Finite Group Databasebeta

← #13159273

Subgroups — #13159273

Order 1728.

968 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 176 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
1C32×#887C211yesmaximal, characteristic, complemented
2#11723517C211yesmaximal, characteristic
3C2×#88795C211yesmaximal, characteristic, complemented
4C2×#88793C211yesmaximal, characteristic, complemented
5#11722179C211yesmaximal, characteristic, complemented
6C3×#63636C211yesmaximal, characteristic, complemented
7Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592726—31nomaximal
9#10590549—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6292—21no—
14C2×#6292—21no—
15C33×C4⋊C4C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×Dic3⋊C4—21no—
19C32×C4×Dic3—21no—
20C32×C4×Dic3C2211yescharacteristic
21#88795—22no—
22#88793—22no—
23C32×C4×Q8—31no—
24C2×#3755—31no—
25C2×#3755—31no—
26#64083—31no—
27C3×#887D314yescomplemented
28#63911—34no—
29C2×#3717—34no—
30C2×#3715—34no—
31#63636—31no—
32#63636—34no—
33#63636—64no—
34#63636D311yescharacteristic, complemented
35C4⋊(C3⋊(C3⋊C8))—31no—
36C3×#809—34no—
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—
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C2×C32×Dic3—21no—
47C2×C32×Dic3D411yescharacteristic
48C2×C32×Dic3D411yescharacteristic
49C32×Dic6—21no—
50C32×Dic6D412yescomplemented
51#6292—42no—
52#6292—42no—
53C3⋊(C3⋊(C3⋊C8))—21no—
54C3×C3⋊(C3⋊C8)—21no—
55#5027—94no—
56#4762—91no—
57C2×C32×Q8—31no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4⋊(C3⋊D3)—34no—
60C2×C4⋊(C3⋊D3)—64no—
61C2×C4⋊(C3⋊D3)—61no—
62C2×C4⋊(C3⋊D3)—61no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C3×D12—34no—
65C2×C3×Dic6C2×D314yes—
66C2×#626—64no—
67C2×#626—64no—
68C32×C4⋊C4—24no—
69C32×C4⋊C4—61no—
70C32×C4⋊C4C2×D314yes—
71C32×C4⋊C4C2×D311yescharacteristic
72C32×C42—31no—
73C32×C42—61no—
74#3755—62no—
75#3755—62no—
76C2×C3⋊(C3⋊C8)—31no—
77C2×C3⋊(C3⋊C8)—34no—
78C2×C3⋊(C3⋊C8)—64no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80C3×Dic3⋊C4—24no—
81C3×C4×Dic3—24no—
82C3×C4×Dic3C2×D314yes—
83C2×C3×C3⋊C8—34no—
84#3717—68no—
85#3715—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C33×C4—41no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C32×Dic3—22no—
97C32×Dic3—21no—
98C32×Dic3SD1612yescomplemented
99C3×C4×Q8—34no—
100C2×#215—94no—
101C2×#215—94no—
102#940—91no—
103C2×#214—91no—
104C2×#204—91no—
105#887C3⋊D311yescharacteristic, complemented
106#874—94no—
107C3×#102—91no—
108Dic3⋊C8—91no—
109#809—34no—
110#809—94no—
111#809—91no—
112#809—184no—
113C4⋊(C3⋊C8)—94no—
114C22×C3⋊D3—21no—
115C22×C3⋊D3—61no—
116C22×C3⋊D3—64no—
117C22×C3⋊D3—124no—
118C22×C3×D3—64no—
119C32×Q8—32no—
120C32×Q8—61no—
121C2×C32×C4—21no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—31no—
125C2×C32×C4—31no—
126C2×C32×C4—44no—
127C2×C32×C4—61no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)—21no—
132C4⋊(C3⋊D3)—68no—
133C4⋊(C3⋊D3)—64no—
134C4⋊(C3⋊D3)—61no—
135C4⋊(C3⋊D3)—61no—
136C4⋊(C3⋊D3)—64no—
137C4⋊(C3⋊D3)—124no—
138C4⋊(C3⋊D3)—122no—
139C4⋊(C3⋊D3)—122no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3#6714yes—
142C2×C3×Dic3#6714yes—
143C3×D12—64no—
144C3×D12—64no—
145C3×Dic6—24no—
146C3×Dic6D1218yescomplemented
147#626—128no—
148#626—128no—
149C3⋊(C3⋊C8)—21no—
150C3⋊(C3⋊C8)—61no—
151C3⋊(C3⋊C8)—64no—
152C3⋊(C3⋊C8)—124no—
153C3×C3⋊C8—64no—
154#460—271noSylow
155C2×C33#12111yescharacteristic
156C2×C33C2×SD1611yescharacteristic
157C2×C33AGL(1,ℤ/8)11yescharacteristic
158C3⋊(C3⋊D3)—81no—
159C3×C3⋊D3—81no—
160C2×C3×Q8—34no—
161C2×C3×D4—91no—
162C2×#67—181no—
163C2×#67—181no—
164C2×D12—34no—
165C2×D12—94no—
166C2×D12—91no—
167C2×D12—184no—
168C2×D12—184no—
169C2×D12—184no—
170C2×Dic6C2×C3⋊D311yescharacteristic
171C2×C3×C8—91no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—64no—
174C3×C4⋊C4D3214yes—
175C3×C4⋊C4C2×C3⋊D311yescharacteristic
176C3×C42—34no—
177C3×C42—64no—
178#215—188no—
179#215—188no—
180#214—182no—
181Dic3⋊C4—21no—
182C4×Dic3—21no—
183C4×Dic3C2×C3⋊D311yescharacteristic
184C2×C3⋊C8—34no—
185C2×C3⋊C8—94no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—184no—
188#204—182no—
189C22×C32—24no—
190C22×C32D3×D411yescharacteristic
191C22×C32C2×#6714yes—
192C2×C3⋊D3—41no—
193C2×C3⋊D3—41no—
194C2×C3⋊D3—41no—
195C2×C3⋊D3—128no—
196C2×C3⋊D3—121no—
197C2×C3⋊D3—121no—
198C2×C3⋊D3—121no—
199C2×C3⋊D3—124no—
200C2×C3⋊D3—124no—
201C2×C3⋊D3—124no—
202C2×C3⋊D3—244no—
203C2×C3⋊D3—244no—
204C2×C3×D3—124no—
205C2×C3×D3—124no—
206C2×C3×D3—124no—
207C32×C4—24no—
208C32×C4—24no—
209C32×C4—32no—
210C32×C4—41no—
211C32×C4—44no—
212C32×C4—62no—
213C32×C4—61no—
214C32×C4—84no—
215C32×C4C2×D1214yes—
216C32×C4#23514yes—
217C32×C4#23511yescharacteristic
218C32×C4D3×D411yescharacteristic
219C3×Dic3—28no—
220C3×Dic3—24no—
221C3×Dic3#21518yescomplemented
222C2×SD16—271no—
223C2×SD16—271no—
224#127—271no—
225C4×Q8—31no—
226C4⋊C8—271no—
227#102—91no—
228#102—271no—
229C33#46011yescharacteristic, complemented, Sylow
230C23×C3—181no—
231C22×D3—64no—
232C22×D3—184no—
233C22×D3—181no—
234C22×D3—364no—
235C3×Q8—38no—
236C3×Q8—64no—
237C3×D4—181no—
238C3×D4—181no—
239C2×C3×C4—24no—
240C2×C3×C4—21no—
241C2×C3×C4—24no—
242C2×C3×C4—34no—
243C2×C3×C4—34no—
244C2×C3×C4—44no—
245C2×C3×C4—64no—
246C2×C3×C4C2×D3214yes—
247C2×C3×C4C22×C3⋊D311yescharacteristic
248#67—362no—
249#67—362no—
250C2×Dic3—21no—
251C2×Dic3#63811yescharacteristic
252C2×Dic3#63811yescharacteristic
253D12—64no—
254D12—64no—
255D12—188no—
256D12—184no—
257D12—184no—
258D12—181no—
259D12—181no—
260D12—364no—
261D12—368no—
262D12—368no—
263Dic6—21no—
264Dic6C4⋊(C3⋊D3)12yescomplemented
265C3×C8—181no—
266C3⋊C8—64no—
267C3⋊C8—184no—
268C3⋊C8—181no—
269C3⋊C8—364no—
270C2×C32—24no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32#88411yescharacteristic
274C2×C32#91111yescharacteristic
275C2×C32D3×SD1611yescharacteristic
276C2×C32#93014yes—
277C2×C32C2×#21514yes—
278C2×C32#94914yes—
279C3⋊D3—81no—
280C3⋊D3—248no—
281C3⋊D3—241no—
282C3⋊D3—244no—
283C3×D3—244no—
284C2×Q8—31no—
285C2×D4—91no—
286C2×D4—271no—
287C2×D4—541no—
288C2×D4—541no—
289SD16—542no—
290SD16—542no—
291C2×C8—91no—
292C2×C8—271no—
293C4⋊C4—61no—
294C4⋊C4D3×C3⋊D311yescharacteristic
295C42—31no—
296C42—61no—
297C22×C3—24no—
298C22×C3—361no—
299C22×C3—361no—
300C22×C3—361no—
301C22×C3D3×#6714yes—
302C22×C3C2×#63811yescharacteristic
303C2×D3—124no—
304C2×D3—124no—
305C2×D3—124no—
306C2×D3—368no—
307C2×D3—361no—
308C2×D3—361no—
309C2×D3—361no—
310C2×D3—364no—
311C2×D3—364no—
312C2×D3—364no—
313C2×D3—724no—
314C2×D3—724no—
315C3×C4—24no—
316C3×C4—24no—
317C3×C4—38no—
318C3×C4—41no—
319C3×C4—44no—
320C3×C4—68no—
321C3×C4—64no—
322C3×C4—84no—
323C3×C4#379814yes—
324C3×C4D3×D1214yes—
325C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
326C3×C4#382811yescharacteristic
327Dic3—22no—
328Dic3—21no—
329Dic3#375512yescomplemented
330C32—24no—
331C32#476211yescharacteristic, complemented
332C32#502714yescomplemented
333C23—181no—
334C23—541no—
335Q8—32no—
336Q8—61no—
337D4—181no—
338D4—181no—
339D4—541no—
340D4—541no—
341D4—1082no—
342D4—1082no—
343C2×C4—21no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—61no—
347C2×C4C2×D3×C3⋊D311yescharacteristic
348C8—181no—
349C8—541no—
350C2×C3—24no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—721no—
354C2×C3#6390614yes—
355C2×C3#6392514yes—
356C2×C3D3×#21514yes—
357C2×C3#6413911yescharacteristic
358C2×C3C2×#375511yescharacteristic
359C2×C3#6415811yescharacteristic
360D3—244no—
361D3—728no—
362D3—724no—
363D3—721no—
364C22—361no—
365C22—361no—
366C22—361no—
367C22—1081no—
368C22—1081no—
369C22—1081no—
370C22D3×#63811yescharacteristic
371C4—32no—
372C4—41no—
373C4—61no—
374C4—62no—
375C4#8902011yescharacteristic
376C4D3×C4⋊(C3⋊D3)11yescharacteristic
377C3—24no—
378C3#1059054914yescomplemented
379C3#1059272611yescharacteristic, complemented
380C2—721no—
381C2—2161no—
382C2#1172351411yescharacteristic
383C2#1172354411yescharacteristic
384C2D3×#375511yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723517C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#460
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. C32×#887C2semidirect
3. C2×#88795C2semidirect
4. C2×#88793C2semidirect
5. #11722179C2semidirect
6. C3×#63636C2semidirect
7. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
27. C3×#887D3semidirect
34. #63636D3semidirect
50. C32×Dic6D4semidirect
98. C32×Dic3SD16semidirect
105. #887C3⋊D3semidirect
146. C3×Dic6D12semidirect
221. C3×Dic3#215semidirect
229. C33#460semidirect
264. Dic6C4⋊(C3⋊D3)semidirect
329. Dic3#3755semidirect
331. C32#4762semidirect
332. C32#5027semidirect
378. C3#10590549semidirect
379. C3#10592726semidirect

Maximal subgroups

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

subgroupindex
C32×#8872
#117235172
C2×#887952
C2×#887932
#117221792
C3×#636362
Dic3⋊(C3⋊(C3⋊C8))2
#105927263
#105905493