Finite Group Databasebeta

← #13159223

Subgroups — #13159223

Order 1728.

944 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 148 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
1C3×#64102C211yesmaximal, characteristic, complemented
2#11723440C211yesmaximal, characteristic, complemented
3C2×#88793C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic
5#11722179C211yesmaximal, characteristic, complemented
6C32×#810C211yesmaximal, characteristic, complemented
7#11722051C211yesmaximal, characteristic, complemented
8#10592518—31nomaximal
9#10590534—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3—21no—
13C2×C32×Dic6C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3752—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20#88804—21no—
21#88793—22no—
22#88791—22no—
23#64156—31no—
24C2×#3755—31no—
25C2×#3745—31no—
26#64102D311yescharacteristic, complemented
27C3×#893—34no—
28#63864—34no—
29C2×#3715—34no—
30C2×#3714—34no—
31C32×#103—31no—
32#63641—31no—
33#63636—31no—
34#63636—34no—
35#63636—64no—
36C3×#810D314yescomplemented
37Dic3⋊(C3⋊C8)—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C2×C3×C3⋊Dic3D411yescharacteristic
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)D412yes—
48C3×C4×C3⋊D3—42no—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6—21no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C32×C3⋊C8—21no—
54#5016—91no—
55#4819—94no—
56C2×C32×Q8—31no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C4⋊(C3⋊D3)—34no—
59C2×C4⋊(C3⋊D3)—64no—
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C4×C3⋊D3—21no—
62C2×C3×D12—34no—
63C2×C3×C4×D3—64no—
64C2×C3×Dic6C2×D314yes—
65C2×C32×C8—31no—
66C32×C4⋊C4—24no—
67C32×C4⋊C4C2×D314yes—
68C32×C4⋊C4C2×D311yescharacteristic
69#3755—62no—
70#3752—21no—
71#3752—61no—
72#3752—61no—
73C4×C3⋊Dic3C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—64no—
77#3745—62no—
78C3×(C2×D3)⋊C4—64no—
79C3×C4×Dic3—34no—
80C2×C3×C3⋊C8C2×D314yes—
81#3722—64no—
82#3722—64no—
83#3715—68no—
84#3714—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3⋊(C3⋊D3)—41no—
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C33×C4—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3×C3⋊Dic3—21no—
96C3×C3⋊Dic3—21no—
97C32×Dic3—41no—
98C3×C4×D4—91no—
99#947—94no—
100C2×#215—94no—
101C2×#214—91no—
102C2×#213—91no—
103C2×D24—94no—
104#893—34no—
105#875—91no—
106C3×#103—34no—
107Dic3⋊C8—94no—
108#810C3⋊D311yescharacteristic, complemented
109#809—94no—
110#809—91no—
111#809—184no—
112C4⋊(C3⋊C8)—91no—
113C22×C3⋊D3—21no—
114C22×C3⋊D3—61no—
115C22×C3⋊D3—64no—
116C22×C3⋊D3—124no—
117C22×C3×D3—64no—
118C32×Q8—61no—
119C32×Q8—61no—
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—44no—
124C2×C32×C4—61no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3#6711yescharacteristic
128C2×C3⋊Dic3#6711yescharacteristic
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)—68no—
131C4⋊(C3⋊D3)—61no—
132C4⋊(C3⋊D3)—64no—
133C4⋊(C3⋊D3)—61no—
134C4⋊(C3⋊D3)—64no—
135C4⋊(C3⋊D3)—124no—
136C4⋊(C3⋊D3)D1212yes—
137C4×C3⋊D3—42no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3—34no—
141C3×D12—38no—
142C3×D12—64no—
143C3×C4×D3—128no—
144C3×Dic6—24no—
145C3×Dic6—24no—
146C32×C8—61no—
147C3⋊(C3⋊C8)—61no—
148C3⋊(C3⋊C8)—64no—
149C3⋊(C3⋊C8)—124no—
150C3×C3⋊C8—24no—
151#463—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33#13511yescharacteristic
154C2×C33AGL(1,ℤ/8)11yescharacteristic
155C3⋊(C3⋊D3)—81no—
156C3×C3⋊D3—42no—
157C2×C3×Q8—34no—
158C2×C3×D4—91no—
159C22×C3×C4—181no—
160C2×D12—34no—
161C2×D12—94no—
162C2×D12—91no—
163C2×D12—184no—
164C2×C4×D3—64no—
165C2×Dic6C2×C3⋊D311yescharacteristic
166C2×C3×C8—34no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4D3214yes—
169C3×C4⋊C4C2×C3⋊D311yescharacteristic
170C3×C22⋊C4—181no—
171C3×C42—91no—
172#215—188no—
173#214—182no—
174#213—182no—
175(C2×D3)⋊C4—64no—
176(C2×D3)⋊C4—181no—
177(C2×D3)⋊C4—181no—
178(C2×D3)⋊C4—184no—
179(C2×D3)⋊C4—184no—
180C4×Dic3—34no—
181C2×C3⋊C8—94no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—184no—
184C2×C3⋊C8C2×C3⋊D311yescharacteristic
185D24—188no—
186C22×C32—24no—
187C22×C32D3×D414yes—
188C22×C32C2×#6711yescharacteristic
189C2×C3⋊D3—22no—
190C2×C3⋊D3—41no—
191C2×C3⋊D3—41no—
192C2×C3⋊D3—128no—
193C2×C3⋊D3—124no—
194C2×C3⋊D3—121no—
195C2×C3⋊D3—121no—
196C2×C3⋊D3—121no—
197C2×C3⋊D3—124no—
198C2×C3⋊D3—124no—
199C2×C3⋊D3—244no—
200C2×C3⋊D3—244no—
201C2×C3×D3—68no—
202C2×C3×D3—124no—
203C2×C3×D3—124no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—41no—
207C32×C4—44no—
208C32×C4—84no—
209C32×C4—121no—
210C32×C4C2×D1211yescharacteristic
211C32×C4#23514yes—
212C32×C4#23511yescharacteristic
213C32×C4D3×D414yes—
214C3⋊Dic3—21no—
215C3⋊Dic3—21no—
216C3×Dic3—44no—
217C3×Dic3—64no—
218C3×Dic3—64no—
219C2×SD16—271no—
220C2×D8—271no—
221#124—271no—
222C4×D4—91no—
223C4⋊C8—271no—
224#103—31no—
225#102—271no—
226C33#46311yescharacteristic, complemented, Sylow
227C23×C3—181no—
228C22×D3—64no—
229C22×D3—184no—
230C22×D3—181no—
231C22×D3—364no—
232C3×Q8—64no—
233C3×Q8—64no—
234C3×D4—92no—
235C3×D4—181no—
236C2×C3×C4—24no—
237C2×C3×C4—21no—
238C2×C3×C4—24no—
239C2×C3×C4—44no—
240C2×C3×C4—64no—
241C2×C3×C4—91no—
242C2×C3×C4—91no—
243C2×C3×C4—362no—
244C2×C3×C4C2×D3214yes—
245C2×C3×C4C22×C3⋊D311yescharacteristic
246C2×Dic3—21no—
247C2×Dic3—34no—
248C2×Dic3—34no—
249D12—38no—
250D12—64no—
251D12—188no—
252D12—184no—
253D12—184no—
254D12—181no—
255D12—181no—
256D12—364no—
257C4×D3—128no—
258Dic6—21no—
259Dic6—21no—
260C3×C8—64no—
261C3⋊C8—21no—
262C3⋊C8—184no—
263C3⋊C8—181no—
264C3⋊C8—364no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32#88414yes—
269C2×C32#91414yes—
270C2×C32#91914yes—
271C2×C32#93011yescharacteristic
272C2×C32#93211yescharacteristic
273C2×C32#95011yescharacteristic
274C3⋊D3—42no—
275C3⋊D3—248no—
276C3⋊D3—241no—
277C3⋊D3—244no—
278C3×D3—128no—
279C2×Q8—31no—
280C2×D4—91no—
281C2×D4—271no—
282C22×C4—181no—
283SD16—542no—
284D8—542no—
285C2×C8—31no—
286C2×C8—271no—
287C4⋊C4D3×C3⋊D311yescharacteristic
288C22⋊C4—181no—
289C22⋊C4—541no—
290C22⋊C4—541no—
291C42—91no—
292C22×C3—24no—
293C22×C3—182no—
294C22×C3—361no—
295C22×C3—361no—
296C22×C3D3×#6714yes—
297C22×C3D4×C3⋊D311yescharacteristic
298C2×D3—68no—
299C2×D3—124no—
300C2×D3—124no—
301C2×D3—368no—
302C2×D3—361no—
303C2×D3—361no—
304C2×D3—361no—
305C2×D3—364no—
306C2×D3—364no—
307C2×D3—364no—
308C2×D3—724no—
309C2×D3—724no—
310C3×C4—24no—
311C3×C4—24no—
312C3×C4—41no—
313C3×C4—44no—
314C3×C4—84no—
315C3×C4—124no—
316C3×C4—181no—
317C3×C4—181no—
318C3×C4#379814yes—
319C3×C4D3×D1214yes—
320C3×C4#382811yescharacteristic
321C3×C4D4×C3⋊D311yescharacteristic
322Dic3—41no—
323Dic3—64no—
324Dic3—64no—
325C32—24no—
326C32#481914yescomplemented
327C32#501611yescharacteristic, complemented
328C23—181no—
329C23—541no—
330Q8—61no—
331Q8—61no—
332D4—92no—
333D4—181no—
334D4—541no—
335D4—541no—
336C2×C4—21no—
337C2×C4—61no—
338C2×C4—91no—
339C2×C4—91no—
340C2×C4—362no—
341C2×C4C2×D3×C3⋊D311yescharacteristic
342C8—61no—
343C8—541no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—362no—
348C2×C3#6390614yes—
349C2×C3#6393414yes—
350C2×C3#6394914yes—
351C2×C3#6409311yescharacteristic
352C2×C3#6412311yescharacteristic
353C2×C3#6412811yescharacteristic
354D3—128no—
355D3—728no—
356D3—721no—
357D3—724no—
358C22—182no—
359C22—361no—
360C22—361no—
361C22—1081no—
362C22—1081no—
363C22—1081no—
364C22C3⋊D3×#6711yescharacteristic
365C4—41no—
366C4—121no—
367C4—181no—
368C4—181no—
369C4#8902011yescharacteristic
370C4C3⋊D3×D1211yescharacteristic
371C3—24no—
372C3#1059053414yescomplemented
373C3#1059251811yescharacteristic, complemented
374C2—362no—
375C2—2161no—
376C2#1172352011yescharacteristic
377C2#1172356211yescharacteristic
378C2#1172358511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#88791C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#463
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64102C2semidirect
2. #11723440C2semidirect
3. C2×#88793C2semidirect
5. #11722179C2semidirect
6. C32×#810C2semidirect
7. #11722051C2semidirect
26. #64102D3semidirect
36. C3×#810D3semidirect
108. #810C3⋊D3semidirect
226. C33#463semidirect
326. C32#4819semidirect
327. C32#5016semidirect
372. C3#10590534semidirect
373. C3#10592518semidirect

Maximal subgroups

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

subgroupindex
C3×#641022
#117234402
C2×#887932
C2×#887912
#117221792
C32×#8102
#117220512
#105925183
#105905343