Finite Group Databasebeta

← #13137330

Subgroups — #13137330

Order 1728.

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

normal subgroups only

This group has 404 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
3C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic
4#11720309C212yesmaximal
5#11720306C212yesmaximal
6#10590310—31nomaximal
7#10590284—31nomaximal
8C2×#6229—21no—
9C2×#634⋊C3C2211yescharacteristic
10C42×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084—21no—
12C4⋊#1084C2212yes—
13C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
14#3750⋊C3—21no—
15C4×(C3⋊Dic3)⋊C3—21no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C4×#634D311yescharacteristic
18C3×#869—31no—
19C3×#869—31no—
20C3×#869—61no—
21C3×C4×Dic6—31no—
22C4⋊(C3⋊(C3⋊C8))D311yescharacteristic
23C3×C4⋊(C3⋊C8)—31no—
24#63576—32no—
25#63575—32no—
26#63567—32no—
27#63565—32no—
28C2×C4×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×#1084—22no—
32#6229—42no—
33C2×(C3⋊Dic3)⋊C3—21no—
34C2×(C3⋊Dic3)⋊C3—21no—
35#634⋊C3—21no—
36#634⋊C3Q812yescomplemented
37(C3⋊(C3⋊C8))⋊C3—22no—
38#5091—91no—
39#5021—91no—
40#4850—91no—
41C2×#634C2×D311yescharacteristic
42C2×C3×Dic6—31no—
43C2×C3×Dic6—62no—
44C2×C3×Dic6—61no—
45C2×C3×Dic6—61no—
46C32×C42—21no—
47C32×C42C2×D311yescharacteristic
48C32×C42C2×D311yescharacteristic
49C4⋊(C3⋊Dic3)C2×D311yescharacteristic
50#3750—21no—
51C4×C3⋊Dic3—21no—
52C2×C3⋊(C3⋊C8)C2×D312yes—
53C3×C4⋊Dic3—32no—
54C3×C4⋊Dic3—32no—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3—62no—
57C3×C4⋊Dic3—62no—
58C3×C4⋊Dic3—61no—
59C3×C4⋊Dic3—61no—
60C3×Dic3⋊C4—61no—
61C3×C4×Dic3—61no—
62C2×C3×C3⋊C8—32no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C4×H(3,𝔽3)—21no—
65C4×H(3,𝔽3)Q1612yes—
66C4×H(3,𝔽3)C2×Q811yescharacteristic
67C4×H(3,𝔽3)#4111yescharacteristic
68#1084—42no—
69(C3⋊Dic3)⋊C3—22no—
70(C3⋊Dic3)⋊C3—41no—
71C3×C4⋊Q8—91no—
72C3×C4×Q8—91no—
73#869—31no—
74#869—31no—
75#869—61no—
76C4×Dic6—31no—
77C4×Dic6—91no—
78C3×C4⋊C8—91no—
79#835—92no—
80#818—92no—
81#816—92no—
82#810—92no—
83#807—92no—
84#807—92no—
85C4⋊(C3⋊C8)—31no—
86C4⋊(C3⋊C8)—91no—
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4D1211yescharacteristic
91C2×C32×C4D1211yescharacteristic
92C2×C32×C4#6711yescharacteristic
93C2×C32×C4#6711yescharacteristic
94C2×C32×C4C22×D311yescharacteristic
95C2×C32×C4C22×D311yescharacteristic
96C2×C3⋊Dic3—21no—
97C2×C3⋊Dic3—21no—
98#634—21no—
99#634Dic612yescomplemented
100C2×C3×Dic3—61no—
101C2×C3×Dic3—64no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—62no—
105C3×Dic6—32no—
106C3×Dic6—61no—
107C3×Dic6—124no—
108C3×Dic6—122no—
109C3×Dic6—122no—
110C3⋊(C3⋊C8)—22no—
111C3×C3⋊C8—62no—
112#477—271noSylow
113C2×H(3,𝔽3)C22⋊Q811yescharacteristic
114C2×H(3,𝔽3)C2×Q1611yescharacteristic
115C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
116C2×C3×Q8—91no—
117C2×C3×Q8—181no—
118C2×Dic6—31no—
119C2×Dic6—61no—
120C2×Dic6—61no—
121C2×Dic6—62no—
122C2×Dic6—91no—
123C2×C3×C8—92no—
124C3×C4⋊C4—91no—
125C3×C4⋊C4—92no—
126C3×C4⋊C4—181no—
127C3×C4⋊C4—181no—
128C3×C42—31no—
129C3×C42—31no—
130C3×C42—61no—
131C3×C42—181no—
132C3×C42D3211yescharacteristic
133C4⋊Dic3—32no—
134C4⋊Dic3—32no—
135C4⋊Dic3—31no—
136C4⋊Dic3—62no—
137C4⋊Dic3—61no—
138C4⋊Dic3—61no—
139C4⋊Dic3—62no—
140C4⋊Dic3—91no—
141Dic3⋊C4—61no—
142Dic3⋊C4—181no—
143C4×Dic3—61no—
144C4×Dic3—181no—
145C2×C3⋊C8—32no—
146C2×C3⋊C8—92no—
147C22×C32—21no—
148C22×C32C2×D1211yescharacteristic
149C22×C32C2×#6711yescharacteristic
150C32×C4—22no—
151C32×C4—21no—
152C32×C4—21no—
153C32×C4—22no—
154C32×C4—21no—
155C32×C4—21no—
156C32×C4Dic1212yes—
157C32×C4#21612yes—
158C32×C4C2×Dic611yescharacteristic
159C32×C4#23511yescharacteristic
160C32×C4#23711yescharacteristic
161C32×C4D3×Q811yescharacteristic
162C3⋊Dic3—22no—
163C3⋊Dic3—41no—
164C3×Dic3—62no—
165C3×Dic3—124no—
166C3×Dic3—121no—
167C3×Dic3—122no—
168C3×Dic3—122no—
169C4⋊Q8—91no—
170C4×Q8—91no—
171#107—272no—
172C4⋊C8—91no—
173#103—272no—
174H(3,𝔽3)#47711yescharacteristic, complemented, Sylow
175C3×Q8—92no—
176C3×Q8—181no—
177C3×Q8—362no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—61no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—181no—
188C2×C3×C4—181no—
189C2×C3×C4—182no—
190C2×C3×C4#62611yescharacteristic
191C2×C3×C4#62611yescharacteristic
192C2×C3×C4C2×D3211yescharacteristic
193C2×Dic3—61no—
194C2×Dic3—62no—
195C2×Dic3—62no—
196C2×Dic3—64no—
197C2×Dic3—61no—
198C2×Dic3—181no—
199C2×Dic3—181no—
200Dic6—32no—
201Dic6—61no—
202Dic6—92no—
203Dic6—122no—
204Dic6—122no—
205Dic6—124no—
206Dic6—181no—
207C3×C8—182no—
208C3⋊C8—62no—
209C3⋊C8—182no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—21no—
213C2×C32#89711yescharacteristic
214C2×C32C2×Dic1211yescharacteristic
215C2×C32#90811yescharacteristic
216C2×C32#92411yescharacteristic
217C2×C32C2×#21611yescharacteristic
218C2×C32#94911yescharacteristic
219C2×Q8—91no—
220C2×Q8—181no—
221C2×C8—92no—
222C4⋊C4—92no—
223C4⋊C4—91no—
224C4⋊C4—181no—
225C4⋊C4—181no—
226C42—181no—
227C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
228C22×C3—31no—
229C22×C3—31no—
230C22×C3—61no—
231C22×C3C2×#62611yescharacteristic
232C3×C4—21no—
233C3×C4—31no—
234C3×C4—31no—
235C3×C4—31no—
236C3×C4—31no—
237C3×C4—32no—
238C3×C4—32no—
239C3×C4—61no—
240C3×C4—61no—
241C3×C4—62no—
242C3×C4—61no—
243C3×C4—61no—
244C3×C4—62no—
245C3×C4—182no—
246C3×C4—361no—
247C3×C4—362no—
248C3×C4#371912yes—
249C3×C4D3×Dic611yescharacteristic
250C3×C4#379511yescharacteristic
251Dic3—62no—
252Dic3—122no—
253Dic3—122no—
254Dic3—124no—
255Dic3—121no—
256Dic3—182no—
257Dic3—361no—
258C32—21no—
259C32#485011yescharacteristic, complemented
260C32#502111yescharacteristic, complemented
261Q8—92no—
262Q8—181no—
263Q8—362no—
264C2×C4—181no—
265C2×C4—181no—
266C2×C4—182no—
267C2×C4#620011yescharacteristic
268C2×C4#620011yescharacteristic
269C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
270C8—182no—
271C2×C3—31no—
272C2×C3—31no—
273C2×C3—31no—
274C2×C3—31no—
275C2×C3—31no—
276C2×C3—31no—
277C2×C3—61no—
278C2×C3—61no—
279C2×C3—61no—
280C2×C3#6382511yescharacteristic
281C2×C3C2×#371911yescharacteristic
282C2×C3#6385711yescharacteristic
283C22C2×#620011yescharacteristic
284C4—21no—
285C4—182no—
286C4—362no—
287C4—361no—
288C4#8843912yes—
289C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
290C4#8864811yescharacteristic
291C3—31no—
292C3—31no—
293C3—61no—
294C3#1059028311yescharacteristic
295C2#1172106811yescharacteristic
296C2C2×#8843911yescharacteristic
297C2#1172110711yescharacteristic
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)#477
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
36. #634⋊C3Q8semidirect
99. #634Dic6semidirect
174. H(3,𝔽3)#477semidirect
259. C32#4850semidirect
260. C32#5021semidirect

Maximal subgroups

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

subgroupindex
#117217152
C4×#634⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203092
#117203062
#105903103
#105902843