Finite Group Databasebeta

← #13137389

Subgroups — #13137389

Order 1728.

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

normal subgroups only

This group has 486 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#11721385C211yesmaximal, characteristic
2C4×#634⋊C3C211yesmaximal, characteristic, complemented
3C2×#88431C212yesmaximal, complemented
4C4⋊#6180C211yesmaximal, characteristic, complemented
5#11720318C212yesmaximal, complemented
6#10590377—31nomaximal
7#10590367—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)—21no—
9C2×C4⋊((C3⋊D3)⋊C3)—21no—
10C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
11C2×#634⋊C3C2211yescharacteristic
12C42×H(3,𝔽3)C2211yescharacteristic
13C2×#6180C2212yes—
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15#3750⋊C3—21no—
16C4×(C3⋊Dic3)⋊C3—21no—
17#88431—24no—
18#64083D311yescharacteristic
19C4×#634D311yescharacteristic, complemented
20C3×#874—31no—
21C3×C4×Dic6—31no—
22C2×#3717—32no—
23C2×#3716—32no—
24C3×C4⋊(C3⋊C8)—31no—
25C3×C4⋊(C3⋊C8)—31no—
26C3×C4⋊(C3⋊C8)—61no—
27#63564—32no—
28#63563—32no—
29C22×(C3⋊D3)⋊C3—22no—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×(C3⋊Dic3)⋊C3—21no—
34C2×(C3⋊Dic3)⋊C3—21no—
35C4⋊((C3⋊D3)⋊C3)—22no—
36C4⋊((C3⋊D3)⋊C3)—22no—
37C4⋊((C3⋊D3)⋊C3)—42no—
38C4⋊((C3⋊D3)⋊C3)—42no—
39#634⋊C3—21no—
40#634⋊C3D412yescomplemented
41#6180—22no—
42#5075—91no—
43#5027—91no—
44#4848—91no—
45C2×C4⋊(C3⋊D3)—21no—
46C2×C4⋊(C3⋊D3)—21no—
47C2×C4⋊(C3⋊D3)C2×D312yes—
48C2×#634C2×D311yescharacteristic
49C2×C3×D12—32no—
50C2×C3×D12—61no—
51C2×C3×D12—61no—
52C2×C3×Dic6—31no—
53C32×C42—21no—
54C32×C42C2×D311yescharacteristic
55C32×C42C2×D311yescharacteristic
56C4⋊(C3⋊Dic3)C2×D311yescharacteristic
57#3750—21no—
58C4×C3⋊Dic3—21no—
59C3×C4⋊Dic3—31no—
60C3×Dic3⋊C4—61no—
61C3×C4×Dic3—61no—
62C2×C3×C3⋊C8—32no—
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—62no—
65#3717—64no—
66#3716—64no—
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C2×(C3⋊D3)⋊C3—42no—
69C2×(C3⋊D3)⋊C3—42no—
70C2×(C3⋊D3)⋊C3—42no—
71C4×H(3,𝔽3)—21no—
72C4×H(3,𝔽3)SD1612yescomplemented
73C4×H(3,𝔽3)C2×D411yescharacteristic
74C4×H(3,𝔽3)#4111yescharacteristic
75(C3⋊Dic3)⋊C3—22no—
76(C3⋊Dic3)⋊C3—41no—
77C3×#127—91no—
78C3×C4×Q8—91no—
79C2×#215—92no—
80C2×#214—92no—
81C2×#204—92no—
82#874—31no—
83#874—91no—
84C4×Dic6—31no—
85C4×Dic6—91no—
86C3×C4⋊C8—91no—
87#832—92no—
88#821—92no—
89#809—92no—
90C4⋊(C3⋊C8)—31no—
91C4⋊(C3⋊C8)—31no—
92C4⋊(C3⋊C8)—61no—
93C22×C3⋊D3—22no—
94C22×C3×D3—62no—
95C2×C32×C4—21no—
96C2×C32×C4—21no—
97C2×C32×C4—21no—
98C2×C32×C4#6711yescharacteristic
99C2×C32×C4#6711yescharacteristic
100C2×C32×C4#6711yescharacteristic
101C2×C32×C4#6711yescharacteristic
102C2×C32×C4C22×D311yescharacteristic
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—21no—
106C4⋊(C3⋊D3)—22no—
107C4⋊(C3⋊D3)—22no—
108C4⋊(C3⋊D3)—42no—
109C4⋊(C3⋊D3)—42no—
110#634—21no—
111#634D1212yescomplemented
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C3×D12—62no—
115C3×D12—62no—
116C3×D12—122no—
117C3×D12—122no—
118C3×Dic6—32no—
119C3×Dic6—61no—
120C3×C3⋊C8—62no—
121C3×C3⋊C8—62no—
122C3×C3⋊C8—122no—
123#460—271noSylow
124C2×H(3,𝔽3)#12111yescharacteristic
125C2×H(3,𝔽3)C2×SD1611yescharacteristic
126C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
127(C3⋊D3)⋊C3—82no—
128C2×C3×Q8—91no—
129C2×C3×D4—92no—
130C2×C3×D4—181no—
131C2×C3×D4—181no—
132C2×D12—32no—
133C2×D12—61no—
134C2×D12—61no—
135C2×D12—92no—
136C2×D12—181no—
137C2×D12—181no—
138C2×Dic6—31no—
139C2×Dic6—91no—
140C2×C3×C8—92no—
141C3×C4⋊C4—91no—
142C3×C4⋊C4—181no—
143C3×C42—31no—
144C3×C42—31no—
145C3×C42—61no—
146C3×C42—181no—
147C3×C42D3211yescharacteristic
148#215—184no—
149#214—184no—
150C4⋊Dic3—31no—
151C4⋊Dic3—91no—
152Dic3⋊C4—61no—
153Dic3⋊C4—181no—
154C4×Dic3—61no—
155C4×Dic3—181no—
156C2×C3⋊C8—32no—
157C2×C3⋊C8—32no—
158C2×C3⋊C8—62no—
159#204—184no—
160C22×C32—21no—
161C22×C32C2×#6711yescharacteristic
162C22×C32C2×#6711yescharacteristic
163C2×C3⋊D3—42no—
164C2×C3⋊D3—42no—
165C2×C3⋊D3—42no—
166C2×C3×D3—122no—
167C2×C3×D3—122no—
168C2×C3×D3—122no—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4—22no—
172C32×C4—21no—
173C32×C4—21no—
174C32×C4—41no—
175C32×C4#21412yescomplemented
176C32×C4#21512yescomplemented
177C32×C4C2×D1211yescharacteristic
178C32×C4#23511yescharacteristic
179C32×C4D3×D411yescharacteristic
180C32×C4#23711yescharacteristic
181C3⋊Dic3—22no—
182C3⋊Dic3—41no—
183C3×Dic3—62no—
184C3×Dic3—121no—
185C2×SD16—272no—
186#127—91no—
187C4×Q8—91no—
188C4⋊C8—91no—
189#102—272no—
190H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
191C23×C3—182no—
192C22×D3—62no—
193C22×D3—182no—
194C3×Q8—92no—
195C3×Q8—181no—
196C3×D4—182no—
197C3×D4—182no—
198C3×D4—362no—
199C3×D4—362no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—61no—
207C2×C3×C4—61no—
208C2×C3×C4—61no—
209C2×C3×C4—181no—
210C2×C3×C4—181no—
211C2×C3×C4#62511yescharacteristic
212C2×C3×C4#62511yescharacteristic
213C2×C3×C4C2×D3211yescharacteristic
214C2×Dic3—61no—
215C2×Dic3—61no—
216C2×Dic3—181no—
217C2×Dic3—181no—
218D12—62no—
219D12—62no—
220D12—122no—
221D12—122no—
222D12—182no—
223D12—182no—
224D12—362no—
225D12—362no—
226Dic6—32no—
227Dic6—61no—
228Dic6—92no—
229Dic6—181no—
230C3×C8—182no—
231C3⋊C8—62no—
232C3⋊C8—62no—
233C3⋊C8—122no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32#93011yescharacteristic
238C2×C32#93211yescharacteristic
239C2×C32C2×#21411yescharacteristic
240C2×C32#93811yescharacteristic
241C2×C32C2×#21511yescharacteristic
242C2×C32#94911yescharacteristic
243C3⋊D3—82no—
244C3×D3—242no—
245C2×Q8—91no—
246C2×D4—92no—
247C2×D4—181no—
248C2×D4—181no—
249SD16—544no—
250C2×C8—92no—
251C4⋊C4—91no—
252C4⋊C4—181no—
253C42—181no—
254C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—61no—
258C22×C3—362no—
259C22×C3—362no—
260C22×C3—362no—
261C22×C3C2×#62511yescharacteristic
262C2×D3—122no—
263C2×D3—122no—
264C2×D3—122no—
265C2×D3—362no—
266C2×D3—362no—
267C2×D3—362no—
268C3×C4—21no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—32no—
274C3×C4—32no—
275C3×C4—61no—
276C3×C4—61no—
277C3×C4—61no—
278C3×C4—61no—
279C3×C4—62no—
280C3×C4—121no—
281C3×C4—182no—
282C3×C4—361no—
283C3×C4#371512yes—
284C3×C4#379511yescharacteristic
285C3×C4D3×D1211yescharacteristic
286Dic3—62no—
287Dic3—121no—
288Dic3—182no—
289Dic3—361no—
290C32—21no—
291C32#502711yescharacteristic, complemented
292C32#507511yescharacteristic, complemented
293C23—182no—
294Q8—92no—
295Q8—181no—
296D4—182no—
297D4—182no—
298D4—362no—
299D4—362no—
300C2×C4—181no—
301C2×C4—181no—
302C2×C4#619811yescharacteristic
303C2×C4#619811yescharacteristic
304C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
305C8—182no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—61no—
313C2×C3—61no—
314C2×C3—61no—
315C2×C3—722no—
316C2×C3#6382311yescharacteristic
317C2×C3C2×#371511yescharacteristic
318C2×C3#6390811yescharacteristic
319D3—242no—
320D3—722no—
321C22—362no—
322C22—362no—
323C22—362no—
324C22C2×#619811yescharacteristic
325C4—21no—
326C4—182no—
327C4—361no—
328C4#8843112yescomplemented
329C4#8864811yescharacteristic
330C4#8865411yescharacteristic
331C3—31no—
332C3—31no—
333C3—61no—
334C3#1059036411yescharacteristic
335C2—722no—
336C2C2×#8843111yescharacteristic
337C2#1172106111yescharacteristic
338C2#1172114111yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radical#11721385C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#460
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
2. C4×#634⋊C3C2semidirect
3. C2×#88431C2semidirect
4. C4⋊#6180C2semidirect
5. #11720318C2semidirect
19. C4×#634D3semidirect
40. #634⋊C3D4semidirect
72. C4×H(3,𝔽3)SD16semidirect
111. #634D12semidirect
175. C32×C4#214semidirect
176. C32×C4#215semidirect
190. H(3,𝔽3)#460semidirect
291. C32#5027semidirect
292. C32#5075semidirect
328. C4#88431semidirect

Maximal subgroups

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

subgroupindex
#117213852
C4×#634⋊C32
C2×#884312
C4⋊#61802
#117203182
#105903773
#105903673