Finite Group Databasebeta

← #13137438

Subgroups — #13137438

Order 1728.

448 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C4×#6229C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic
3C2×#88439C212yesmaximal
4C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic
5#11720309C212yesmaximal
6#10590385—31nomaximal
7#10590384—31nomaximal
8C2×#6229C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084C2211yescharacteristic
12#88540—21no—
13C4×#1084—21no—
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)—21no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17#88439—24no—
18#64078D311yescharacteristic
19C3×#869—31no—
20C3×C4×Dic6—31no—
21C3×C4×Dic6—31no—
22C3×C4×Dic6—61no—
23C2×#3719—32no—
24C2×#3718—32no—
25C4⋊(C3⋊(C3⋊C8))D311yescharacteristic
26C3×C4⋊(C3⋊C8)—31no—
27#63567—32no—
28#63565—32no—
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—21no—
33C2×#1084—21no—
34#6229—21no—
35#6229D412yes—
36C2×(C3⋊Dic3)⋊C3—22no—
37#634⋊C3—22no—
38#634⋊C3—22no—
39(C3⋊(C3⋊C8))⋊C3—22no—
40#5090—91no—
41#5031—91no—
42#4849—91no—
43C2×#634C2×D312yes—
44C2×C3×Dic6—31no—
45C2×C3×Dic6—31no—
46C2×C3×Dic6—32no—
47C2×C3×Dic6—61no—
48C32×C42—21no—
49C32×C42C2×D311yescharacteristic
50C32×C42C2×D311yescharacteristic
51C4⋊(C3⋊Dic3)—21no—
52C4⋊(C3⋊Dic3)—21no—
53C2×C3⋊(C3⋊C8)C2×D312yes—
54C3×C4⋊Dic3—31no—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3—61no—
57C3×C4⋊Dic3—61no—
58C3×C4⋊Dic3—61no—
59C3×Dic3⋊C4—62no—
60C3×Dic3⋊C4—61no—
61C3×Dic3⋊C4—61no—
62C3×C4×Dic3—62no—
63C3×C4×Dic3—61no—
64C3×C4×Dic3—61no—
65C2×C3×C3⋊C8—32no—
66#3719—64no—
67#3718—64no—
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C4×H(3,𝔽3)—21no—
70C4×H(3,𝔽3)Q1612yescomplemented
71C4×H(3,𝔽3)C2×D411yescharacteristic
72C4×H(3,𝔽3)#4111yescharacteristic
73#1084—22no—
74#1084—41no—
75(C3⋊Dic3)⋊C3—42no—
76C3×C4⋊Q8—91no—
77C3×C4×Q8—91no—
78C2×#216—92no—
79C2×#216—92no—
80C2×Dic12—92no—
81#869—31no—
82#869—91no—
83C4×Dic6—31no—
84C4×Dic6—31no—
85C4×Dic6—61no—
86C3×C4⋊C8—91no—
87#835—92no—
88#816—92no—
89#810—92no—
90C4⋊(C3⋊C8)—31no—
91C4⋊(C3⋊C8)—91no—
92C2×C32×C4—21no—
93C2×C32×C4—21no—
94C2×C32×C4—21no—
95C2×C32×C4D1211yescharacteristic
96C2×C32×C4D1211yescharacteristic
97C2×C32×C4#6711yescharacteristic
98C2×C32×C4#6711yescharacteristic
99C2×C32×C4C22×D311yescharacteristic
100C2×C32×C4C22×D311yescharacteristic
101C2×C3⋊Dic3—22no—
102#634—22no—
103#634—22no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C3×Dic6—32no—
112C3×Dic6—32no—
113C3×Dic6—62no—
114C3×Dic6—62no—
115C3×Dic6—61no—
116C3×Dic6—61no—
117C3×Dic6—62no—
118C3×Dic6—62no—
119C3⋊(C3⋊C8)—22no—
120C3×C3⋊C8—62no—
121#462—271noSylow
122C2×H(3,𝔽3)#12111yescharacteristic
123C2×H(3,𝔽3)C2×Q1611yescharacteristic
124C2×H(3,𝔽3)#13711yescharacteristic
125C2×C3×Q8—92no—
126C2×C3×Q8—91no—
127C2×Dic6—31no—
128C2×Dic6—31no—
129C2×Dic6—32no—
130C2×Dic6—61no—
131C2×Dic6—92no—
132C2×C3×C8—92no—
133C3×C4⋊C4—91no—
134C3×C4⋊C4—181no—
135C3×C4⋊C4—181no—
136C3×C4⋊C4—181no—
137C3×C42—31no—
138C3×C42—31no—
139C3×C42—61no—
140C3×C42—181no—
141C3×C42D3211yescharacteristic
142#216—184no—
143#216—184no—
144C4⋊Dic3—31no—
145C4⋊Dic3—31no—
146C4⋊Dic3—61no—
147C4⋊Dic3—61no—
148C4⋊Dic3—61no—
149C4⋊Dic3—181no—
150C4⋊Dic3—181no—
151Dic3⋊C4—61no—
152Dic3⋊C4—61no—
153Dic3⋊C4—62no—
154C4×Dic3—61no—
155C4×Dic3—61no—
156C4×Dic3—62no—
157C2×C3⋊C8—32no—
158C2×C3⋊C8—92no—
159Dic12—184no—
160C22×C32—21no—
161C22×C32C2×D1211yescharacteristic
162C22×C32C2×#6711yescharacteristic
163C32×C4—21no—
164C32×C4—21no—
165C32×C4—22no—
166C32×C4—21no—
167C32×C4—21no—
168C32×C4—41no—
169C32×C4Dic1212yescomplemented
170C32×C4#21612yescomplemented
171C32×C4D3×D411yescharacteristic
172C32×C4D3×D411yescharacteristic
173C32×C4#23711yescharacteristic
174C32×C4#23911yescharacteristic
175C3⋊Dic3—42no—
176C3×Dic3—64no—
177C3×Dic3—62no—
178C3×Dic3—62no—
179C3×Dic3—122no—
180C3×Dic3—122no—
181C3×Dic3—121no—
182C3×Dic3—121no—
183C2×Q16—272no—
184C4⋊Q8—91no—
185C4×Q8—91no—
186C4⋊C8—91no—
187#103—272no—
188H(3,𝔽3)#46211yescharacteristic, complemented, Sylow
189C3×Q8—92no—
190C3×Q8—182no—
191C3×Q8—182no—
192C3×Q8—181no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—61no—
202C2×C3×C4—181no—
203C2×C3×C4—182no—
204C2×C3×C4—181no—
205C2×C3×C4#62611yescharacteristic
206C2×C3×C4#62611yescharacteristic
207C2×C3×C4C2×D3211yescharacteristic
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—62no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C2×Dic3—62no—
214C2×Dic3—62no—
215C2×Dic3—182no—
216Dic6—32no—
217Dic6—32no—
218Dic6—62no—
219Dic6—61no—
220Dic6—61no—
221Dic6—62no—
222Dic6—62no—
223Dic6—62no—
224Dic6—182no—
225Dic6—182no—
226C3×C8—182no—
227C3⋊C8—62no—
228C3⋊C8—182no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32#89511yescharacteristic
233C2×C32C2×Dic1211yescharacteristic
234C2×C32#90911yescharacteristic
235C2×C32#93811yescharacteristic
236C2×C32#94211yescharacteristic
237C2×C32C2×#21611yescharacteristic
238C2×Q8—91no—
239C2×Q8—92no—
240Q16—544no—
241C2×C8—92no—
242C4⋊C4—91no—
243C4⋊C4—181no—
244C4⋊C4—181no—
245C4⋊C4—181no—
246C42—181no—
247C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
248C22×C3—31no—
249C22×C3—31no—
250C22×C3—61no—
251C22×C3C2×#62611yescharacteristic
252C3×C4—21no—
253C3×C4—31no—
254C3×C4—31no—
255C3×C4—31no—
256C3×C4—31no—
257C3×C4—32no—
258C3×C4—32no—
259C3×C4—61no—
260C3×C4—61no—
261C3×C4—61no—
262C3×C4—61no—
263C3×C4—62no—
264C3×C4—121no—
265C3×C4—182no—
266C3×C4—362no—
267C3×C4—361no—
268C3×C4#371912yes—
269C3×C4#379611yescharacteristic
270C3×C4#380211yescharacteristic
271Dic3—62no—
272Dic3—62no—
273Dic3—64no—
274Dic3—121no—
275Dic3—121no—
276Dic3—122no—
277Dic3—122no—
278Dic3—362no—
279C32—21no—
280C32#484911yescharacteristic, complemented
281C32#509011yescharacteristic, complemented
282Q8—92no—
283Q8—181no—
284Q8—182no—
285Q8—182no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4—182no—
289C2×C4#620011yescharacteristic
290C2×C4#620011yescharacteristic
291C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
292C8—182no—
293C2×C3—31no—
294C2×C3—31no—
295C2×C3—31no—
296C2×C3—31no—
297C2×C3—31no—
298C2×C3—31no—
299C2×C3—61no—
300C2×C3—61no—
301C2×C3—61no—
302C2×C3#6383311yescharacteristic
303C2×C3C2×#371911yescharacteristic
304C2×C3#6391611yescharacteristic
305C22C2×#620011yescharacteristic
306C4—21no—
307C4—182no—
308C4—361no—
309C4—362no—
310C4#8843912yescomplemented
311C4#8864911yescharacteristic
312C4#8865511yescharacteristic
313C3—31no—
314C3—31no—
315C3—61no—
316C3#1059038811yescharacteristic
317C2#1172106911yescharacteristic
318C2C2×#8843911yescharacteristic
319C2#1172115911yescharacteristic
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
70. C4×H(3,𝔽3)Q16semidirect
169. C32×C4Dic12semidirect
170. C32×C4#216semidirect
188. H(3,𝔽3)#462semidirect
280. C32#4849semidirect
281. C32#5090semidirect
310. C4#88439semidirect

Maximal subgroups

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

subgroupindex
C4×#62292
#64078⋊C32
C2×#884392
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203092
#105903853
#105903843