Finite Group Databasebeta

← #13137325

Subgroups — #13137325

Order 1728.

418 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 75 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, complemented
2#64084⋊C3C211yesmaximal, characteristic
3#63630⋊C3C211yesmaximal, characteristic, complemented
4#11720309C212yesmaximal, complemented
5#11720308C212yesmaximal
6#10590294—31nomaximal
7#10590280—31nomaximal
8C2×#6229—21no—
9C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
10C2×#634⋊C3C2211yescharacteristic
11C42×H(3,𝔽3)C2211yescharacteristic
12C4⋊#1084—21no—
13C4⋊#1084C2212yes—
14#3752⋊C3—21no—
15#3752⋊C3—21no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17#64084D311yescharacteristic
18C3×#875—31no—
19C3×#869—31no—
20C3×#869—31no—
21C3×#869—61no—
22#63630D311yescharacteristic, complemented
23C3×#803—31no—
24#63567—32no—
25#63565—32no—
26#63564—32no—
27#63562—32no—
28C22×(C3⋊D3)⋊C3—21no—
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—22no—
33#6229—42no—
34C2×(C3⋊Dic3)⋊C3—21no—
35C4⋊((C3⋊D3)⋊C3)—22no—
36#634⋊C3—22no—
37(C3⋊(C3⋊C8))⋊C3—22no—
38#5089—91no—
39#5055—91no—
40#4714—91no—
41C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
42C2×#634C2×D311yescharacteristic
43C2×C3×D12—31no—
44C2×C3×Dic6—31no—
45C2×C3×Dic6—62no—
46C2×C3×Dic6—61no—
47C2×C3×Dic6—61no—
48C32×C42—21no—
49C32×C42C2×D311yescharacteristic
50C32×C42C2×D311yescharacteristic
51#3752—21no—
52#3752—21no—
53C2×C3⋊(C3⋊C8)C2×D312yes—
54C3×(C2×D3)⋊C4—61no—
55C3×(C2×D3)⋊C4—61no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—32no—
58C3×C4⋊Dic3—62no—
59C3×C4⋊Dic3—62no—
60C3×C4⋊Dic3—61no—
61C3×C4⋊Dic3—61no—
62C2×C3×C3⋊C8—32no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×(C3⋊D3)⋊C3—41no—
65C2×(C3⋊D3)⋊C3—41no—
66C2×(C3⋊D3)⋊C3—41no—
67C4×H(3,𝔽3)—21no—
68C4×H(3,𝔽3)—21no—
69C4×H(3,𝔽3)#4112yes—
70#1084—42no—
71(C3⋊Dic3)⋊C3—41no—
72C3×C4⋊Q8—91no—
73C3×#124—91no—
74#875—31no—
75#875—91no—
76#869—31no—
77#869—31no—
78#869—61no—
79C3×#97—91no—
80#835—92no—
81#832—92no—
82#821—92no—
83#816—92no—
84#810—92no—
85#809—92no—
86#803—31no—
87#803—91no—
88C22×C3⋊D3—21no—
89C22×C3×D3—61no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4D1211yescharacteristic
94C2×C32×C4D1211yescharacteristic
95C2×C32×C4#6711yescharacteristic
96C2×C32×C4#6711yescharacteristic
97C2×C32×C4C22×D311yescharacteristic
98C2×C32×C4C22×D311yescharacteristic
99C2×C3⋊Dic3—21no—
100C4⋊(C3⋊D3)—22no—
101#634—22no—
102C2×C3×Dic3—64no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C3×D12—62no—
107C3×Dic6—62no—
108C3×Dic6—124no—
109C3×Dic6—122no—
110C3×Dic6—122no—
111C3⋊(C3⋊C8)—22no—
112C3×C3⋊C8—62no—
113#489—271noSylow
114C2×H(3,𝔽3)#12411yescharacteristic
115C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
116C2×H(3,𝔽3)#13711yescharacteristic
117(C3⋊D3)⋊C3—81no—
118C2×C3×Q8—91no—
119C2×C3×Q8—181no—
120C2×C3×D4—91no—
121C2×D12—31no—
122C2×D12—91no—
123C2×Dic6—31no—
124C2×Dic6—61no—
125C2×Dic6—61no—
126C2×Dic6—62no—
127C2×Dic6—91no—
128C2×C3×C8—92no—
129C3×C4⋊C4—92no—
130C3×C4⋊C4—181no—
131C3×C22⋊C4—181no—
132C3×C22⋊C4—181no—
133C3×C42—31no—
134C3×C42—31no—
135C3×C42—61no—
136C3×C42D3211yescharacteristic
137(C2×D3)⋊C4—61no—
138(C2×D3)⋊C4—61no—
139(C2×D3)⋊C4—181no—
140(C2×D3)⋊C4—181no—
141C4⋊Dic3—32no—
142C4⋊Dic3—32no—
143C4⋊Dic3—62no—
144C4⋊Dic3—61no—
145C4⋊Dic3—61no—
146C4⋊Dic3—62no—
147C2×C3⋊C8—32no—
148C2×C3⋊C8—92no—
149C22×C32—21no—
150C22×C32C2×D1211yescharacteristic
151C22×C32C2×#6711yescharacteristic
152C2×C3⋊D3—41no—
153C2×C3⋊D3—41no—
154C2×C3⋊D3—41no—
155C2×C3×D3—121no—
156C2×C3×D3—121no—
157C2×C3×D3—121no—
158C32×C4—22no—
159C32×C4—22no—
160C32×C4—21no—
161C32×C4—21no—
162C32×C4—21no—
163C32×C4—21no—
164C32×C4—22no—
165C32×C4#23512yes—
166C32×C4#23712yes—
167C3⋊Dic3—41no—
168C3×Dic3—124no—
169C3×Dic3—121no—
170C3×Dic3—122no—
171C3×Dic3—122no—
172C4⋊Q8—91no—
173#124—91no—
174#103—272no—
175#102—272no—
176#97—91no—
177H(3,𝔽3)#48911yescharacteristic, complemented, Sylow
178C23×C3—181no—
179C22×D3—61no—
180C22×D3—181no—
181C3×Q8—182no—
182C3×Q8—362no—
183C3×D4—182no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—181no—
194C2×C3×C4—182no—
195C2×C3×C4#62611yescharacteristic
196C2×C3×C4#62611yescharacteristic
197C2×C3×C4C2×D3211yescharacteristic
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—64no—
201C2×Dic3—61no—
202C2×Dic3—181no—
203D12—62no—
204D12—182no—
205Dic6—62no—
206Dic6—122no—
207Dic6—122no—
208Dic6—124no—
209Dic6—182no—
210C3×C8—182no—
211C3⋊C8—62no—
212C3⋊C8—182no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32#87511yescharacteristic
217C2×C32#90811yescharacteristic
218C2×C32#90911yescharacteristic
219C2×C32#93611yescharacteristic
220C2×C32#94911yescharacteristic
221C2×C32#95111yescharacteristic
222C3⋊D3—81no—
223C3×D3—241no—
224C2×Q8—91no—
225C2×Q8—181no—
226C2×D4—91no—
227C2×C8—92no—
228C4⋊C4—92no—
229C4⋊C4—181no—
230C22⋊C4—181no—
231C22⋊C4—181no—
232C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
233C22×C3—31no—
234C22×C3—31no—
235C22×C3—61no—
236C22×C3—361no—
237C22×C3—361no—
238C22×C3—361no—
239C22×C3C2×#62611yescharacteristic
240C2×D3—121no—
241C2×D3—121no—
242C2×D3—121no—
243C2×D3—361no—
244C2×D3—361no—
245C2×D3—361no—
246C3×C4—21no—
247C3×C4—21no—
248C3×C4—32no—
249C3×C4—32no—
250C3×C4—61no—
251C3×C4—61no—
252C3×C4—61no—
253C3×C4—61no—
254C3×C4—62no—
255C3×C4—62no—
256C3×C4—62no—
257C3×C4—361no—
258C3×C4—362no—
259C3×C4#379512yes—
260Dic3—122no—
261Dic3—122no—
262Dic3—124no—
263Dic3—121no—
264Dic3—361no—
265C32—21no—
266C32#471411yescharacteristic, complemented
267C32#505511yescharacteristic, complemented
268C23—181no—
269Q8—182no—
270Q8—362no—
271D4—182no—
272C2×C4—181no—
273C2×C4—182no—
274C2×C4#620011yescharacteristic
275C2×C4#620011yescharacteristic
276C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
277C8—182no—
278C2×C3—31no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—31no—
283C2×C3—31no—
284C2×C3—61no—
285C2×C3—61no—
286C2×C3—61no—
287C2×C3—721no—
288C2×C3#6382511yescharacteristic
289C2×C3#6383111yescharacteristic
290C2×C3#6386011yescharacteristic
291D3—241no—
292D3—721no—
293C22—361no—
294C22—361no—
295C22—361no—
296C22C2×#620011yescharacteristic
297C4—21no—
298C4—21no—
299C4—362no—
300C4—361no—
301C4#8864812yes—
302C3—31no—
303C3—31no—
304C3—61no—
305C3#1059027611yescharacteristic
306C2—721no—
307C2#1172106811yescharacteristic
308C2#1172107411yescharacteristic
309C2#1172111011yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#489
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
1. #11721715C2semidirect
3. #63630⋊C3C2semidirect
4. #11720309C2semidirect
22. #63630D3semidirect
177. H(3,𝔽3)#489semidirect
266. C32#4714semidirect
267. C32#5055semidirect

Maximal subgroups

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

subgroupindex
#117217152
#64084⋊C32
#63630⋊C32
#117203092
#117203082
#105902943
#105902803