Finite Group Databasebeta

← #13138940

Subgroups — #13138940

Order 1728.

523 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 77 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×#88527C211yesmaximal, characteristic, complemented
2C2×#3745⋊C3C211yesmaximal, characteristic, complemented
3#11721159C212yesmaximal, complemented
4#63644⋊C3C211yesmaximal, characteristic, complemented
5#11720318C212yesmaximal, complemented
6#10590836—31nomaximal
7#10590835—31nomaximal
8C2×C4×#272C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×#6200—22no—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13#88527—22no—
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15#3745⋊C3—22no—
16#88450—22no—
17C2×#3745D311yescharacteristic, complemented
18C2×C3×D24—31no—
19C2×C3×ΓL(1,𝔽25)—31no—
20C2×C3×ΓL(1,𝔽25)—31no—
21C2×C3×ΓL(1,𝔽25)—61no—
22#63909—32no—
23#63908—32no—
24#63644D311yescharacteristic, complemented
25C3×#817—31no—
26#63564—32no—
27#63563—32no—
28C22×#272D411yescharacteristic
29C22×(C3⋊D3)⋊C3—22no—
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×#1084D411yescharacteristic
32C4×#272—21no—
33C4×#272—21no—
34C2×(C3⋊Dic3)⋊C3—22no—
35C4⋊((C3⋊D3)⋊C3)—22no—
36C4⋊((C3⋊D3)⋊C3)—22no—
37#6200—44no—
38C8×H(3,𝔽3)D412yescomplemented
39#6180—21no—
40#5159—91no—
41#5135—91no—
42#4865—91no—
43C2×C4⋊(C3⋊D3)C2×D312yes—
44C2×C3×D12—32no—
45C2×C3×C4×D3—31no—
46C2×C3×C4×D3—31no—
47C2×C3×C4×D3—61no—
48C2×#626—62no—
49C2×#625—62no—
50C2×C32×C8—21no—
51C2×C32×C8C2×D311yescharacteristic
52C2×C32×C8C2×D311yescharacteristic
53C4⋊(C3⋊Dic3)C2×D312yes—
54#3745—22no—
55C3×C4⋊Dic3—32no—
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8—31no—
58C2×C3×C3⋊C8—61no—
59C3×D24—62no—
60C3×ΓL(1,𝔽25)—64no—
61C3×ΓL(1,𝔽25)—62no—
62C3×ΓL(1,𝔽25)—62no—
63#3722—62no—
64(C2×D3)⋊Dic3—62no—
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C2×#272—21no—
67C2×#272—22no—
68C2×(C3⋊D3)⋊C3—42no—
69C2×(C3⋊D3)⋊C3—42no—
70C2×(C3⋊D3)⋊C3—42no—
71C4×H(3,𝔽3)C2×D411yescharacteristic
72C4×H(3,𝔽3)#4111yescharacteristic
73#1084—21no—
74(C3⋊Dic3)⋊C3—42no—
75C2×C3×D8—91no—
76C2×C3×M16—91no—
77#938—92no—
78#930—92no—
79C2×D24—31no—
80C2×D24—91no—
81C2×ΓL(1,𝔽25)—31no—
82C2×ΓL(1,𝔽25)—31no—
83C2×ΓL(1,𝔽25)—61no—
84#895—92no—
85C3×#106—91no—
86#832—92no—
87#821—92no—
88#817—31no—
89#817—91no—
90#809—92no—
91C22×C3⋊D3—22no—
92C22×C3×D3—31no—
93C22×C3×D3—31no—
94C22×C3×D3—61no—
95C22×C3×D3—62no—
96C2×C32×C4—21no—
97C2×C32×C4C22×D311yescharacteristic
98C2×C32×C4C22×D311yescharacteristic
99C2×C3⋊Dic3—22no—
100C4⋊(C3⋊D3)—22no—
101C4⋊(C3⋊D3)—22no—
102C2×C3×Dic3—31no—
103C2×C3×Dic3—31no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—62no—
106C3×D12—62no—
107C3×D12—62no—
108C3×C4×D3—62no—
109C3×C4×D3—62no—
110C3×C4×D3—61no—
111C3×C4×D3—61no—
112C3×C4×D3—61no—
113C3×C4×D3—61no—
114#626—124no—
115#625—124no—
116C32×C8—22no—
117C32×C8D1212yescomplemented
118C32×C8#6712yescomplemented
119C3×C3⋊C8—62no—
120C3×C3⋊C8—61no—
121C3×C3⋊C8—61no—
122#469—271noSylow
123C2×H(3,𝔽3)#12111yescharacteristic
124C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
125#272—41no—
126(C3⋊D3)⋊C3—82no—
127C2×C3×D4—92no—
128C22×C3×C4—91no—
129C2×#67—182no—
130C2×#67—182no—
131C2×D12—32no—
132C2×D12—92no—
133C2×D12—182no—
134C2×C4×D3—31no—
135C2×C4×D3—31no—
136C2×C4×D3—61no—
137C3×D8—182no—
138C3×M16—182no—
139C2×C3×C8—31no—
140C2×C3×C8—31no—
141C2×C3×C8—61no—
142C2×C3×C8—91no—
143C2×C3×C8D3211yescharacteristic
144C3×C4⋊C4—92no—
145#217—182no—
146(C2×D3)⋊C4—182no—
147(C2×D3)⋊C4—182no—
148C4⋊Dic3—32no—
149C4⋊Dic3—92no—
150C2×C3⋊C8—31no—
151C2×C3⋊C8—31no—
152C2×C3⋊C8—61no—
153D24—62no—
154D24—182no—
155ΓL(1,𝔽25)—62no—
156ΓL(1,𝔽25)—62no—
157ΓL(1,𝔽25)—64no—
158C22×C32—21no—
159C22×C32D3×D411yescharacteristic
160C22×C32D3×D411yescharacteristic
161C2×C3⋊D3—42no—
162C2×C3⋊D3—42no—
163C2×C3⋊D3—42no—
164C2×C3×D3—62no—
165C2×C3×D3—61no—
166C2×C3×D3—62no—
167C2×C3×D3—62no—
168C2×C3×D3—61no—
169C2×C3×D3—122no—
170C2×C3×D3—122no—
171C2×C3×D3—122no—
172C2×C3×D3—122no—
173C32×C4—21no—
174C32×C4—21no—
175C32×C4C2×D1211yescharacteristic
176C32×C4#23711yescharacteristic
177C32×C4#23911yescharacteristic
178C32×C4C2×#6711yescharacteristic
179C3⋊Dic3—42no—
180C3×Dic3—62no—
181C3×Dic3—61no—
182C3×Dic3—61no—
183C3×Dic3—122no—
184C2×D8—91no—
185C2×M16—91no—
186#121—272no—
187#106—91no—
188#102—272no—
189H(3,𝔽3)#46911yescharacteristic, complemented, Sylow
190C23×C3—91no—
191C23×C3—182no—
192C22×D3—31no—
193C22×D3—31no—
194C22×D3—61no—
195C22×D3—62no—
196C22×D3—182no—
197C3×D4—182no—
198C3×D4—182no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—61no—
202C2×C3×C4—91no—
203C2×C3×C4—181no—
204C2×C3×C4—181no—
205C2×C3×C4—182no—
206C2×C3×C4C2×D3211yescharacteristic
207#67—364no—
208#67—364no—
209C2×Dic3—31no—
210C2×Dic3—31no—
211C2×Dic3—61no—
212C2×Dic3—62no—
213C2×Dic3—182no—
214D12—62no—
215D12—62no—
216D12—182no—
217D12—182no—
218D12—364no—
219C4×D3—61no—
220C4×D3—61no—
221C4×D3—61no—
222C4×D3—61no—
223C4×D3—62no—
224C4×D3—62no—
225C3×C8—32no—
226C3×C8—32no—
227C3×C8—62no—
228C3×C8—181no—
229C3×C8#62612yes—
230C3⋊C8—61no—
231C3⋊C8—61no—
232C3⋊C8—62no—
233C2×C32—22no—
234C2×C32—21no—
235C2×C32#89511yescharacteristic
236C2×C32#91112yes—
237C2×C32#91412yes—
238C2×C32#93811yescharacteristic
239C3⋊D3—82no—
240C3×D3—122no—
241C3×D3—121no—
242C3×D3—121no—
243C3×D3—242no—
244C2×D4—92no—
245C2×D4—542no—
246C22×C4—91no—
247D8—182no—
248M16—182no—
249C2×C8—91no—
250C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
251C4⋊C4—92no—
252C22⋊C4—542no—
253C22×C3—31no—
254C22×C3—31no—
255C22×C3—61no—
256C22×C3—182no—
257C22×C3—181no—
258C22×C3—362no—
259C22×C3—362no—
260C22×C3—362no—
261C22×C3#380211yescharacteristic
262C2×D3—61no—
263C2×D3—61no—
264C2×D3—62no—
265C2×D3—62no—
266C2×D3—62no—
267C2×D3—122no—
268C2×D3—122no—
269C2×D3—122no—
270C2×D3—122no—
271C2×D3—362no—
272C2×D3—362no—
273C2×D3—362no—
274C3×C4—31no—
275C3×C4—31no—
276C3×C4—31no—
277C3×C4—31no—
278C3×C4—61no—
279C3×C4—61no—
280C3×C4—181no—
281C3×C4—362no—
282C3×C4#379611yescharacteristic
283C3×C4C2×#62611yescharacteristic
284Dic3—61no—
285Dic3—61no—
286Dic3—62no—
287Dic3—122no—
288Dic3—362no—
289C32—21no—
290C32#486511yescharacteristic, complemented
291C32#515911yescharacteristic, complemented
292C23—91no—
293C23—182no—
294D4—182no—
295D4—182no—
296D4—1084no—
297C2×C4—91no—
298C2×C4—181no—
299C2×C4—181no—
300C2×C4—182no—
301C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
302C8—181no—
303C8#620012yescomplemented
304C2×C3—32no—
305C2×C3—32no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—62no—
309C2×C3—61no—
310C2×C3—361no—
311C2×C3—722no—
312C2×C3#6379812yes—
313C2×C3#6391611yescharacteristic
314D3—121no—
315D3—121no—
316D3—122no—
317D3—242no—
318D3—722no—
319C22—182no—
320C22—181no—
321C22—362no—
322C22—362no—
323C22—362no—
324C22#8865511yescharacteristic
325C4—181no—
326C4—362no—
327C4#8864911yescharacteristic
328C4C2×#620011yescharacteristic
329C3—31no—
330C3—31no—
331C3—61no—
332C3#1059084611yescharacteristic
333C2—361no—
334C2—722no—
335C2#1172083912yes—
336C2#1172115911yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
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×#88527C2semidirect
2. C2×#3745⋊C3C2semidirect
3. #11721159C2semidirect
4. #63644⋊C3C2semidirect
5. #11720318C2semidirect
17. C2×#3745D3semidirect
24. #63644D3semidirect
38. C8×H(3,𝔽3)D4semidirect
117. C32×C8D12semidirect
118. C32×C8#67semidirect
189. H(3,𝔽3)#469semidirect
290. C32#4865semidirect
291. C32#5159semidirect
303. C8#6200semidirect

Maximal subgroups

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

subgroupindex
C2×#885272
C2×#3745⋊C32
#117211592
#63644⋊C32
#117203182
#105908363
#105908353