Finite Group Databasebeta

← #13156565

Subgroups — #13156565

Order 1728.

702 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 98 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#11723680C211yesmaximal, characteristic, complemented
2#11723524C211yesmaximal, characteristic
3#11722036C212yesmaximal, complemented
4C3×#63563C212yesmaximal, complemented
5C3×#63555C211yesmaximal, characteristic, complemented
6#10591480—31nomaximal
7#10590882—32nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×C32×D12C2211yescharacteristic
10C2×#6291—22no—
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14#88815—22no—
15#64149—31no—
16#63911—32no—
17#63911—61no—
18#63872—31no—
19#63869—32no—
20#63648—32no—
21C3×#832—32no—
22C3×#821—32no—
23C3×#815—32no—
24#63564—32no—
25#63564—32no—
26#63564—62no—
27#63563D312yescomplemented
28#63555D311yescharacteristic, complemented
29C22×C3⋊(C3⋊D3)—21no—
30C22×C32×D3—21no—
31C2×C33×C4C2311yescharacteristic
32C4⋊(C3⋊(C3⋊D3))—22no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3D412yes—
35C32×D12—22no—
36#6291—44no—
37C32×C3⋊C8—22no—
38#4764—92no—
39#4713—91no—
40C2×C32×D4—31no—
41C2×#638—62no—
42C2×C4⋊(C3⋊D3)—31no—
43C2×C4⋊(C3⋊D3)—32no—
44C2×C4⋊(C3⋊D3)—62no—
45C2×C4⋊(C3⋊D3)—61no—
46C2×C4⋊(C3⋊D3)—121no—
47C2×C3×D12—21no—
48C2×C3×D12C2×D312yes—
49C2×#626—62no—
50C2×#626—62no—
51C2×#626—64no—
52C2×C32×C8—32no—
53C4⋊(C3⋊Dic3)C2×D312yes—
54C4×C3⋊Dic3C2×D311yescharacteristic
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—62no—
58C3×C4×Dic3—32no—
59C3×C4×Dic3—61no—
60C2×C3×C3⋊C8—22no—
61C2×C3×C3⋊C8C2×D312yes—
62C2×C3×C3⋊C8C2×D312yes—
63#3724—62no—
64Dic3⋊Dic3—62no—
65Dic3⋊Dic3—62no—
66C22×C33C2×D411yescharacteristic
67C2×C3⋊(C3⋊D3)—41no—
68C2×C3⋊(C3⋊D3)—42no—
69C2×C32×D3—41no—
70C2×C32×D3—42no—
71C33×C4#4112yes—
72C3×C3⋊Dic3—22no—
73C3×C3⋊Dic3—42no—
74#940—92no—
75#940—181no—
76#890—92no—
77#874—91no—
78#870—91no—
79C3×#102—92no—
80C3×#97—91no—
81#832—32no—
82#821—32no—
83#821—92no—
84#821—92no—
85#821—182no—
86#815—32no—
87#809—92no—
88C23×C32—61no—
89C22×C3⋊D3—61no—
90C22×C3⋊D3—62no—
91C22×C3⋊D3—62no—
92C22×C3⋊D3—122no—
93C22×C3⋊D3—122no—
94C22×C3×D3—22no—
95C22×C3×D3—22no—
96C32×D4—62no—
97C2×C32×C4—22no—
98C2×C32×C4—21no—
99C2×C32×C4—41no—
100C2×C32×C4C22×D311yescharacteristic
101C2×C32×C4C22×D312yes—
102#638—124no—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3D1212yes—
105C4⋊(C3⋊D3)—62no—
106C4⋊(C3⋊D3)—64no—
107C4⋊(C3⋊D3)—62no—
108C4⋊(C3⋊D3)—62no—
109C4⋊(C3⋊D3)—122no—
110C4⋊(C3⋊D3)—122no—
111C4⋊(C3⋊D3)—124no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—122no—
118C3×D12—22no—
119C3×D12—22no—
120C3×D12—42no—
121#626—128no—
122#626—124no—
123#626—124no—
124C32×C8—62no—
125C3×C3⋊C8—22no—
126C3×C3⋊C8—22no—
127C3×C3⋊C8—42no—
128#490—271noSylow
129C2×C33#12411yescharacteristic
130C2×C33AGL(1,ℤ/8)12yes—
131C3⋊(C3⋊D3)—81no—
132C32×D3—81no—
133C2×C3×D4—32no—
134C2×C3×D4—61no—
135C2×#67—182no—
136C2×#67—182no—
137C2×#67—184no—
138C2×D12—92no—
139C2×D12—91no—
140C2×D12—182no—
141C2×D12—181no—
142C2×D12—182no—
143C2×D12—361no—
144C2×D12D3211yescharacteristic
145C2×C3×C8—32no—
146C2×C3×C8—32no—
147C2×C3×C8—62no—
148C3×C4⋊C4—92no—
149C3×C42—91no—
150C4⋊Dic3—32no—
151C4⋊Dic3—32no—
152C4⋊Dic3—62no—
153C4⋊Dic3—182no—
154Dic3⋊C4—182no—
155Dic3⋊C4—182no—
156C4×Dic3—32no—
157C4×Dic3—61no—
158C2×C3⋊C8D3212yes—
159C22×C32—22no—
160C22×C32—21no—
161C22×C32—41no—
162C22×C32—121no—
163C22×C32—122no—
164C22×C32C2×D1211yescharacteristic
165C22×C32D3×D412yes—
166C2×C3⋊D3—124no—
167C2×C3⋊D3—122no—
168C2×C3⋊D3—124no—
169C2×C3⋊D3—121no—
170C2×C3⋊D3—122no—
171C2×C3⋊D3—122no—
172C2×C3⋊D3—124no—
173C2×C3⋊D3—122no—
174C2×C3⋊D3—122no—
175C2×C3⋊D3—242no—
176C2×C3⋊D3—242no—
177C2×C3⋊D3—244no—
178C2×C3×D3—44no—
179C2×C3×D3—42no—
180C2×C3×D3—42no—
181C2×C3×D3—42no—
182C2×C3×D3—42no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—42no—
187C32×C4#23512yes—
188C32×C4#23512yes—
189C32×C4#23712yes—
190C3⋊Dic3—22no—
191C3⋊Dic3—42no—
192C3×Dic3—64no—
193C3×Dic3—62no—
194C3×Dic3—62no—
195C3×Dic3—122no—
196C3×Dic3—122no—
197C3×Dic3—242no—
198#127—271no—
199#125—271no—
200#102—92no—
201#102—272no—
202#97—91no—
203C33#49011yescharacteristic, complemented, Sylow
204C23×C3—62no—
205C23×C3—62no—
206C22×D3—21no—
207C22×D3—182no—
208C22×D3—182no—
209C22×D3—181no—
210C22×D3—362no—
211C22×D3—362no—
212C3×D4—62no—
213C3×D4—62no—
214C3×D4—122no—
215C2×C3×C4—21no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4—92no—
219C2×C3×C4—182no—
220C2×C3×C4C2×D3212yes—
221C2×C3×C4C2×D3211yescharacteristic
222#67—368no—
223#67—364no—
224#67—364no—
225C2×Dic3—32no—
226C2×Dic3—32no—
227C2×Dic3—62no—
228C2×Dic3—62no—
229C2×Dic3—62no—
230C2×Dic3—122no—
231D12—22no—
232D12—184no—
233D12—182no—
234D12—182no—
235D12—182no—
236D12—362no—
237D12—362no—
238D12—364no—
239D12—364no—
240C3×C8—62no—
241C3×C8—62no—
242C3×C8—122no—
243C3⋊C8—22no—
244C2×C32—22no—
245C2×C32—22no—
246C2×C32—22no—
247C2×C32—22no—
248C2×C32—21no—
249C2×C32—41no—
250C2×C32—42no—
251C2×C32—241no—
252C2×C32#87511yescharacteristic
253C2×C32#88512yes—
254C2×C32#90812yes—
255C2×C32#91112yes—
256C2×C32#91412yes—
257C3⋊D3—242no—
258C3⋊D3—244no—
259C3⋊D3—244no—
260C3⋊D3—241no—
261C3⋊D3—242no—
262C3×D3—82no—
263C3×D3—82no—
264C2×D4—31no—
265C2×D4—271no—
266C2×D4—542no—
267C2×C8—32no—
268C4⋊C4—92no—
269C4⋊C4—542no—
270C42—91no—
271C22×C3—21no—
272C22×C3—22no—
273C22×C3—41no—
274C22×C3—124no—
275C22×C3—122no—
276C22×C3—122no—
277C22×C3—122no—
278C22×C3—122no—
279C22×C3D3×D1212yes—
280C22×C3#381111yescharacteristic
281C2×D3—42no—
282C2×D3—41no—
283C2×D3—364no—
284C2×D3—362no—
285C2×D3—364no—
286C2×D3—364no—
287C2×D3—362no—
288C2×D3—362no—
289C2×D3—362no—
290C2×D3—362no—
291C2×D3—361no—
292C2×D3—722no—
293C2×D3—722no—
294C2×D3—724no—
295C3×C4—22no—
296C3×C4—22no—
297C3×C4—22no—
298C3×C4—42no—
299C3×C4—182no—
300C3×C4—362no—
301C3×C4#379812yes—
302C3×C4#380412yes—
303C3×C4#380412yes—
304Dic3—64no—
305Dic3—62no—
306Dic3—62no—
307Dic3—122no—
308Dic3—122no—
309Dic3—242no—
310C32—22no—
311C32—21no—
312C32—41no—
313C32#471311yescharacteristic, complemented
314C32#476412yescomplemented
315C23—61no—
316C23—541no—
317D4—62no—
318D4—542no—
319D4—1084no—
320C2×C4—92no—
321C2×C4—182no—
322C2×C4D3311yescharacteristic
323C8—62no—
324C2×C3—21no—
325C2×C3—22no—
326C2×C3—22no—
327C2×C3—22no—
328C2×C3—22no—
329C2×C3—41no—
330C2×C3—42no—
331C2×C3—242no—
332C2×C3—242no—
333C2×C3#6379412yes—
334C2×C3#6379512yes—
335C2×C3#6385212yes—
336C2×C3#6393712yes—
337C2×C3#6395311yescharacteristic
338D3—81no—
339D3—722no—
340D3—724no—
341D3—724no—
342D3—722no—
343D3—721no—
344C22—121no—
345C22—122no—
346C22—1081no—
347C22—1082no—
348C22#8895511yescharacteristic
349C4—182no—
350C4—362no—
351C4#8895912yes—
352C3—21no—
353C3—22no—
354C3—41no—
355C3#1059088212yescomplemented
356C3#1059148011yescharacteristic, complemented
357C2—241no—
358C2—2161no—
359C2#1172287912yes—
360C2#1172300811yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723524C2
hypocenterC33#490
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. #11723680C2semidirect
3. #11722036C2semidirect
4. C3×#63563C2semidirect
5. C3×#63555C2semidirect
27. #63563D3semidirect
28. #63555D3semidirect
203. C33#490semidirect
313. C32#4713semidirect
314. C32#4764semidirect
355. C3#10590882semidirect
356. C3#10591480semidirect

Maximal subgroups

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

subgroupindex
#117236802
#117235242
#117220362
C3×#635632
C3×#635552
#105914803
#105908823