Finite Group Databasebeta

← #13138974

Subgroups — #13138974

Order 1728.

531 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#88529C211yesmaximal, characteristic, complemented
2C2×#3743⋊C3C211yesmaximal, characteristic, complemented
3#11721156C212yesmaximal, complemented
4#63644⋊C3C211yesmaximal, characteristic, complemented
5#11720324C212yesmaximal, complemented
6#10590846—31nomaximal
7#10590836—31nomaximal
8C2×C4⋊#272C2212yes—
9C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
10C2×#6200—22no—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12#88529—22no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15#3743⋊C3—22no—
16#88451—22no—
17C2×#3743D311yescharacteristic, complemented
18C2×C3×D24—31no—
19C2×C3×D24—31no—
20C2×C3×D24—61no—
21C2×C3×ΓL(1,𝔽25)—31no—
22#63916—32no—
23#63908—32no—
24#63644D311yescharacteristic, complemented
25C3×#817—31no—
26#63563—32no—
27#63562—32no—
28C22×#272—22no—
29C22×(C3⋊D3)⋊C3D411yescharacteristic
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C4⋊#272—22no—
32C4⋊#272—22no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3D411yescharacteristic
35C4×(C3⋊D3)⋊C3—21no—
36C4×(C3⋊D3)⋊C3—21no—
37#6200—44no—
38C8×H(3,𝔽3)D412yescomplemented
39(C3⋊(C3⋊C8))⋊C3—21no—
40#5159—91no—
41#5135—91no—
42#4865—91no—
43C2×C4×C3⋊D3C2×D311yescharacteristic
44C2×C3×D12—32no—
45C2×C3×D12—32no—
46C2×C3×D12—62no—
47C2×C3×C4×D3—31no—
48C2×#626—62no—
49C2×#625—62no—
50C2×C32×C8—21no—
51C2×C32×C8C2×D311yescharacteristic
52C2×C32×C8C2×D311yescharacteristic
53C4⋊(C3⋊Dic3)C2×D312yes—
54C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
55#3743—22no—
56C3×C4⋊Dic3—32no—
57C2×C3×C3⋊C8—31no—
58C3×D24—64no—
59C3×D24—62no—
60C3×D24—62no—
61C3×ΓL(1,𝔽25)—62no—
62(C2×D3)⋊Dic3—62no—
63(C2×D3)⋊Dic3—62no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C2×#272—42no—
66C2×#272—42no—
67C2×#272—42no—
68C2×(C3⋊D3)⋊C3—21no—
69C2×(C3⋊D3)⋊C3—22no—
70C4×H(3,𝔽3)C2×D411yescharacteristic
71C4×H(3,𝔽3)#4111yescharacteristic
72(C3⋊Dic3)⋊C3—21no—
73(C3⋊Dic3)⋊C3—42no—
74C2×C3×D8—91no—
75C2×C3×M16—91no—
76#938—92no—
77#930—92no—
78C2×D24—31no—
79C2×D24—31no—
80C2×D24—61no—
81C2×ΓL(1,𝔽25)—31no—
82C2×ΓL(1,𝔽25)—91no—
83#895—92no—
84C3×#106—91no—
85#832—92no—
86#821—92no—
87#817—31no—
88#817—91no—
89#809—92no—
90C22×C3⋊D3D1211yescharacteristic
91C22×C3×D3—31no—
92C22×C3×D3—62no—
93C22×C3×D3—62no—
94C22×C3×D3—122no—
95C2×C32×C4—21no—
96C2×C32×C4C22×D311yescharacteristic
97C2×C32×C4C22×D311yescharacteristic
98C2×C3⋊Dic3—22no—
99C2×C3⋊Dic3D1211yescharacteristic
100C4×C3⋊D3—21no—
101C4×C3⋊D3—21no—
102C2×C3×Dic3—31no—
103C2×C3×Dic3—62no—
104C3×D12—64no—
105C3×D12—62no—
106C3×D12—62no—
107C3×D12—62no—
108C3×D12—62no—
109C3×D12—122no—
110C3×C4×D3—61no—
111C3×C4×D3—61no—
112#626—124no—
113#625—124no—
114C32×C8—22no—
115C32×C8D1212yescomplemented
116C32×C8#6712yescomplemented
117C3⋊(C3⋊C8)—21no—
118C3×C3⋊C8—61no—
119#469—271noSylow
120C2×H(3,𝔽3)#12111yescharacteristic
121C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
122#272—82no—
123(C3⋊D3)⋊C3—41no—
124C2×C3×D4—92no—
125C22×C3×C4—91no—
126C2×#67—182no—
127C2×#67—182no—
128C2×D12—32no—
129C2×D12—32no—
130C2×D12—62no—
131C2×D12—182no—
132C2×C4×D3—31no—
133C2×C4×D3—91no—
134C3×D8—182no—
135C3×M16—182no—
136C2×C3×C8—31no—
137C2×C3×C8—31no—
138C2×C3×C8—61no—
139C2×C3×C8—91no—
140C2×C3×C8D3211yescharacteristic
141C3×C4⋊C4—92no—
142#217—182no—
143(C2×D3)⋊C4—182no—
144(C2×D3)⋊C4—182no—
145C4⋊Dic3—32no—
146C4⋊Dic3—92no—
147C2×C3⋊C8—31no—
148C2×C3⋊C8—91no—
149D24—62no—
150D24—62no—
151D24—64no—
152ΓL(1,𝔽25)—62no—
153ΓL(1,𝔽25)—182no—
154C22×C32—21no—
155C22×C32C2×D1211yescharacteristic
156C22×C32D3×D411yescharacteristic
157C2×C3⋊D3—22no—
158C2×C3⋊D3—21no—
159C2×C3×D3—61no—
160C2×C3×D3—62no—
161C2×C3×D3—124no—
162C2×C3×D3—122no—
163C2×C3×D3—122no—
164C2×C3×D3—122no—
165C2×C3×D3—122no—
166C2×C3×D3—122no—
167C2×C3×D3—122no—
168C2×C3×D3—242no—
169C2×C3×D3—242no—
170C32×C4—21no—
171C32×C4—21no—
172C32×C4C2×D1211yescharacteristic
173C32×C4#23511yescharacteristic
174C32×C4#23911yescharacteristic
175C32×C4C2×#6711yescharacteristic
176C3⋊Dic3—21no—
177C3⋊Dic3—42no—
178C3×Dic3—61no—
179C3×Dic3—122no—
180C2×D8—91no—
181C2×M16—91no—
182#121—272no—
183#106—91no—
184#102—272no—
185H(3,𝔽3)#46911yescharacteristic, complemented, Sylow
186C23×C3—91no—
187C23×C3—182no—
188C22×D3—31no—
189C22×D3—62no—
190C22×D3—62no—
191C22×D3—91no—
192C22×D3—122no—
193C3×D4—182no—
194C3×D4—182no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—61no—
198C2×C3×C4—91no—
199C2×C3×C4—181no—
200C2×C3×C4—181no—
201C2×C3×C4—182no—
202C2×C3×C4C2×D3211yescharacteristic
203#67—364no—
204#67—364no—
205C2×Dic3—31no—
206C2×Dic3—62no—
207C2×Dic3—91no—
208C2×Dic3—182no—
209D12—62no—
210D12—62no—
211D12—62no—
212D12—62no—
213D12—64no—
214D12—122no—
215D12—364no—
216C4×D3—61no—
217C4×D3—61no—
218C4×D3—181no—
219C4×D3—181no—
220C3×C8—32no—
221C3×C8—32no—
222C3×C8—62no—
223C3×C8—181no—
224C3×C8#62612yes—
225C3⋊C8—61no—
226C3⋊C8—181no—
227C2×C32—22no—
228C2×C32—21no—
229C2×C32#89511yescharacteristic
230C2×C32#90812yes—
231C2×C32#91412yes—
232C2×C32#93011yescharacteristic
233C3⋊D3—41no—
234C3×D3—121no—
235C3×D3—244no—
236C3×D3—242no—
237C3×D3—242no—
238C2×D4—92no—
239C2×D4—542no—
240C22×C4—91no—
241D8—182no—
242M16—182no—
243C2×C8—91no—
244C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
245C4⋊C4—92no—
246C22⋊C4—542no—
247C22×C3—31no—
248C22×C3—31no—
249C22×C3—61no—
250C22×C3—182no—
251C22×C3—181no—
252C22×C3—362no—
253C22×C3—362no—
254C22×C3—362no—
255C22×C3D3×D1211yescharacteristic
256C2×D3—62no—
257C2×D3—61no—
258C2×D3—122no—
259C2×D3—122no—
260C2×D3—122no—
261C2×D3—122no—
262C2×D3—122no—
263C2×D3—122no—
264C2×D3—124no—
265C2×D3—181no—
266C2×D3—182no—
267C2×D3—242no—
268C2×D3—242no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—61no—
274C3×C4—61no—
275C3×C4—181no—
276C3×C4—362no—
277C3×C4#379911yescharacteristic
278C3×C4C2×#62611yescharacteristic
279Dic3—61no—
280Dic3—122no—
281Dic3—181no—
282Dic3—362no—
283C32—21no—
284C32#486511yescharacteristic, complemented
285C32#513511yescharacteristic, complemented
286C23—91no—
287C23—182no—
288D4—182no—
289D4—182no—
290D4—1084no—
291C2×C4—91no—
292C2×C4—181no—
293C2×C4—181no—
294C2×C4—182no—
295C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
296C8—181no—
297C8#620012yescomplemented
298C2×C3—32no—
299C2×C3—32no—
300C2×C3—31no—
301C2×C3—31no—
302C2×C3—62no—
303C2×C3—61no—
304C2×C3—361no—
305C2×C3—722no—
306C2×C3#6379412yes—
307C2×C3#6390911yescharacteristic
308D3—121no—
309D3—242no—
310D3—242no—
311D3—244no—
312D3—361no—
313C22—182no—
314C22—181no—
315C22—362no—
316C22—362no—
317C22—362no—
318C22#8865411yescharacteristic
319C4—181no—
320C4—362no—
321C4#8865211yescharacteristic
322C4C2×#620011yescharacteristic
323C3—31no—
324C3—31no—
325C3—61no—
326C3#1059083511yescharacteristic
327C2—361no—
328C2—722no—
329C2#1172084812yes—
330C2#1172115611yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#469
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#88529C2semidirect
2. C2×#3743⋊C3C2semidirect
3. #11721156C2semidirect
4. #63644⋊C3C2semidirect
5. #11720324C2semidirect
17. C2×#3743D3semidirect
24. #63644D3semidirect
38. C8×H(3,𝔽3)D4semidirect
115. C32×C8D12semidirect
116. C32×C8#67semidirect
185. H(3,𝔽3)#469semidirect
284. C32#4865semidirect
285. C32#5135semidirect
297. C8#6200semidirect

Maximal subgroups

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

subgroupindex
C2×#885292
C2×#3743⋊C32
#117211562
#63644⋊C32
#117203242
#105908463
#105908363