Finite Group Databasebeta

← #13160377

Subgroups — #13160377

Order 1728.

1228 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#235C211yesmaximal, characteristic
2#11723967C211yesmaximal, characteristic, complemented
3C2×C3×#3748C211yesmaximal, characteristic, complemented
4#11722043C214yesmaximal, complemented
5#10592935—31nomaximal
6#10591110—34nomaximal
7C22×C33×C4C2211yescharacteristic
8C2×C32×#67—21no—
9C32×#235—21no—
10C32×#235—21no—
11C32×#235C412yescomplemented
12C32×#235C2212yes—
13C2×C32×D12C411yescharacteristic, complemented
14C2×C32×C4×D3—21no—
15C2×C32×Dic6C411yescharacteristic, complemented
16#88864—21no—
17C4⋊(C3⋊(C3⋊Dic3))—21no—
18C4×C3⋊(C3⋊Dic3)C2212yescomplemented
19C3×#3748—21no—
20C3×#3748C2212yescomplemented
21C2×C3×C3⋊(C3⋊C8)—21no—
22C2×C32×#41—31no—
23C2×C3×#235D314yes—
24#64135—31no—
25#64135—34no—
26#64135—64no—
27C2×#3748D311yescharacteristic, complemented
28C2×C3×#208—34no—
29#63664—34no—
30#63568—316no—
31C23×C33D411yescharacteristic
32C22×C32×D3—21no—
33C2×C33×C4C2×C411yescharacteristic
34C2×C33×C4C2×C411yescharacteristic
35C2×C33×C4D411yescharacteristic
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—22no—
40C32×#67—22no—
41C32×#67—22no—
42C2×C32×Dic3—21no—
43C32×D12—21no—
44C32×D12C2×C412yes—
45C32×C4×D3—22no—
46C32×C4×D3—22no—
47C32×Dic6—21no—
48C32×Dic6C2×C412yes—
49C3×C3⋊(C3⋊C8)—22no—
50#5236—94no—
51#5139—91no—
52C32×#41—32no—
53C32×#41—32no—
54C32×#41—61no—
55C32×#41—61no—
56C2×C32×Q8—31no—
57C2×C32×D4—31no—
58C2×C32×D4—61no—
59C22×C32×C4—24no—
60C22×C32×C4—61no—
61C22×C32×C4C2×D311yescharacteristic
62C22×C32×C4C2×D314yes—
63C2×C3×#67—24no—
64C3×#235—24no—
65C3×#235—24no—
66C3×#235Dic318yescomplemented
67C3×#235C2×D318yes—
68C2×C3×D12Dic314yescomplemented
69C2×C3×C4×D3—24no—
70C2×C3×Dic6Dic314yescomplemented
71#3757—68no—
72#3757—61no—
73#3757—64no—
74C4⋊(C3⋊Dic3)—68no—
75C4⋊(C3⋊Dic3)—61no—
76C4⋊(C3⋊Dic3)—64no—
77C4×C3⋊Dic3—32no—
78C4×C3⋊Dic3—38no—
79C4×C3⋊Dic3—68no—
80#3748—21no—
81#3748C2×D312yescomplemented
82C2×C3⋊(C3⋊C8)—21no—
83C3×#208—38no—
84C3×#208—64no—
85C2×C3×C3⋊C8—64no—
86C22×C33—41no—
87C22×C33C22⋊C411yescharacteristic
88C22×C33C22⋊C411yescharacteristic
89C22×C33C2×D411yescharacteristic
90C2×C32×D3—22no—
91C2×C32×D3—42no—
92C33×C4C22⋊C411yescharacteristic
93C33×C4C22⋊C411yescharacteristic
94C33×C4C22×C411yescharacteristic
95C33×C4C2×D411yescharacteristic
96C3⋊(C3⋊Dic3)—42no—
97C32×Dic3—22no—
98C2×C3×#41—34no—
99C2×#235C3⋊D311yescharacteristic
100C2×C3×M16—91no—
101#926—94no—
102#926—91no—
103#926—184no—
104C2×#208—34no—
105#837—916no—
106#825—94no—
107C23×C32—24no—
108C23×C32—61no—
109C23×C32D1211yescharacteristic
110C23×C32#6714yes—
111C22×C3×D3—24no—
112C32×Q8—32no—
113C32×Q8—61no—
114C32×D4—32no—
115C32×D4—62no—
116C32×D4—62no—
117C32×D4—61no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—61no—
125C2×C32×C4—62no—
126C2×C32×C4—62no—
127C2×C32×C4C4×D311yescharacteristic
128C2×C32×C4C4×D311yescharacteristic
129C2×C32×C4D1211yescharacteristic
130C2×C32×C4C2×Dic314yes—
131C2×C32×C4C2×Dic314yes—
132C2×C32×C4#6714yes—
133C2×C32×C4#6714yes—
134C2×C32×C4#6714yes—
135C2×C32×C4#6711yescharacteristic
136C2×C32×C4#6711yescharacteristic
137C2×C32×C4C22×D314yes—
138C2×C32×C4C22×D311yescharacteristic
139C2×C3⋊Dic3—616no—
140C2×C3⋊Dic3—62no—
141C2×C3⋊Dic3—68no—
142C3×#67—28no—
143C3×#67—28no—
144C2×C3×Dic3—24no—
145C3×D12—24no—
146C3×D12C2×Dic318yes—
147C3×C4×D3—28no—
148C3×C4×D3—28no—
149C3×Dic6—24no—
150C3×Dic6C2×Dic318yes—
151C3⋊(C3⋊C8)—22no—
152C3×C3⋊C8—68no—
153#421—271noSylow
154C2×C33—21no—
155C2×C33—21no—
156C2×C33—21no—
157C2×C33C2×C22⋊C411yescharacteristic
158C32×D3—42no—
159C3×#41—38no—
160C3×#41—38no—
161C3×#41—64no—
162C3×#41—64no—
163C2×C3×Q8—34no—
164C2×C3×D4—34no—
165C2×C3×D4—64no—
166C22×C3×C4—24no—
167C22×C3×C4—64no—
168C22×C3×C4D3214yes—
169C22×C3×C4C2×C3⋊D311yescharacteristic
170C2×#67—21no—
171#235—21no—
172#235—21no—
173#235C3⋊Dic312yescomplemented
174#235C2×C3⋊D312yes—
175C2×D12C3⋊Dic311yescharacteristic, complemented
176C2×C4×D3—21no—
177C2×Dic6C3⋊Dic311yescharacteristic, complemented
178C3×M16—92no—
179C3×M16—181no—
180C2×C3×C8—181no—
181#217—188no—
182#217—184no—
183#217—181no—
184C4⋊Dic3—188no—
185C4⋊Dic3—184no—
186C4⋊Dic3—181no—
187C4×Dic3—98no—
188C4×Dic3—92no—
189C4×Dic3—188no—
190#208—38no—
191#208—64no—
192C2×C3⋊C8—64no—
193C22×C32—24no—
194C22×C32—24no—
195C22×C32—24no—
196C22×C32—41no—
197C22×C32—44no—
198C22×C32—48no—
199C22×C32—62no—
200C22×C32—122no—
201C22×C32(C2×D3)⋊C411yescharacteristic
202C22×C32(C2×D3)⋊C411yescharacteristic
203C22×C32#21714yes—
204C22×C32#21714yes—
205C22×C32C2×D1211yescharacteristic
206C22×C32C2×#6714yes—
207C2×C3×D3—28no—
208C2×C3×D3—48no—
209C32×C4—24no—
210C32×C4—24no—
211C32×C4—24no—
212C32×C4—24no—
213C32×C4—62no—
214C32×C4(C2×D3)⋊C411yescharacteristic
215C32×C4(C2×D3)⋊C411yescharacteristic
216C32×C4#21714yes—
217C32×C4#21714yes—
218C32×C4C2×C4×D311yescharacteristic
219C32×C4C22×Dic314yes—
220C32×C4C2×#6714yes—
221C32×C4C2×#6711yescharacteristic
222C3⋊Dic3—1216no—
223C3⋊Dic3—122no—
224C3⋊Dic3—128no—
225C3×Dic3—28no—
226C2×#41—31no—
227C2×M16—91no—
228#117—271no—
229#104—274no—
230C33#42111yescharacteristic, complemented, Sylow
231C23×C3—24no—
232C23×C3—64no—
233C23×C3#62614yes—
234C23×C3#63811yescharacteristic
235C22×D3—21no—
236C3×Q8—38no—
237C3×Q8—64no—
238C3×D4—38no—
239C3×D4—68no—
240C3×D4—68no—
241C3×D4—64no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—64no—
249C2×C3×C4—68no—
250C2×C3×C4—68no—
251C2×C3×C4D3×Dic314yes—
252C2×C3×C4D3×Dic314yes—
253C2×C3×C4#62514yes—
254C2×C3×C4#62514yes—
255C2×C3×C4#62614yes—
256C2×C3×C4C2×C3⋊Dic311yescharacteristic
257C2×C3×C4C2×C3⋊Dic311yescharacteristic
258C2×C3×C4#63811yescharacteristic
259C2×C3×C4#63811yescharacteristic
260C2×C3×C4#63811yescharacteristic
261C2×C3×C4C2×D3214yes—
262C2×C3×C4C22×C3⋊D311yescharacteristic
263#67—22no—
264#67—22no—
265C2×Dic3—21no—
266C2×Dic3—1816no—
267C2×Dic3—188no—
268C2×Dic3—182no—
269D12—21no—
270D12C2×C3⋊Dic312yes—
271C4×D3—22no—
272C4×D3—22no—
273Dic6—21no—
274Dic6C2×C3⋊Dic312yes—
275C3×C8—182no—
276C3⋊C8—68no—
277C2×C32—28no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—24no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32—24no—
285C2×C32—44no—
286C2×C32—44no—
287C2×C32—122no—
288C2×C32C2×(C2×D3)⋊C411yescharacteristic
289C2×C32C2×#21714yes—
290C3×D3—48no—
291#41—32no—
292#41—32no—
293#41—61no—
294#41—61no—
295C2×Q8—31no—
296C2×D4—31no—
297C2×D4—61no—
298C22×C4—61no—
299C22×C4D3×C3⋊D311yescharacteristic
300M16—92no—
301M16—181no—
302C2×C8—181no—
303C4⋊C4—541no—
304C22⋊C4—541no—
305C42—272no—
306C22×C3—24no—
307C22×C3—24no—
308C22×C3—24no—
309C22×C3—44no—
310C22×C3—41no—
311C22×C3—48no—
312C22×C3—68no—
313C22×C3—128no—
314C22×C3(C2×D3)⋊Dic314yes—
315C22×C3(C2×D3)⋊Dic314yes—
316C22×C3#375711yescharacteristic
317C22×C3#375711yescharacteristic
318C22×C3C2×#62614yes—
319C22×C3C2×#63811yescharacteristic
320C2×D3—22no—
321C2×D3—42no—
322C3×C4—24no—
323C3×C4—24no—
324C3×C4—24no—
325C3×C4—24no—
326C3×C4—68no—
327C3×C4(C2×D3)⋊Dic314yes—
328C3×C4(C2×D3)⋊Dic314yes—
329C3×C4#375711yescharacteristic
330C3×C4#375711yescharacteristic
331C3×C4C2×D3×Dic314yes—
332C3×C4C2×#62514yes—
333C3×C4C22×C3⋊Dic311yescharacteristic
334C3×C4C2×#63811yescharacteristic
335Dic3—22no—
336Dic3—3616no—
337Dic3—368no—
338Dic3—362no—
339C32—24no—
340C32#513911yescharacteristic, complemented
341C32#523614yescomplemented
342C23—61no—
343C23#629111yescharacteristic
344Q8—32no—
345Q8—61no—
346D4—32no—
347D4—62no—
348D4—62no—
349D4—61no—
350C2×C4—61no—
351C2×C4—62no—
352C2×C4—62no—
353C2×C4—542no—
354C2×C4D3×C3⋊Dic311yescharacteristic
355C2×C4D3×C3⋊Dic311yescharacteristic
356C2×C4#629011yescharacteristic
357C2×C4#629011yescharacteristic
358C2×C4#629111yescharacteristic
359C2×C4C2×D3×C3⋊D311yescharacteristic
360C8—182no—
361C2×C3—28no—
362C2×C3—24no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—24no—
369C2×C3—44no—
370C2×C3—44no—
371C2×C3—128no—
372C2×C3C2×(C2×D3)⋊Dic314yes—
373C2×C3C2×#375711yescharacteristic
374D3—42no—
375C22—41no—
376C22—62no—
377C22—122no—
378C22(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
379C22(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
380C22C2×#629111yescharacteristic
381C4—62no—
382C4—1082no—
383C4(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
384C4(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
385C4C2×D3×C3⋊Dic311yescharacteristic
386C4C2×#629011yescharacteristic
387C3—24no—
388C3#1059111014yescomplemented
389C3#1059293511yescharacteristic, complemented
390C2—21no—
391C2—21no—
392C2—21no—
393C2—122no—
394C2C2×(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
subgroupG/H
centerC4C2×#6290
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C32×#235C2
cosocleC2×C33×C4C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#421
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11723967C2semidirect
3. C2×C3×#3748C2semidirect
4. #11722043C2semidirect
11. C32×#235C4semidirect
13. C2×C32×D12C4semidirect
15. C2×C32×Dic6C4semidirect
18. C4×C3⋊(C3⋊Dic3)C22semidirect
20. C3×#3748C22semidirect
27. C2×#3748D3semidirect
66. C3×#235Dic3semidirect
68. C2×C3×D12Dic3semidirect
70. C2×C3×Dic6Dic3semidirect
81. #3748C2×D3semidirect
173. #235C3⋊Dic3semidirect
175. C2×D12C3⋊Dic3semidirect
177. C2×Dic6C3⋊Dic3semidirect
230. C33#421semidirect
340. C32#5139semidirect
341. C32#5236semidirect
388. C3#10591110semidirect
389. C3#10592935semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2352
#117239672
C2×C3×#37482
#117220432
#105929353
#105911103