Finite Group Databasebeta

← #13137420

Subgroups — #13137420

Order 1728.

580 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721721C211yesmaximal, characteristic, complemented
2#11721385C211yesmaximal, characteristic, complemented
3C2×#88438C212yesmaximal, complemented
4C2×#88436C212yesmaximal, complemented
5#63630⋊C3C211yesmaximal, characteristic, complemented
6#10590354—31nomaximal
7#10590351—31nomaximal
8C2×C4⋊#272C2211yescharacteristic
9C2×#6229C2211yescharacteristic
10C2×C4⋊((C3⋊D3)⋊C3)—22no—
11C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
12C42×H(3,𝔽3)C2211yescharacteristic
13#88542—22no—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88438—24no—
16#88436—24no—
17#64083D311yescharacteristic, complemented
18C3×#875—31no—
19C3×#875—31no—
20C3×#875—61no—
21C3×#874—31no—
22C2×#3717—32no—
23C2×#3715—32no—
24C2×#3714—32no—
25C2×#3713—32no—
26#63630D311yescharacteristic, complemented
27C3×#803—31no—
28C22×#272—21no—
29C22×(C3⋊D3)⋊C3—22no—
30C2×C4×H(3,𝔽3)D412yes—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—21no—
33C4⋊#272—22no—
34#6229—22no—
35C4⋊((C3⋊D3)⋊C3)—22no—
36C4⋊((C3⋊D3)⋊C3)—22no—
37C4⋊((C3⋊D3)⋊C3)—44no—
38(C3⋊(C3⋊C8))⋊C3D414yescomplemented
39#5074—91no—
40#5058—91no—
41#4712—91no—
42C2×C4⋊(C3⋊D3)—22no—
43C2×C4⋊(C3⋊D3)C2×D312yes—
44C2×C3×D12—31no—
45C2×C3×D12—32no—
46C2×C3×D12—31no—
47C2×C3×D12—61no—
48C2×C3×D12—62no—
49C2×C3×Dic6—31no—
50C2×C3×Dic6—31no—
51C2×C3×Dic6—61no—
52C32×C42—21no—
53C32×C42C2×D311yescharacteristic
54C32×C42C2×D311yescharacteristic
55C2×C3⋊(C3⋊C8)C2×D312yes—
56C3×(C2×D3)⋊C4—64no—
57C3×(C2×D3)⋊C4—62no—
58C3×(C2×D3)⋊C4—62no—
59C2×C3×C3⋊C8—32no—
60#3717—64no—
61#3715—64no—
62#3714—64no—
63#3713—64no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C2×#272—41no—
66C2×#272—42no—
67C2×(C3⋊D3)⋊C3—42no—
68C2×(C3⋊D3)⋊C3—44no—
69C4×H(3,𝔽3)—22no—
70C4×H(3,𝔽3)C2×D412yes—
71#1084—41no—
72C3×#127—91no—
73C3×#124—91no—
74C2×#215—92no—
75C2×#214—92no—
76C2×#213—92no—
77C2×#213—92no—
78C2×D24—92no—
79C2×#204—92no—
80#875—31no—
81#875—31no—
82#875—61no—
83#874—31no—
84#874—91no—
85C3×#97—91no—
86#803—31no—
87#803—91no—
88C22×C3⋊D3—22no—
89C22×C3×D3—62no—
90C22×C3×D3—62no—
91C22×C3×D3—61no—
92C22×C3×D3—61no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4D1212yes—
96C2×C32×C4#6712yes—
97C2×C32×C4C22×D311yescharacteristic
98C2×C32×C4C22×D311yescharacteristic
99C4⋊(C3⋊D3)—22no—
100C4⋊(C3⋊D3)—22no—
101C4⋊(C3⋊D3)—44no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C3×D12—62no—
106C3×D12—62no—
107C3×D12—62no—
108C3×D12—62no—
109C3×D12—122no—
110C3×D12—124no—
111C3×Dic6—62no—
112C3×Dic6—62no—
113C3×Dic6—122no—
114C3⋊(C3⋊C8)D1214yescomplemented
115C3×C3⋊C8—34no—
116#496—271noSylow
117C2×H(3,𝔽3)#12711yescharacteristic
118C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
119#272—81no—
120(C3⋊D3)⋊C3—82no—
121C2×C3×Q8—91no—
122C2×C3×D4—92no—
123C2×C3×D4—91no—
124C2×C3×D4—182no—
125C2×D12—31no—
126C2×D12—31no—
127C2×D12—32no—
128C2×D12—61no—
129C2×D12—62no—
130C2×D12—92no—
131C2×D12—182no—
132C2×Dic6—31no—
133C2×Dic6—31no—
134C2×Dic6—61no—
135C2×C3×C8—92no—
136C3×C22⋊C4—182no—
137C3×C42—31no—
138C3×C42—31no—
139C3×C42—61no—
140C3×C42D3211yescharacteristic
141#215—184no—
142#214—184no—
143#213—184no—
144#213—184no—
145(C2×D3)⋊C4—62no—
146(C2×D3)⋊C4—62no—
147(C2×D3)⋊C4—64no—
148C2×C3⋊C8—32no—
149C2×C3⋊C8—92no—
150D24—184no—
151#204—184no—
152C22×C32—21no—
153C22×C32C2×D1211yescharacteristic
154C22×C32C2×#6711yescharacteristic
155C2×C3⋊D3—44no—
156C2×C3⋊D3—42no—
157C2×C3×D3—122no—
158C2×C3×D3—124no—
159C2×C3×D3—122no—
160C2×C3×D3—121no—
161C2×C3×D3—124no—
162C2×C3×D3—122no—
163C2×C3×D3—122no—
164C2×C3×D3—121no—
165C32×C4—22no—
166C32×C4—22no—
167C32×C4—22no—
168C32×C4—42no—
169C32×C4C2×D1212yes—
170C32×C4D3×D412yes—
171C3×Dic3—122no—
172C3×Dic3—121no—
173C3×Dic3—121no—
174C2×SD16—272no—
175C2×D8—272no—
176#127—91no—
177#124—91no—
178#97—91no—
179H(3,𝔽3)#49611yescharacteristic, complemented, Sylow
180C23×C3—182no—
181C23×C3—181no—
182C22×D3—61no—
183C22×D3—61no—
184C22×D3—62no—
185C22×D3—62no—
186C22×D3—182no—
187C3×Q8—182no—
188C3×D4—182no—
189C3×D4—182no—
190C3×D4—182no—
191C3×D4—364no—
192C2×C3×C4—32no—
193C2×C3×C4—32no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—62no—
197C2×C3×C4—61no—
198C2×C3×C4—181no—
199C2×C3×C4#62612yes—
200C2×C3×C4C2×D3211yescharacteristic
201C2×Dic3—61no—
202C2×Dic3—61no—
203C2×Dic3—62no—
204D12—62no—
205D12—62no—
206D12—62no—
207D12—62no—
208D12—122no—
209D12—124no—
210D12—182no—
211D12—182no—
212D12—364no—
213Dic6—62no—
214Dic6—62no—
215Dic6—122no—
216C3×C8—94no—
217C3⋊C8—34no—
218C3⋊C8—94no—
219C2×C32—22no—
220C2×C32—21no—
221C2×C32#87411yescharacteristic
222C2×C32#90812yes—
223C2×C32#93212yes—
224C2×C32#94011yescharacteristic
225C3⋊D3—82no—
226C3×D3—242no—
227C3×D3—242no—
228C3×D3—241no—
229C3×D3—241no—
230C2×Q8—91no—
231C2×D4—91no—
232C2×D4—92no—
233C2×D4—182no—
234SD16—544no—
235D8—544no—
236C2×C8—92no—
237C22⋊C4—182no—
238C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
239C22×C3—31no—
240C22×C3—31no—
241C22×C3—61no—
242C22×C3—362no—
243C22×C3—364no—
244C22×C3—362no—
245C22×C3—361no—
246C22×C3C2×#62611yescharacteristic
247C2×D3—121no—
248C2×D3—121no—
249C2×D3—122no—
250C2×D3—122no—
251C2×D3—122no—
252C2×D3—124no—
253C2×D3—124no—
254C2×D3—122no—
255C2×D3—362no—
256C2×D3—364no—
257C3×C4—22no—
258C3×C4—32no—
259C3×C4—32no—
260C3×C4—62no—
261C3×C4—62no—
262C3×C4—62no—
263C3×C4—122no—
264C3×C4—361no—
265C3×C4D3×D1212yes—
266Dic3—121no—
267Dic3—121no—
268Dic3—122no—
269C32—21no—
270C32#471211yescharacteristic, complemented
271C32#507411yescharacteristic, complemented
272C23—181no—
273C23—182no—
274Q8—182no—
275D4—182no—
276D4—182no—
277D4—182no—
278D4—364no—
279C2×C4—181no—
280C2×C4#620012yes—
281C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
282C8—94no—
283C2×C3—32no—
284C2×C3—32no—
285C2×C3—31no—
286C2×C3—31no—
287C2×C3—62no—
288C2×C3—61no—
289C2×C3—722no—
290C2×C3—721no—
291C2×C3#6383012yes—
292C2×C3#6391111yescharacteristic
293D3—241no—
294D3—241no—
295D3—242no—
296D3—242no—
297D3—722no—
298C22—364no—
299C22—362no—
300C22—361no—
301C22—362no—
302C22C2×#620011yescharacteristic
303C4—22no—
304C4—361no—
305C4#8865412yes—
306C3—31no—
307C3—31no—
308C3—61no—
309C3#1059037111yescharacteristic
310C2—721no—
311C2—722no—
312C2#1172107112yes—
313C2#1172115811yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#496
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. #11721721C2semidirect
2. #11721385C2semidirect
3. C2×#88438C2semidirect
4. C2×#88436C2semidirect
5. #63630⋊C3C2semidirect
17. #64083D3semidirect
26. #63630D3semidirect
38. (C3⋊(C3⋊C8))⋊C3D4semidirect
114. C3⋊(C3⋊C8)D12semidirect
179. H(3,𝔽3)#496semidirect
270. C32#4712semidirect
271. C32#5074semidirect

Maximal subgroups

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

subgroupindex
#117217212
#117213852
C2×#884382
C2×#884362
#63630⋊C32
#105903543
#105903513