Finite Group Databasebeta

← #13159269

Subgroups — #13159269

Order 1728.

968 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 152 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×#64096C211yesmaximal, characteristic, complemented
2#11723524C211yesmaximal, characteristic
3C2×#88797C211yesmaximal, characteristic, complemented
4C2×#88792C211yesmaximal, characteristic, complemented
5#11722179C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7#11722051C211yesmaximal, characteristic, complemented
8#10592461—31nomaximal
9#10590549—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6291—21no—
14C2×#6291—21no—
15C33×C4⋊C4C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3750—21no—
18C3×C4×C3⋊Dic3—21no—
19C3×C4×C3⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88797—22no—
22#88792—22no—
23#64149—31no—
24C2×#3754—31no—
25C2×#3744—31no—
26#64096D311yescharacteristic, complemented
27C3×#887—34no—
28#63911—34no—
29C2×#3717—34no—
30C2×#3715—34no—
31C32×#102—31no—
32#63641—31no—
33#63636—31no—
34#63636—34no—
35#63636—64no—
36C3×#809D314yescomplemented
37Dic3⋊(C3⋊C8)—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C32×D3—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3D411yescharacteristic
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×#634—21no—
48C3×#634D412yescomplemented
49C32×D12—21no—
50C32×D12—21no—
51#6291—42no—
52#6291—42no—
53C3⋊(C3⋊(C3⋊C8))—21no—
54C32×C3⋊C8—21no—
55#5027—91no—
56#4762—94no—
57C2×C32×D4—31no—
58C2×#638—61no—
59C2×#638—61no—
60C2×C4⋊(C3⋊D3)—31no—
61C2×C4⋊(C3⋊D3)—34no—
62C2×C4⋊(C3⋊D3)—64no—
63C2×#634C2×D311yescharacteristic
64C2×C3×D12C2×D314yes—
65C2×C3×Dic6—34no—
66C2×#626—64no—
67C2×#626—64no—
68C2×C32×C8—31no—
69C32×C4⋊C4—24no—
70C32×C4⋊C4C2×D314yes—
71C32×C4⋊C4C2×D311yescharacteristic
72#3754—62no—
73#3750—21no—
74C4×C3⋊Dic3—21no—
75C4×C3⋊Dic3C2×D311yescharacteristic
76C2×C3⋊(C3⋊C8)—31no—
77C2×C3⋊(C3⋊C8)—34no—
78C2×C3⋊(C3⋊C8)—64no—
79#3744—62no—
80C3×Dic3⋊C4—64no—
81C3×C4×Dic3—34no—
82C3×C4×Dic3—64no—
83C2×C3×C3⋊C8C2×D314yes—
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×C32×D3—41no—
91C2×C32×D3—41no—
92C2×C32×D3—41no—
93C33×C4—41no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3×C3⋊Dic3—22no—
97C3×C3⋊Dic3—21no—
98C3×C3⋊Dic3SD1612yescomplemented
99C3×C4×Q8—91no—
100C2×#215—91no—
101C2×#215—91no—
102#940—94no—
103C2×#214—94no—
104C2×#204—94no—
105#887—34no—
106#874—91no—
107C3×#102—34no—
108Dic3⋊C8—94no—
109#809—94no—
110#809—91no—
111#809—184no—
112#809C3⋊D311yescharacteristic, complemented
113C4⋊(C3⋊C8)—91no—
114C23×C32—61no—
115C22×C3⋊D3—61no—
116C22×C3⋊D3—64no—
117C22×C3⋊D3—124no—
118C22×C3×D3—24no—
119C32×D4—61no—
120C32×D4—61no—
121C2×C32×C4—21no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—44no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D311yescharacteristic
127#638—122no—
128#638—122no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3#6711yescharacteristic
131C2×C3⋊Dic3#6711yescharacteristic
132C4⋊(C3⋊D3)—68no—
133C4⋊(C3⋊D3)—61no—
134C4⋊(C3⋊D3)—64no—
135C4⋊(C3⋊D3)—61no—
136C4⋊(C3⋊D3)—64no—
137C4⋊(C3⋊D3)—124no—
138#634—21no—
139#634D1212yescomplemented
140C2×C3×Dic3—34no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—64no—
143C3×D12—24no—
144C3×D12—24no—
145C3×Dic6—38no—
146C3×Dic6—64no—
147#626—128no—
148#626—128no—
149C32×C8—61no—
150C3⋊(C3⋊C8)—61no—
151C3⋊(C3⋊C8)—64no—
152C3⋊(C3⋊C8)—124no—
153C3×C3⋊C8—24no—
154#460—271noSylow
155C2×C33#12111yescharacteristic
156C2×C33C2×SD1611yescharacteristic
157C2×C33AGL(1,ℤ/8)11yescharacteristic
158C3⋊(C3⋊D3)—81no—
159C32×D3—81no—
160C2×C3×Q8—91no—
161C2×C3×D4—34no—
162C2×#67—184no—
163C2×#67—184no—
164C2×D12—94no—
165C2×D12—91no—
166C2×D12—184no—
167C2×D12—181no—
168C2×D12—181no—
169C2×D12C2×C3⋊D311yescharacteristic
170C2×Dic6—34no—
171C2×C3×C8—34no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—181no—
174C3×C4⋊C4D3214yes—
175C3×C4⋊C4C2×C3⋊D311yescharacteristic
176C3×C42—91no—
177C3×C42—181no—
178#215—182no—
179#215—182no—
180#214—188no—
181Dic3⋊C4—64no—
182C4×Dic3—34no—
183C4×Dic3—64no—
184C2×C3⋊C8—94no—
185C2×C3⋊C8—91no—
186C2×C3⋊C8—184no—
187C2×C3⋊C8C2×C3⋊D311yescharacteristic
188#204—188no—
189C22×C32—24no—
190C22×C32—121no—
191C22×C32—121no—
192C22×C32—121no—
193C22×C32D3×D414yes—
194C22×C32C2×#6711yescharacteristic
195C2×C3⋊D3—128no—
196C2×C3⋊D3—124no—
197C2×C3⋊D3—121no—
198C2×C3⋊D3—121no—
199C2×C3⋊D3—121no—
200C2×C3⋊D3—124no—
201C2×C3⋊D3—124no—
202C2×C3⋊D3—244no—
203C2×C3⋊D3—244no—
204C2×C3×D3—44no—
205C2×C3×D3—44no—
206C2×C3×D3—44no—
207C32×C4—24no—
208C32×C4—24no—
209C32×C4—41no—
210C32×C4—44no—
211C32×C4—84no—
212C32×C4C2×D1211yescharacteristic
213C32×C4#23514yes—
214C32×C4#23511yescharacteristic
215C32×C4D3×D414yes—
216C3⋊Dic3—22no—
217C3⋊Dic3—21no—
218C3⋊Dic3#21512yescomplemented
219C3×Dic3—38no—
220C3×Dic3—68no—
221C3×Dic3—64no—
222C2×SD16—271no—
223C2×SD16—271no—
224#127—271no—
225C4×Q8—91no—
226C4⋊C8—271no—
227#102—31no—
228#102—271no—
229C33#46011yescharacteristic, complemented, Sylow
230C23×C3—64no—
231C22×D3—21no—
232C22×D3—184no—
233C22×D3—181no—
234C22×D3—364no—
235C3×Q8—92no—
236C3×Q8—181no—
237C3×D4—64no—
238C3×D4—64no—
239C2×C3×C4—24no—
240C2×C3×C4—21no—
241C2×C3×C4—24no—
242C2×C3×C4—44no—
243C2×C3×C4—91no—
244C2×C3×C4—91no—
245C2×C3×C4—181no—
246C2×C3×C4C2×D3214yes—
247C2×C3×C4C22×C3⋊D311yescharacteristic
248#67—368no—
249#67—368no—
250C2×Dic3—34no—
251C2×Dic3—34no—
252C2×Dic3—64no—
253D12—21no—
254D12—21no—
255D12—188no—
256D12—184no—
257D12—184no—
258D12—181no—
259D12—181no—
260D12—364no—
261D12—362no—
262D12—362no—
263Dic6—38no—
264Dic6—64no—
265C3×C8—64no—
266C3⋊C8—21no—
267C3⋊C8—184no—
268C3⋊C8—181no—
269C3⋊C8—364no—
270C2×C32—24no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32—241no—
274C2×C32#88414yes—
275C2×C32#91114yes—
276C2×C32D3×SD1614yes—
277C2×C32#93011yescharacteristic
278C2×C32C2×#21511yescharacteristic
279C2×C32#94911yescharacteristic
280C3⋊D3—248no—
281C3⋊D3—244no—
282C3⋊D3—241no—
283C3×D3—84no—
284C2×Q8—91no—
285C2×D4—31no—
286C2×D4—271no—
287C2×D4—541no—
288C2×D4—541no—
289SD16—542no—
290SD16—542no—
291C2×C8—31no—
292C2×C8—271no—
293C4⋊C4—181no—
294C4⋊C4D3×C3⋊D311yescharacteristic
295C42—91no—
296C42—181no—
297C22×C3—24no—
298C22×C3—124no—
299C22×C3—124no—
300C22×C3—124no—
301C22×C3D3×#6714yes—
302C22×C3D4×C3⋊D311yescharacteristic
303C2×D3—41no—
304C2×D3—41no—
305C2×D3—41no—
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—41no—
318C3×C4—44no—
319C3×C4—84no—
320C3×C4—92no—
321C3×C4—182no—
322C3×C4—181no—
323C3×C4#379814yes—
324C3×C4D3×D1214yes—
325C3×C4#382811yescharacteristic
326C3×C4D4×C3⋊D311yescharacteristic
327Dic3—38no—
328Dic3—68no—
329Dic3—64no—
330C32—24no—
331C32#476214yescomplemented
332C32#502711yescharacteristic, complemented
333C23—61no—
334C23—541no—
335Q8—92no—
336Q8—181no—
337D4—61no—
338D4—61no—
339D4—541no—
340D4—541no—
341D4—1082no—
342D4—1082no—
343C2×C4—21no—
344C2×C4—91no—
345C2×C4—91no—
346C2×C4—181no—
347C2×C4C2×D3×C3⋊D311yescharacteristic
348C8—61no—
349C8—541no—
350C2×C3—24no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—244no—
354C2×C3#6390614yes—
355C2×C3#6392514yes—
356C2×C3D3×#21514yes—
357C2×C3#6409311yescharacteristic
358C2×C3#6412011yescharacteristic
359C2×C3SD16×C3⋊D311yescharacteristic
360D3—81no—
361D3—728no—
362D3—724no—
363D3—721no—
364C22—121no—
365C22—121no—
366C22—121no—
367C22—1081no—
368C22—1081no—
369C22—1081no—
370C22C3⋊D3×#6711yescharacteristic
371C4—41no—
372C4—92no—
373C4—182no—
374C4—181no—
375C4#8902011yescharacteristic
376C4C3⋊D3×D1211yescharacteristic
377C3—24no—
378C3#1059054914yescomplemented
379C3#1059246111yescharacteristic, complemented
380C2—241no—
381C2—2161no—
382C2#1172352011yescharacteristic
383C2#1172354611yescharacteristic
384C2C3⋊D3×#21511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723524C2
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. C3×#64096C2semidirect
3. C2×#88797C2semidirect
4. C2×#88792C2semidirect
5. #11722179C2semidirect
6. C32×#809C2semidirect
7. #11722051C2semidirect
26. #64096D3semidirect
36. C3×#809D3semidirect
48. C3×#634D4semidirect
98. C3×C3⋊Dic3SD16semidirect
112. #809C3⋊D3semidirect
139. #634D12semidirect
218. C3⋊Dic3#215semidirect
229. C33#460semidirect
331. C32#4762semidirect
332. C32#5027semidirect
378. C3#10590549semidirect
379. C3#10592461semidirect

Maximal subgroups

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

subgroupindex
C3×#640962
#117235242
C2×#887972
C2×#887922
#117221792
C32×#8092
#117220512
#105924613
#105905493