Finite Group Databasebeta

← #10592815⋊C3 #13143122

Subgroups — #10592815⋊C3 #13143122

Order 1728.

475 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C22×C8×H(3,𝔽3)C211yesmaximal, characteristic
2#64133⋊C3C212yesmaximal
3C2×#3746⋊C3C211yesmaximal, characteristic, complemented
4#63645⋊C3C211yesmaximal, characteristic, complemented
5#63643⋊C3C212yesmaximal, complemented
6#10592815C311yesmaximal, characteristic, complemented
7C3×#5116—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10C2×C8×H(3,𝔽3)—22no—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13#3757⋊C3—22no—
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15#3750⋊C3—22no—
16#3746⋊C3—22no—
17C22×C32×C8—21no—
18C22×C32×C8D311yescharacteristic
19C22×C32×C8C2×C311yescharacteristic
20#64133C2×C312yes—
21C2×#3746C2×C311yescharacteristic, complemented
22C3×#924—32no—
23C2×C3×Dic12—31no—
24#63645C2×C311yescharacteristic, complemented
25#63643C2×C312yescomplemented
26C3×#818—31no—
27C3×#816—32no—
28C23×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—22no—
35#634⋊C3—22no—
36#634⋊C3—22no—
37C8×H(3,𝔽3)—21no—
38C8×H(3,𝔽3)D412yes—
39C3×#467—91no—
40#5116—31no—
41#5116—91no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C22×C311yescharacteristic
45C2×#634C22×C312yes—
46C2×C3×Dic6—32no—
47C2×C32×C8—24no—
48C2×C32×C8—21no—
49C2×C32×C8—21no—
50C2×C32×C8—22no—
51C2×C32×C8—22no—
52C2×C32×C8C2×D311yescharacteristic
53C2×C32×C8C2×D311yescharacteristic
54C2×C32×C8C22×C311yescharacteristic
55C2×C32×C8C22×C311yescharacteristic
56#3757—22no—
57C4⋊(C3⋊Dic3)C22×C312yes—
58#3750—22no—
59#3746—22no—
60C3×#217—62no—
61C3×C4⋊Dic3—32no—
62C3×Dic3⋊C4—62no—
63C3×Dic12—62no—
64C22×H(3,𝔽3)—21no—
65C22×H(3,𝔽3)—21no—
66C22×H(3,𝔽3)Q1612yescomplemented
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C4×H(3,𝔽3)—21no—
69C4×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)#4111yescharacteristic
71(C3⋊Dic3)⋊C3—42no—
72(C3⋊Dic3)⋊C3—42no—
73C2×C3×Q16—91no—
74C22×C3×C8—31no—
75C22×C3×C8—31no—
76C22×C3×C8—61no—
77C22×C3×C8C3×D311yescharacteristic
78C3×C22⋊Q8—92no—
79#924—32no—
80#924—92no—
81C2×Dic12—31no—
82C2×Dic12—91no—
83C3×#107—91no—
84C3×#103—92no—
85#818—31no—
86#818—91no—
87#816—32no—
88#816—92no—
89C23×C32—21no—
90C23×C32D1211yescharacteristic
91C23×C32C3×D411yescharacteristic
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—21no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4D1211yescharacteristic
101C2×C32×C4C3×D411yescharacteristic
102C2×C32×C4C22×D311yescharacteristic
103C2×C32×C4C23×C311yescharacteristic
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3—22no—
106#634—22no—
107#634—22no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—62no—
110C3×Dic6—62no—
111C3×Dic6—62no—
112C32×C8—22no—
113C32×C8—21no—
114C32×C8—21no—
115C32×C8—22no—
116C32×C8#6712yes—
117C32×C8C3×D412yes—
118#467—91noSylow
119C2×H(3,𝔽3)—22no—
120C2×H(3,𝔽3)#12111yescharacteristic
121C2×H(3,𝔽3)C2×Q1611yescharacteristic
122C2×H(3,𝔽3)#13511yescharacteristic
123C2×C3×Q8—92no—
124C22×C3×C4—31no—
125C22×C3×C4—31no—
126C22×C3×C4—61no—
127C22×C3×C4C2×C3×D311yescharacteristic
128C2×Dic6—32no—
129C2×Dic6—92no—
130C3×Q16—182no—
131C2×C3×C8—22no—
132C2×C3×C8—31no—
133C2×C3×C8—31no—
134C2×C3×C8—31no—
135C2×C3×C8—31no—
136C2×C3×C8—62no—
137C2×C3×C8—62no—
138C2×C3×C8—64no—
139C2×C3×C8—61no—
140C2×C3×C8—61no—
141C2×C3×C8C2×C3×D311yescharacteristic
142C2×C3×C8C2×C3×D311yescharacteristic
143C3×C4⋊C4—92no—
144C3×C4⋊C4—182no—
145C3×C22⋊C4—182no—
146#217—62no—
147#217—182no—
148C4⋊Dic3—32no—
149C4⋊Dic3—92no—
150Dic3⋊C4—62no—
151Dic3⋊C4—182no—
152Dic12—62no—
153Dic12—182no—
154C22×C32—22no—
155C22×C32—22no—
156C22×C32—21no—
157C22×C32—21no—
158C22×C32—21no—
159C22×C32—21no—
160C22×C32—22no—
161C22×C32—21no—
162C22×C32Dic1212yescomplemented
163C22×C32C3×Q1612yescomplemented
164C22×C32C2×D1211yescharacteristic
165C22×C32C2×C3×D411yescharacteristic
166C32×C4—22no—
167C32×C4—21no—
168C32×C4—21no—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4#23511yescharacteristic
172C32×C4C2×#6711yescharacteristic
173C32×C4C2×C3×D411yescharacteristic
174C32×C4C3×#4111yescharacteristic
175C3⋊Dic3—42no—
176C3⋊Dic3—42no—
177C3×Dic3—122no—
178C3×Dic3—122no—
179C2×Q16—91no—
180C22×C8(C3⋊D3)⋊C311yescharacteristic
181C22⋊Q8—92no—
182#107—91no—
183#103—92no—
184H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
185C23×C3—31no—
186C23×C3—31no—
187C23×C3—61no—
188C23×C3C3×D1211yescharacteristic
189C3×Q8—182no—
190C3×Q8—182no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—62no—
202C2×C3×C4—62no—
203C2×C3×C4—61no—
204C2×C3×C4—61no—
205C2×C3×C4—182no—
206C2×C3×C4—182no—
207C2×C3×C4C3×D1211yescharacteristic
208C2×C3×C4C22×C3×D311yescharacteristic
209C2×Dic3—62no—
210C2×Dic3—62no—
211C2×Dic3—182no—
212C2×Dic3—182no—
213Dic6—62no—
214Dic6—62no—
215Dic6—182no—
216Dic6—182no—
217C3×C8—21no—
218C3×C8—32no—
219C3×C8—32no—
220C3×C8—61no—
221C3×C8—61no—
222C3×C8—62no—
223C3×C8—62no—
224C3×C8C3×#6712yes—
225C2×C32—24no—
226C2×C32—22no—
227C2×C32—22no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32#90411yescharacteristic
232C2×C32C2×Dic1211yescharacteristic
233C2×C32#93011yescharacteristic
234C2×C32C3×#12111yescharacteristic
235C2×C32C2×C3×Q1611yescharacteristic
236C2×C32C3×#13511yescharacteristic
237C2×Q8—92no—
238C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
239Q16—182no—
240C2×C8—22no—
241C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
242C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
243C4⋊C4—92no—
244C4⋊C4—182no—
245C22⋊C4—182no—
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—62no—
257C22×C3—62no—
258C22×C3—61no—
259C22×C3—62no—
260C22×C3C3×Dic1212yes—
261C22×C3C2×C3×D1211yescharacteristic
262C3×C4—21no—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—61no—
268C3×C4—61no—
269C3×C4—62no—
270C3×C4—61no—
271C3×C4—61no—
272C3×C4—362no—
273C3×C4—362no—
274C3×C4C3×#23511yescharacteristic
275C3×C4C2×C3×#6711yescharacteristic
276Dic3—122no—
277Dic3—122no—
278Dic3—362no—
279Dic3—362no—
280C32—21no—
281C32#511611yescharacteristic, complemented
282C32C3×#46711yescharacteristic, complemented
283C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
284Q8—182no—
285Q8—182no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—182no—
289C2×C4—182no—
290C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
291C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
292C8—21no—
293C8#638⋊C312yes—
294C2×C3—22no—
295C2×C3—31no—
296C2×C3—31no—
297C2×C3—31no—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—62no—
302C2×C3—62no—
303C2×C3—64no—
304C2×C3—61no—
305C2×C3—61no—
306C2×C3—61no—
307C2×C3C3×#90411yescharacteristic
308C2×C3C2×C3×Dic1211yescharacteristic
309C2×C3C3×#93011yescharacteristic
310C22—21no—
311C22—21no—
312C22#3746⋊C312yescomplemented
313C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
314C4—21no—
315C4—362no—
316C4—362no—
317C4#3828⋊C311yescharacteristic
318C4C2×#638⋊C311yescharacteristic
319C3—31no—
320C3—31no—
321C3—61no—
322C3C3×#511611yescharacteristic
323C2—22no—
324C2C2×#3746⋊C311yescharacteristic
325C2#64113⋊C311yescharacteristic
326C2#64139⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC22×C8×H(3,𝔽3)C2
socleC22×C3C2×C3×D12
prime radicalC23×H(3,𝔽3)D4
hypercenterC22×C8(C3⋊D3)⋊C3
hypocenterC32C3×#467
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
3. C2×#3746⋊C3C2semidirect
4. #63645⋊C3C2semidirect
5. #63643⋊C3C2semidirect
6. #10592815C3semidirect
21. C2×#3746C2×C3semidirect
24. #63645C2×C3semidirect
25. #63643C2×C3semidirect
66. C22×H(3,𝔽3)Q16semidirect
162. C22×C32Dic12semidirect
163. C22×C32C3×Q16semidirect
184. H(3,𝔽3)#467semidirect
281. C32#5116semidirect
282. C32C3×#467semidirect
312. C22#3746⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C8×H(3,𝔽3)2
#64133⋊C32
C2×#3746⋊C32
#63645⋊C32
#63643⋊C32
#105928153
C3×#51163