Finite Group Databasebeta

← #13137439

Subgroups — #13137439

Order 1728.

446 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#11721715C211yesmaximal, characteristic
2C4×#634⋊C3C211yesmaximal, characteristic
3C2×#88439C212yesmaximal
4C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic
5#11720326C212yesmaximal
6#10590388—31nomaximal
7#10590384—31nomaximal
8C2×#6229C2212yes—
9C2×#634⋊C3C2211yescharacteristic
10C42×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084—21no—
12C4⋊#1084—21no—
13C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
14#3750⋊C3—21no—
15C4×(C3⋊Dic3)⋊C3—21no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17#88439—24no—
18C4×#634D311yescharacteristic
19C3×#869—31no—
20C3×#869—31no—
21C3×#869—61no—
22C3×C4×Dic6—31no—
23C2×#3719—32no—
24C2×#3718—32no—
25C4⋊(C3⋊(C3⋊C8))D311yescharacteristic
26C3×C4⋊(C3⋊C8)—31no—
27#63566—32no—
28#63565—32no—
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—22no—
33#6229—22no—
34#6229—22no—
35C2×(C3⋊Dic3)⋊C3—21no—
36C2×(C3⋊Dic3)⋊C3—21no—
37#634⋊C3—21no—
38#634⋊C3D412yes—
39(C3⋊(C3⋊C8))⋊C3—22no—
40#5090—91no—
41#5031—91no—
42#4849—91no—
43C2×#634C2×D311yescharacteristic
44C2×C3×Dic6—31no—
45C2×C3×Dic6—32no—
46C2×C3×Dic6—32no—
47C2×C3×Dic6—62no—
48C32×C42—21no—
49C32×C42C2×D311yescharacteristic
50C32×C42C2×D311yescharacteristic
51C4⋊(C3⋊Dic3)C2×D311yescharacteristic
52#3750—21no—
53C4×C3⋊Dic3—21no—
54C2×C3⋊(C3⋊C8)C2×D312yes—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3—62no—
57C3×C4⋊Dic3—62no—
58C3×C4⋊Dic3—61no—
59C3×C4⋊Dic3—61no—
60C3×C4⋊Dic3—61no—
61C3×C4⋊Dic3—61no—
62C3×Dic3⋊C4—61no—
63C3×C4×Dic3—61no—
64C2×C3×C3⋊C8—32no—
65#3719—64no—
66#3718—64no—
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C4×H(3,𝔽3)—21no—
69C4×H(3,𝔽3)Q1612yescomplemented
70C4×H(3,𝔽3)C2×D411yescharacteristic
71C4×H(3,𝔽3)#4111yescharacteristic
72#1084—42no—
73(C3⋊Dic3)⋊C3—22no—
74(C3⋊Dic3)⋊C3—41no—
75C3×C4⋊Q8—91no—
76C3×C4×Q8—91no—
77C2×#216—92no—
78C2×#216—92no—
79C2×Dic12—92no—
80#869—31no—
81#869—31no—
82#869—61no—
83C4×Dic6—31no—
84C4×Dic6—91no—
85C3×C4⋊C8—91no—
86#835—92no—
87#816—92no—
88#810—92no—
89C4⋊(C3⋊C8)—31no—
90C4⋊(C3⋊C8)—91no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4—21no—
94C2×C32×C4D1211yescharacteristic
95C2×C32×C4D1211yescharacteristic
96C2×C32×C4#6711yescharacteristic
97C2×C32×C4#6711yescharacteristic
98C2×C32×C4C22×D311yescharacteristic
99C2×C32×C4C22×D311yescharacteristic
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—21no—
102#634—21no—
103#634D1212yes—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—64no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C3×Dic6—32no—
110C3×Dic6—61no—
111C3×Dic6—62no—
112C3×Dic6—62no—
113C3×Dic6—62no—
114C3×Dic6—62no—
115C3×Dic6—122no—
116C3×Dic6—122no—
117C3⋊(C3⋊C8)—22no—
118C3×C3⋊C8—62no—
119#462—271noSylow
120C2×H(3,𝔽3)#12111yescharacteristic
121C2×H(3,𝔽3)C2×Q1611yescharacteristic
122C2×H(3,𝔽3)#13711yescharacteristic
123C2×C3×Q8—91no—
124C2×C3×Q8—92no—
125C2×Dic6—31no—
126C2×Dic6—32no—
127C2×Dic6—32no—
128C2×Dic6—62no—
129C2×Dic6—91no—
130C2×C3×C8—92no—
131C3×C4⋊C4—91no—
132C3×C4⋊C4—181no—
133C3×C4⋊C4—181no—
134C3×C4⋊C4—181no—
135C3×C42—31no—
136C3×C42—31no—
137C3×C42—61no—
138C3×C42—181no—
139C3×C42D3211yescharacteristic
140#216—184no—
141#216—184no—
142C4⋊Dic3—31no—
143C4⋊Dic3—61no—
144C4⋊Dic3—61no—
145C4⋊Dic3—61no—
146C4⋊Dic3—61no—
147C4⋊Dic3—62no—
148C4⋊Dic3—62no—
149C4⋊Dic3—91no—
150Dic3⋊C4—61no—
151Dic3⋊C4—181no—
152C4×Dic3—61no—
153C4×Dic3—181no—
154C2×C3⋊C8—32no—
155C2×C3⋊C8—92no—
156Dic12—184no—
157C22×C32—21no—
158C22×C32C2×D1211yescharacteristic
159C22×C32C2×#6711yescharacteristic
160C32×C4—22no—
161C32×C4—21no—
162C32×C4—21no—
163C32×C4—22no—
164C32×C4—21no—
165C32×C4—21no—
166C32×C4Dic1212yescomplemented
167C32×C4#21612yescomplemented
168C32×C4C2×D1211yescharacteristic
169C32×C4#23511yescharacteristic
170C32×C4D3×D411yescharacteristic
171C32×C4#23911yescharacteristic
172C3⋊Dic3—22no—
173C3⋊Dic3—41no—
174C3×Dic3—62no—
175C3×Dic3—124no—
176C3×Dic3—122no—
177C3×Dic3—121no—
178C3×Dic3—122no—
179C2×Q16—272no—
180C4⋊Q8—91no—
181C4×Q8—91no—
182C4⋊C8—91no—
183#103—272no—
184H(3,𝔽3)#46211yescharacteristic, complemented, Sylow
185C3×Q8—92no—
186C3×Q8—181no—
187C3×Q8—182no—
188C3×Q8—182no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—61no—
196C2×C3×C4—61no—
197C2×C3×C4—61no—
198C2×C3×C4—181no—
199C2×C3×C4—181no—
200C2×C3×C4—182no—
201C2×C3×C4#62611yescharacteristic
202C2×C3×C4#62611yescharacteristic
203C2×C3×C4C2×D3211yescharacteristic
204C2×Dic3—61no—
205C2×Dic3—62no—
206C2×Dic3—62no—
207C2×Dic3—64no—
208C2×Dic3—61no—
209C2×Dic3—181no—
210C2×Dic3—181no—
211Dic6—32no—
212Dic6—61no—
213Dic6—62no—
214Dic6—62no—
215Dic6—62no—
216Dic6—62no—
217Dic6—92no—
218Dic6—122no—
219Dic6—122no—
220Dic6—181no—
221C3×C8—182no—
222C3⋊C8—62no—
223C3⋊C8—182no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32#89511yescharacteristic
228C2×C32C2×Dic1211yescharacteristic
229C2×C32#90911yescharacteristic
230C2×C32#93011yescharacteristic
231C2×C32C2×#21611yescharacteristic
232C2×C32#95111yescharacteristic
233C2×Q8—91no—
234C2×Q8—92no—
235Q16—544no—
236C2×C8—92no—
237C4⋊C4—91no—
238C4⋊C4—181no—
239C4⋊C4—181no—
240C4⋊C4—181no—
241C42—181no—
242C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
243C22×C3—31no—
244C22×C3—31no—
245C22×C3—61no—
246C22×C3C2×#62611yescharacteristic
247C3×C4—21no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—31no—
252C3×C4—32no—
253C3×C4—32no—
254C3×C4—61no—
255C3×C4—61no—
256C3×C4—62no—
257C3×C4—61no—
258C3×C4—61no—
259C3×C4—62no—
260C3×C4—182no—
261C3×C4—361no—
262C3×C4—362no—
263C3×C4#371912yes—
264C3×C4#379911yescharacteristic
265C3×C4D3×D1211yescharacteristic
266Dic3—62no—
267Dic3—122no—
268Dic3—122no—
269Dic3—124no—
270Dic3—121no—
271Dic3—182no—
272Dic3—361no—
273C32—21no—
274C32#484911yescharacteristic, complemented
275C32#503111yescharacteristic, complemented
276Q8—92no—
277Q8—181no—
278Q8—182no—
279Q8—182no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4—182no—
283C2×C4#620011yescharacteristic
284C2×C4#620011yescharacteristic
285C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
286C8—182no—
287C2×C3—31no—
288C2×C3—31no—
289C2×C3—31no—
290C2×C3—31no—
291C2×C3—31no—
292C2×C3—31no—
293C2×C3—61no—
294C2×C3—61no—
295C2×C3—61no—
296C2×C3#6383111yescharacteristic
297C2×C3C2×#371911yescharacteristic
298C2×C3#6390911yescharacteristic
299C22C2×#620011yescharacteristic
300C4—21no—
301C4—182no—
302C4—361no—
303C4—362no—
304C4#8843912yescomplemented
305C4#8865211yescharacteristic
306C4#8865411yescharacteristic
307C3—31no—
308C3—31no—
309C3—61no—
310C3#1059038511yescharacteristic
311C2#1172107411yescharacteristic
312C2C2×#8843911yescharacteristic
313C2#1172115611yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#462
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
69. C4×H(3,𝔽3)Q16semidirect
166. C32×C4Dic12semidirect
167. C32×C4#216semidirect
184. H(3,𝔽3)#462semidirect
274. C32#4849semidirect
275. C32#5031semidirect
304. C4#88439semidirect

Maximal subgroups

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

subgroupindex
#117217152
C4×#634⋊C32
C2×#884392
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203262
#105903883
#105903843