
177C3×Dic3—61no—
178C2×C42—91no—
179C4⋊C8—92no—
180C22⋊C8—91no—
181C22⋊C8(C3⋊D3)⋊C311yescharacteristic
182#97—92no—
183H(3,𝔽3)#43211yescharacteristic, complemented, Sylow
184C23×C3—31no—
185C23×C3—31no—
186C23×C3—61no—
187C23×C3C3×C4×D311yescharacteristic
188C2×C3×C4—22no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—62no—
194C2×C3×C4—62no—
195C2×C3×C4—61no—
196C2×C3×C4—61no—
197C2×C3×C4—91no—
198C2×C3×C4—91no—
199C2×C3×C4—91no—
200C2×C3×C4—91no—
201C2×C3×C4—122no—
202C2×C3×C4—182no—
203C2×C3×C4—182no—
204C2×C3×C4C3×C4×D311yescharacteristic
205C2×C3×C4C22×C3×D311yescharacteristic
206C2×Dic3—31no—
207C2×Dic3—31no—
208C2×Dic3—31no—
209C2×Dic3—31no—
210C2×Dic3—62no—
211C2×Dic3—62no—
212C2×Dic3—91no—
213C2×Dic3—91no—
214C2×Dic3—91no—
215C2×Dic3—91no—
216C2×Dic3—182no—
217C2×Dic3—182no—
218C3×C8—22no—
219C3×C8—62no—
220C3×C8—62no—
221C3×C8—122no—
222C3×C8—182no—
223C3⋊C8—62no—
224C3⋊C8—182no—
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—42no—
231C2×C32#87711yescharacteristic
232C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
233C2×C32D3×M1611yescharacteristic
234C2×C32C3×#11711yescharacteristic
235C2×C32C2×C3×M1611yescharacteristic
236C2×C32C2×C3×M1611yescharacteristic
237C22×C4—91no—
238C22×C4—91no—
239C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
240C2×C8—92no—
241C2×C8C2×(C3⋊D3)⋊C312yes—
242C42—91no—
243C42—91no—
244C42—91no—
245C42—91no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—31no—
249C22×C3—31no—
250C22×C3—32no—
251C22×C3—32no—
252C22×C3—61no—
253C22×C3—61no—
254C22×C3—61no—
255C22×C3—61no—
256C22×C3—61no—
257C22×C3—62no—
258C22×C3—121no—
259C22×C3—121no—
260C22×C3C3×ΓL(1,𝔽25)12yes—
261C22×C3C2×C3×C4×D311yescharacteristic
262C3×C4—21no—
263C3×C4—32no—
264C3×C4—32no—
265C3×C4—61no—
266C3×C4—61no—
267C3×C4—62no—
268C3×C4—92no—
269C3×C4—121no—
270C3×C4—181no—
271C3×C4—181no—
272C3×C4—181no—
273C3×C4C3×#23712yes—
274Dic3—32no—
275Dic3—61no—
276Dic3—61no—
277Dic3—61no—
278Dic3—92no—
279Dic3—181no—
280Dic3—181no—
281Dic3—181no—
282C32—21no—
283C32#471911yescharacteristic, complemented
284C32C3×#43211yescharacteristic, complemented
285C23C4×(C3⋊D3)⋊C311yescharacteristic
286C2×C4—22no—
287C2×C4—91no—
288C2×C4—91no—
289C2×C4—91no—
290C2×C4—91no—
291C2×C4—182no—
292C2×C4—182no—
293C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
294C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
295C8—22no—
296C8—182no—
297C2×C3—22no—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—62no—
305C2×C3—62no—
306C2×C3—61no—
307C2×C3—61no—
308C2×C3—61no—
309C2×C3—122no—
310C2×C3C3×#87711yescharacteristic
311C2×C3C2×C3×ΓL(1,𝔽25)11yescharacteristic
312C2×C3C3×D3×M1611yescharacteristic
313C22—21no—
314C22—21no—
315C22#3743⋊C312yes—
316C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
317C4—21no—
318C4—92no—
319C4—181no—
320C4—181no—
321C4—181no—
322C4#3830⋊C312yes—
323C3—31no—
324C3—31no—
325C3—61no—
326C3C3×#471911yescharacteristic
327C2—22no—
328C2#64086⋊C311yescharacteristic
329C2C2×#3743⋊C311yescharacteristic
330C2M16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊C8C2
socleC22×C3C2×C3×C4×D3
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC22⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#432
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #63642⋊C3C2semidirect
5. #63641⋊C3C2semidirect
6. #10592418C3semidirect
23. #63642C2×C3semidirect
24. #63641C2×C3semidirect
183. H(3,𝔽3)#432semidirect
283. C32#4719semidirect
284. C32C3×#432semidirect

Maximal subgroups

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

subgroupindex
C2×C4×(C3⋊Dic3)⋊C32
H(3,𝔽3)×C22⋊C82
#63657⋊C32
#63642⋊C32
#63641⋊C32
#105924183
C3×#47193