Finite Group Databasebeta

← #13159271

Subgroups — #13159271

Order 1728.

1040 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1040 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×#893C211yesmaximal, characteristic, complemented
2#11723517C211yesmaximal, characteristic, complemented
3C2×#88790C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic, complemented
5#11722179C211yesmaximal, characteristic, complemented
6C3×#63636C211yesmaximal, characteristic, complemented
7Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592714—31nomaximal
9#10590547—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3—21no—
14C2×#6292—21no—
15C2×#6292—21no—
16C33×C4⋊C4C2211yescharacteristic
17C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×(C2×D3)⋊C4—21no—
20C32×C4×Dic3C2211yescharacteristic
21#88790—22no—
22#88789—22no—
23C32×C4×D4—31no—
24C2×#3753—31no—
25C2×#3753—31no—
26#64083—31no—
27C3×#893D314yescomplemented
28#63911—34no—
29C2×#3714—34no—
30C2×#3713—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—
40C22×C32×D3—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C2×C32×Dic3D411yescharacteristic
48C2×C32×Dic3D411yescharacteristic
49C32×D12—21no—
50C32×D12D412yescomplemented
51C32×C4×D3—42no—
52#6292—42no—
53#6292—42no—
54C3⋊(C3⋊(C3⋊C8))—21no—
55C3×C3⋊(C3⋊C8)—21no—
56#5015—94no—
57#4786—91no—
58C2×C32×D4—31no—
59C22×C32×C4—61no—
60C2×C4⋊(C3⋊D3)—31no—
61C2×C4⋊(C3⋊D3)—34no—
62C2×C4⋊(C3⋊D3)—64no—
63C2×C4⋊(C3⋊D3)—61no—
64C2×C4⋊(C3⋊D3)—61no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C3×D12—34no—
67C2×C3×D12C2×D314yes—
68C2×C3×C4×D3—24no—
69C2×#626—64no—
70C2×#626—64no—
71C32×C4⋊C4—24no—
72C32×C4⋊C4C2×D314yes—
73C32×C4⋊C4C2×D311yescharacteristic
74C32×C22⋊C4—61no—
75C32×C42—31no—
76#3753—62no—
77#3753—62no—
78C2×C3⋊(C3⋊C8)—31no—
79C2×C3⋊(C3⋊C8)—34no—
80C2×C3⋊(C3⋊C8)—64no—
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C3×(C2×D3)⋊C4—24no—
83C3×C4×Dic3C2×D314yes—
84C2×C3×C3⋊C8—34no—
85#3714—68no—
86#3713—68no—
87C22×C33C2×D411yescharacteristic
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—
93C2×C3×C3⋊D3—41no—
94C2×C32×D3—22no—
95C2×C32×D3—41no—
96C2×C32×D3—41no—
97C33×C4—41no—
98C33×C4C2×D411yescharacteristic
99C33×C4#4111yescharacteristic
100C32×Dic3—21no—
101C32×Dic3D812yescomplemented
102C3×C4×D4—34no—
103#940—91no—
104C2×#213—94no—
105C2×#213—94no—
106C2×#213—91no—
107C2×D24—91no—
108#893C3⋊D311yescharacteristic, complemented
109#874—94no—
110C3×#102—91no—
111Dic3⋊C8—91no—
112#809—34no—
113#809—94no—
114#809—91no—
115#809—184no—
116C4⋊(C3⋊C8)—94no—
117C23×C32—61no—
118C22×C3⋊D3—21no—
119C22×C3⋊D3—61no—
120C22×C3⋊D3—64no—
121C22×C3⋊D3—124no—
122C22×C3×D3—24no—
123C22×C3×D3—64no—
124C32×D4—32no—
125C32×D4—61no—
126C2×C32×C4—21no—
127C2×C32×C4—24no—
128C2×C32×C4—24no—
129C2×C32×C4—31no—
130C2×C32×C4—31no—
131C2×C32×C4—44no—
132C2×C32×C4—122no—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C22×D311yescharacteristic
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—68no—
138C4⋊(C3⋊D3)—64no—
139C4⋊(C3⋊D3)—61no—
140C4⋊(C3⋊D3)—61no—
141C4⋊(C3⋊D3)—64no—
142C4⋊(C3⋊D3)—124no—
143C4⋊(C3⋊D3)—122no—
144C4⋊(C3⋊D3)—122no—
145C2×C3×Dic3#6714yes—
146C2×C3×Dic3#6714yes—
147C3×D12—24no—
148C3×D12—64no—
149C3×D12—64no—
150C3×D12D1218yescomplemented
151C3×C4×D3—48no—
152#626—128no—
153#626—128no—
154C3⋊(C3⋊C8)—21no—
155C3⋊(C3⋊C8)—61no—
156C3⋊(C3⋊C8)—64no—
157C3⋊(C3⋊C8)—124no—
158C3×C3⋊C8—64no—
159#459—271noSylow
160C2×C33#12111yescharacteristic
161C2×C33C2×D811yescharacteristic
162C2×C33AGL(1,ℤ/8)11yescharacteristic
163C3⋊(C3⋊D3)—81no—
164C3×C3⋊D3—81no—
165C32×D3—42no—
166C2×C3×D4—34no—
167C2×C3×D4—91no—
168C22×C3×C4—64no—
169C2×#67—181no—
170C2×#67—181no—
171C2×D12—34no—
172C2×D12—94no—
173C2×D12—91no—
174C2×D12—184no—
175C2×D12—184no—
176C2×D12—184no—
177C2×D12C2×C3⋊D311yescharacteristic
178C2×C4×D3—21no—
179C2×C3×C8—91no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4D3214yes—
182C3×C4⋊C4C2×C3⋊D311yescharacteristic
183C3×C22⋊C4—64no—
184C3×C42—34no—
185#213—188no—
186#213—188no—
187#213—182no—
188(C2×D3)⋊C4—21no—
189C4×Dic3C2×C3⋊D311yescharacteristic
190C2×C3⋊C8—34no—
191C2×C3⋊C8—94no—
192C2×C3⋊C8—91no—
193C2×C3⋊C8—184no—
194D24—182no—
195C22×C32—24no—
196C22×C32—62no—
197C22×C32—121no—
198C22×C32—121no—
199C22×C32D3×D411yescharacteristic
200C22×C32C2×#6714yes—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3⋊D3—128no—
205C2×C3⋊D3—121no—
206C2×C3⋊D3—121no—
207C2×C3⋊D3—121no—
208C2×C3⋊D3—124no—
209C2×C3⋊D3—124no—
210C2×C3⋊D3—124no—
211C2×C3⋊D3—244no—
212C2×C3⋊D3—244no—
213C2×C3×D3—28no—
214C2×C3×D3—44no—
215C2×C3×D3—44no—
216C2×C3×D3—124no—
217C2×C3×D3—124no—
218C2×C3×D3—124no—
219C32×C4—24no—
220C32×C4—24no—
221C32×C4—32no—
222C32×C4—41no—
223C32×C4—44no—
224C32×C4—61no—
225C32×C4—84no—
226C32×C4C2×D1214yes—
227C32×C4#23514yes—
228C32×C4#23511yescharacteristic
229C32×C4D3×D411yescharacteristic
230C3×Dic3—24no—
231C3×Dic3#21318yescomplemented
232C2×D8—271no—
233C2×D8—271no—
234#127—271no—
235C4×D4—31no—
236C4⋊C8—271no—
237#102—91no—
238#102—271no—
239C33#45911yescharacteristic, complemented, Sylow
240C23×C3—64no—
241C23×C3—181no—
242C22×D3—21no—
243C22×D3—64no—
244C22×D3—184no—
245C22×D3—181no—
246C22×D3—364no—
247C3×D4—38no—
248C3×D4—64no—
249C3×D4—181no—
250C3×D4—181no—
251C2×C3×C4—24no—
252C2×C3×C4—21no—
253C2×C3×C4—24no—
254C2×C3×C4—34no—
255C2×C3×C4—34no—
256C2×C3×C4—44no—
257C2×C3×C4—128no—
258C2×C3×C4C2×D3214yes—
259C2×C3×C4C22×C3⋊D311yescharacteristic
260#67—362no—
261#67—362no—
262C2×Dic3#63811yescharacteristic
263C2×Dic3#63811yescharacteristic
264D12—21no—
265D12—64no—
266D12—64no—
267D12—188no—
268D12—184no—
269D12—184no—
270D12—181no—
271D12—181no—
272D12—364no—
273D12—368no—
274D12—368no—
275D12C4⋊(C3⋊D3)12yescomplemented
276C4×D3—42no—
277C3×C8—181no—
278C3⋊C8—64no—
279C3⋊C8—184no—
280C3⋊C8—181no—
281C3⋊C8—364no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32—24no—
285C2×C32—122no—
286C2×C32#88411yescharacteristic
287C2×C32D3×D811yescharacteristic
288C2×C32#91411yescharacteristic
289C2×C32#93014yes—
290C2×C32C2×#21314yes—
291C2×C32#94914yes—
292C3⋊D3—81no—
293C3⋊D3—248no—
294C3⋊D3—244no—
295C3⋊D3—241no—
296C3×D3—48no—
297C3×D3—244no—
298C2×D4—31no—
299C2×D4—91no—
300C2×D4—271no—
301C2×D4—541no—
302C2×D4—541no—
303C22×C4—61no—
304D8—542no—
305D8—542no—
306C2×C8—91no—
307C2×C8—271no—
308C4⋊C4D3×C3⋊D311yescharacteristic
309C22⋊C4—61no—
310C42—31no—
311C22×C3—24no—
312C22×C3—68no—
313C22×C3—124no—
314C22×C3—124no—
315C22×C3—361no—
316C22×C3—361no—
317C22×C3—361no—
318C22×C3D3×#6714yes—
319C22×C3C2×#63811yescharacteristic
320C2×D3—22no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—124no—
324C2×D3—124no—
325C2×D3—124no—
326C2×D3—368no—
327C2×D3—361no—
328C2×D3—361no—
329C2×D3—361no—
330C2×D3—364no—
331C2×D3—364no—
332C2×D3—364no—
333C2×D3—724no—
334C2×D3—724no—
335C3×C4—24no—
336C3×C4—24no—
337C3×C4—38no—
338C3×C4—41no—
339C3×C4—44no—
340C3×C4—64no—
341C3×C4—84no—
342C3×C4#379814yes—
343C3×C4D3×D1214yes—
344C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
345C3×C4#382811yescharacteristic
346Dic3—21no—
347Dic3#375312yescomplemented
348C32—24no—
349C32#478611yescharacteristic, complemented
350C32#501514yescomplemented
351C23—61no—
352C23—181no—
353C23—541no—
354D4—32no—
355D4—61no—
356D4—181no—
357D4—181no—
358D4—541no—
359D4—541no—
360D4—1082no—
361D4—1082no—
362C2×C4—21no—
363C2×C4—31no—
364C2×C4—31no—
365C2×C4—122no—
366C2×C4C2×D3×C3⋊D311yescharacteristic
367C8—181no—
368C8—541no—
369C2×C3—24no—
370C2×C3—24no—
371C2×C3—24no—
372C2×C3—128no—
373C2×C3—721no—
374C2×C3#6390614yes—
375C2×C3D3×#21314yes—
376C2×C3#6393914yes—
377C2×C3#6413911yescharacteristic
378C2×C3C2×#375311yescharacteristic
379C2×C3#6415811yescharacteristic
380D3—42no—
381D3—244no—
382D3—728no—
383D3—724no—
384D3—721no—
385C22—62no—
386C22—121no—
387C22—121no—
388C22—361no—
389C22—361no—
390C22—361no—
391C22—1081no—
392C22—1081no—
393C22—1081no—
394C22D3×#63811yescharacteristic
395C4—32no—
396C4—41no—
397C4—61no—
398C4#8902011yescharacteristic
399C4D3×C4⋊(C3⋊D3)11yescharacteristic
400C3—24no—
401C3#1059054714yescomplemented
402C3#1059271411yescharacteristic, complemented
403C2—122no—
404C2—721no—
405C2—2161no—
406C2#1172351411yescharacteristic
407C2D3×#375311yescharacteristic
408C2#1172356811yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#459
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×#893C2semidirect
2. #11723517C2semidirect
3. C2×#88790C2semidirect
4. C2×#88789C2semidirect
5. #11722179C2semidirect
6. C3×#63636C2semidirect
7. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
27. C3×#893D3semidirect
34. #63636D3semidirect
50. C32×D12D4semidirect
101. C32×Dic3D8semidirect
108. #893C3⋊D3semidirect
150. C3×D12D12semidirect
231. C3×Dic3#213semidirect
239. C33#459semidirect
275. D12C4⋊(C3⋊D3)semidirect
347. Dic3#3753semidirect
349. C32#4786semidirect
350. C32#5015semidirect
401. C3#10590547semidirect
402. C3#10592714semidirect

Maximal subgroups

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

subgroupindex
C32×#8932
#117235172
C2×#887902
C2×#887892
#117221792
C3×#636362
Dic3⋊(C3⋊(C3⋊C8))2
#105927143
#105905473