Finite Group Databasebeta

← #13156589

Subgroups — #13156589

Order 1728.

866 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 866 conjugacy classes of subgroups — too many to draw. Only its 116 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#11723697C211yesmaximal, characteristic, complemented
2C2×#88789C212yesmaximal, complemented
3C3×#63914C211yesmaximal, characteristic, complemented
4C2×C3×#3714C212yesmaximal, complemented
5C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
6#10591482—32nomaximal
7#10590886—31nomaximal
8C2×C3×C4⋊(C3⋊D3)C2212yes—
9C2×C32×D12C2212yes—
10C2×#6295—22no—
11C2×C3×#625—22no—
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2211yescharacteristic
15#88789—24no—
16C3×#3714—24no—
17C2×#3753—32no—
18C3×#940—32no—
19C2×C3×#213—32no—
20C2×C3×D24—32no—
21#63914—31no—
22#63914D311yescharacteristic, complemented
23#63911—32no—
24C2×#3714D312yescomplemented
25C2×#3713—32no—
26C2×#3713—32no—
27C2×#3713—62no—
28C8×C3⋊Dic3—31no—
29Dic3×C3⋊C8—32no—
30Dic3×C3⋊C8—61no—
31C22×C3×C3⋊D3—22no—
32C22×C32×D3—22no—
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3D412yes—
35C3×C4⋊(C3⋊D3)—22no—
36C3×C4⋊(C3⋊D3)—22no—
37C32×D12—22no—
38C32×D12—22no—
39#6295—44no—
40C3×#625—44no—
41C3⋊(C3⋊(C3⋊C8))D412yescomplemented
42C32×C3⋊C8D412yescomplemented
43#5151—92no—
44#5073—91no—
45C2×C32×D4—32no—
46C2×C4⋊(C3⋊D3)C2×D312yes—
47C2×C3×#67—64no—
48C2×C3×D12—22no—
49C2×C3×D12—32no—
50C2×C3×D12—32no—
51C2×C3×D12—62no—
52C2×C3×D12C2×D312yes—
53C2×C3×D12C2×D312yes—
54C2×#626—64no—
55C2×#625—22no—
56C2×#625—62no—
57C2×C32×C8—31no—
58#3753—64no—
59C4×C3⋊Dic3C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)—32no—
61C2×C3⋊(C3⋊C8)—31no—
62C2×C3⋊(C3⋊C8)—61no—
63C2×C3⋊(C3⋊C8)—62no—
64C2×C3⋊(C3⋊C8)—121no—
65C3×#213—64no—
66C3×C4×Dic3—32no—
67C3×C4×Dic3—61no—
68C2×C3×C3⋊C8—21no—
69C2×C3×C3⋊C8C2×D312yes—
70C3×D24—64no—
71#3714—24no—
72#3713—64no—
73#3713—64no—
74#3713—68no—
75C22×C33C2×D411yescharacteristic
76C2×C3×C3⋊D3—42no—
77C2×C3×C3⋊D3—44no—
78C2×C32×D3—42no—
79C2×C32×D3—44no—
80C33×C4C2×D411yescharacteristic
81C33×C4C2×D411yescharacteristic
82C3×C3⋊Dic3D814yescomplemented
83C2×C3×D8—92no—
84C3×#127—91no—
85#940—32no—
86#940—92no—
87C2×#213—32no—
88C2×#213—92no—
89C2×#213—92no—
90C2×#213—92no—
91C2×#213—182no—
92C2×D24—32no—
93#874—91no—
94C8×Dic3—92no—
95C8×Dic3—181no—
96C4×C3⋊C8—91no—
97C23×C32—62no—
98C22×C3⋊D3—22no—
99C22×C3×D3—22no—
100C22×C3×D3—22no—
101C22×C3×D3—42no—
102C22×C3×D3—62no—
103C22×C3×D3—62no—
104C22×C3×D3—122no—
105C32×D4—62no—
106C32×D4—62no—
107C2×C32×C4—21no—
108C2×C32×C4—22no—
109C2×C32×C4—41no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C32×C4C22×D312yes—
112C2×C3⋊Dic3#6712yes—
113C4⋊(C3⋊D3)—22no—
114C4⋊(C3⋊D3)—22no—
115C3×#67—128no—
116C2×C3×Dic3—34no—
117C2×C3×Dic3—62no—
118C3×D12—24no—
119C3×D12—22no—
120C3×D12—22no—
121C3×D12—22no—
122C3×D12—22no—
123C3×D12—42no—
124C3×D12—64no—
125C3×D12—62no—
126C3×D12—62no—
127C3×D12—62no—
128C3×D12—62no—
129C3×D12—122no—
130#626—128no—
131#625—44no—
132#625—124no—
133C32×C8—32no—
134C3⋊(C3⋊C8)—32no—
135C3⋊(C3⋊C8)—34no—
136C3⋊(C3⋊C8)—64no—
137C3⋊(C3⋊C8)—62no—
138C3⋊(C3⋊C8)—122no—
139C3×C3⋊C8—22no—
140C3×C3⋊C8#6714yescomplemented
141#493—271noSylow
142C2×C33#12711yescharacteristic
143C2×C33C2×D812yes—
144C3×C3⋊D3—82no—
145C32×D3—82no—
146C2×C3×D4—32no—
147C2×C3×D4—32no—
148C2×C3×D4—62no—
149C2×C3×D4—92no—
150C2×C3×D4—182no—
151C2×#67—64no—
152C2×#67—184no—
153C2×D12—32no—
154C2×D12—32no—
155C2×D12—62no—
156C2×D12—182no—
157C2×D12D3212yes—
158C3×D8—184no—
159C2×C3×C8—32no—
160C2×C3×C8—61no—
161C3×C42—91no—
162#213—64no—
163#213—184no—
164#213—184no—
165#213—184no—
166#213—188no—
167C4×Dic3—32no—
168C4×Dic3—61no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—92no—
171C2×C3⋊C8—181no—
172C2×C3⋊C8—182no—
173C2×C3⋊C8—361no—
174C2×C3⋊C8D3211yescharacteristic
175D24—64no—
176C22×C32—21no—
177C22×C32—22no—
178C22×C32—41no—
179C22×C32—122no—
180C22×C32—124no—
181C22×C32D3×D412yes—
182C22×C32C2×#6711yescharacteristic
183C2×C3⋊D3—42no—
184C2×C3⋊D3—44no—
185C2×C3×D3—44no—
186C2×C3×D3—44no—
187C2×C3×D3—44no—
188C2×C3×D3—42no—
189C2×C3×D3—42no—
190C2×C3×D3—84no—
191C2×C3×D3—124no—
192C2×C3×D3—124no—
193C2×C3×D3—124no—
194C2×C3×D3—122no—
195C2×C3×D3—122no—
196C2×C3×D3—244no—
197C32×C4—21no—
198C32×C4—21no—
199C32×C4—22no—
200C32×C4—22no—
201C32×C4—41no—
202C32×C4—41no—
203C32×C4D3×D412yes—
204C32×C4D3×D411yescharacteristic
205C32×C4D3×D411yescharacteristic
206C32×C4C2×#6712yes—
207C3⋊Dic3#21314yescomplemented
208C3×Dic3—38no—
209C3×Dic3—64no—
210C2×D8—92no—
211C2×D8—272no—
212#127—91no—
213#127—271no—
214C4×C8—271no—
215C33#49311yescharacteristic, complemented, Sylow
216C23×C3—62no—
217C23×C3—62no—
218C23×C3—122no—
219C23×C3—182no—
220C22×D3—22no—
221C22×D3—62no—
222C22×D3—62no—
223C22×D3—122no—
224C3×D4—64no—
225C3×D4—62no—
226C3×D4—62no—
227C3×D4—62no—
228C3×D4—62no—
229C3×D4—122no—
230C3×D4—182no—
231C3×D4—182no—
232C3×D4—364no—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4—92no—
237C2×C3×C4C2×D3211yescharacteristic
238C2×C3×C4C2×D3212yes—
239#67—128no—
240#67—368no—
241C2×Dic3—34no—
242C2×Dic3—62no—
243D12—22no—
244D12—22no—
245D12—64no—
246D12—62no—
247D12—62no—
248D12—62no—
249D12—62no—
250D12—122no—
251D12—364no—
252C3×C8—34no—
253C3×C8—62no—
254C3⋊C8—92no—
255C3⋊C8—94no—
256C3⋊C8—182no—
257C3⋊C8—184no—
258C3⋊C8—362no—
259C3⋊C8#62512yescomplemented
260C2×C32—22no—
261C2×C32—21no—
262C2×C32—24no—
263C2×C32—22no—
264C2×C32—41no—
265C2×C32—42no—
266C2×C32—242no—
267C2×C32D3×D814yes—
268C2×C32C2×#21312yes—
269C2×C32#94012yes—
270C2×C32#94011yescharacteristic
271C3⋊D3—82no—
272C3×D3—84no—
273C3×D3—82no—
274C3×D3—82no—
275C3×D3—244no—
276C3×D3—242no—
277C3×D3—242no—
278C2×D4—32no—
279C2×D4—92no—
280C2×D4—182no—
281C2×D4—542no—
282D8—184no—
283D8—544no—
284C2×C8—31no—
285C2×C8—271no—
286C42—91no—
287C22×C3—21no—
288C22×C3—22no—
289C22×C3—41no—
290C22×C3—124no—
291C22×C3—124no—
292C22×C3—122no—
293C22×C3—124no—
294C22×C3—122no—
295C22×C3—244no—
296C22×C3—364no—
297C22×C3—362no—
298C22×C3#380211yescharacteristic
299C22×C3D3×#6712yes—
300C2×D3—44no—
301C2×D3—42no—
302C2×D3—124no—
303C2×D3—124no—
304C2×D3—122no—
305C2×D3—124no—
306C2×D3—122no—
307C2×D3—244no—
308C3×C4—21no—
309C3×C4—22no—
310C3×C4—22no—
311C3×C4—21no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4—94no—
315C3×C4#380211yescharacteristic
316C3×C4C2×#62511yescharacteristic
317C3×C4D3×#6712yes—
318C3×C4#381112yes—
319Dic3—38no—
320Dic3—64no—
321C32—21no—
322C32—22no—
323C32—41no—
324C32#507311yescharacteristic, complemented
325C32#515112yescomplemented
326C23—62no—
327C23—182no—
328D4—62no—
329D4—62no—
330D4—182no—
331D4—182no—
332D4—364no—
333D4—1084no—
334C2×C4—92no—
335C2×C4D3311yescharacteristic
336C8—32no—
337C8—272no—
338C2×C3—22no—
339C2×C3—21no—
340C2×C3—22no—
341C2×C3—24no—
342C2×C3—41no—
343C2×C3—42no—
344C2×C3—244no—
345C2×C3—242no—
346C2×C3—242no—
347C2×C3—722no—
348C2×C3#6379712yes—
349C2×C3#6391411yescharacteristic
350C2×C3D3×#21314yes—
351C2×C3#6397912yes—
352D3—82no—
353D3—244no—
354D3—242no—
355D3—242no—
356C22—122no—
357C22—124no—
358C22—362no—
359C22—364no—
360C22#8895311yescharacteristic
361C4—94no—
362C4D3×#62511yescharacteristic
363C4#8895411yescharacteristic
364C3—22no—
365C3—21no—
366C3—41no—
367C3#1059088611yescharacteristic, complemented
368C3#1059148212yescomplemented
369C2—242no—
370C2—722no—
371C2#1172283112yes—
372C2#1172302911yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#493
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723697C2semidirect
2. C2×#88789C2semidirect
3. C3×#63914C2semidirect
4. C2×C3×#3714C2semidirect
5. C3⋊C8×C3⋊Dic3C2semidirect
22. #63914D3semidirect
24. C2×#3714D3semidirect
41. C3⋊(C3⋊(C3⋊C8))D4semidirect
42. C32×C3⋊C8D4semidirect
82. C3×C3⋊Dic3D8semidirect
140. C3×C3⋊C8#67semidirect
207. C3⋊Dic3#213semidirect
215. C33#493semidirect
259. C3⋊C8#625semidirect
324. C32#5073semidirect
325. C32#5151semidirect
367. C3#10590886semidirect
368. C3#10591482semidirect

Maximal subgroups

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

subgroupindex
#117236972
C2×#887892
C3×#639142
C2×C3×#37142
C3⋊C8×C3⋊Dic32
#105914823
#105908863