Finite Group Databasebeta

← #13161239

Subgroups — #13161239

Order 1728.

1124 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C2×#88862C211yesmaximal, characteristic, complemented
2C3×#64153C211yesmaximal, characteristic, complemented
3C2×C32×#215C211yesmaximal, characteristic, complemented
4#11723524C211yesmaximal, characteristic
5C2×#88797C211yesmaximal, characteristic, complemented
6C2×#88792C211yesmaximal, characteristic, complemented
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592874—31nomaximal
9#10591618—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6291—22no—
15#88862—22no—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3750—22no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C32×#215—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21#88797—22no—
22#88792—22no—
23C2×C32×SD16—31no—
24#64153D311yescharacteristic, complemented
25C2×#3755—31no—
26C2×#3755—34no—
27C2×#3755—64no—
28#64149—31no—
29C2×#3754—31no—
30C2×#3744—31no—
31C3×#944—34no—
32C2×C3×#215D314yescomplemented
33#63911—34no—
34C2×#3717—34no—
35C2×#3715—34no—
36C8×C3⋊Dic3—31no—
37Dic3×C3⋊C8—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C32×D3—21no—
40C33×Q8—21no—
41C33×Q8—21no—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3D412yes—
48C3×#634—21no—
49C3×#634—21no—
50C32×D12—21no—
51C32×D12—21no—
52#6291—44no—
53C3⋊(C3⋊(C3⋊C8))D412yescomplemented
54C32×C3⋊C8D412yescomplemented
55#5175—94no—
56#5085—91no—
57C2×C32×Q8—24no—
58C2×C32×Q8C2×D314yes—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C32×D4—31no—
61C2×#638—62no—
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4⋊(C3⋊D3)—34no—
64C2×C4⋊(C3⋊D3)—64no—
65C2×#634C2×D311yescharacteristic
66C2×C3×D12C2×D314yes—
67C2×C3×Dic6—34no—
68C2×#626—68no—
69C32×SD16—62no—
70C2×C32×C8—31no—
71#3755—62no—
72#3755—68no—
73#3755—616no—
74#3754—62no—
75#3750—22no—
76C4×C3⋊Dic3C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—34no—
79C2×C3⋊(C3⋊C8)—64no—
80#3744—62no—
81C3×#215—28no—
82C3×Dic3⋊C4—68no—
83C3×C4×Dic3—34no—
84C2×C3×C3⋊C8C2×D314yes—
85#3717—68no—
86#3715—68no—
87C22×C33C2×D411yescharacteristic
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—42no—
90C2×C32×D3—41no—
91C2×C32×D3—42no—
92C33×C4—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4C2×D411yescharacteristic
95C3×C3⋊Dic3—41no—
96C3×C3⋊Dic3SD1614yescomplemented
97C2×C3×SD16—34no—
98C3×C4⋊Q8—91no—
99#944—34no—
100C2×#215—94no—
101C2×#215—91no—
102C2×#215—91no—
103C2×#215—91no—
104C2×#215—184no—
105C2×#215C3⋊D311yescharacteristic, complemented
106#940—94no—
107C2×#214—94no—
108C2×#204—94no—
109#874—91no—
110C8×Dic3—94no—
111C4×C3⋊C8—91no—
112C23×C32—61no—
113C22×C3⋊D3—61no—
114C22×C3⋊D3—64no—
115C22×C3⋊D3—124no—
116C22×C3×D3—24no—
117C32×Q8—28no—
118C32×Q8—24no—
119C32×Q8—21no—
120C32×Q8—21no—
121C32×Q8—24no—
122C32×Q8—44no—
123C32×D4—61no—
124C32×D4—61no—
125C2×C32×C4—21no—
126C2×C32×C4—24no—
127C2×C32×C4—24no—
128C2×C32×C4—44no—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D311yescharacteristic
131#638—124no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3#6712yes—
134C4⋊(C3⋊D3)—68no—
135C4⋊(C3⋊D3)—61no—
136C4⋊(C3⋊D3)—64no—
137C4⋊(C3⋊D3)—61no—
138C4⋊(C3⋊D3)—64no—
139C4⋊(C3⋊D3)—124no—
140#634—21no—
141#634—21no—
142C2×C3×Dic3—38no—
143C2×C3×Dic3—64no—
144C3×D12—24no—
145C3×D12—24no—
146C3×Dic6—64no—
147C3×Dic6—64no—
148#626—1216no—
149C32×C8—32no—
150C3⋊(C3⋊C8)—32no—
151C3⋊(C3⋊C8)—38no—
152C3⋊(C3⋊C8)—68no—
153C3×C3⋊C8#6718yescomplemented
154#492—271noSylow
155C2×C33#12711yescharacteristic
156C2×C33C2×SD1612yes—
157C3⋊(C3⋊D3)—81no—
158C32×D3—81no—
159C2×C3×Q8—24no—
160C2×C3×Q8—91no—
161C2×C3×Q8D3214yes—
162C2×C3×Q8C2×C3⋊D311yescharacteristic
163C2×C3×D4—34no—
164C2×#67—188no—
165C2×D12—94no—
166C2×D12—91no—
167C2×D12—184no—
168C2×D12—182no—
169C2×D12C2×C3⋊D311yescharacteristic
170C2×Dic6—34no—
171C3×SD16—68no—
172C2×C3×C8—34no—
173C3×C4⋊C4—182no—
174C3×C42—91no—
175#215—22no—
176#215—182no—
177#215—188no—
178#215—182no—
179#215—1816no—
180#215—182no—
181#214—188no—
182Dic3⋊C4—68no—
183C4×Dic3—34no—
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—122no—
192C22×C32D3×D414yes—
193C22×C32C2×#6711yescharacteristic
194C2×C3⋊D3—128no—
195C2×C3⋊D3—124no—
196C2×C3⋊D3—121no—
197C2×C3⋊D3—122no—
198C2×C3⋊D3—128no—
199C2×C3⋊D3—248no—
200C2×C3×D3—48no—
201C2×C3×D3—44no—
202C32×C4—24no—
203C32×C4—24no—
204C32×C4—48no—
205C32×C4—44no—
206C32×C4—41no—
207C32×C4D3×D414yes—
208C32×C4D3×D411yescharacteristic
209C32×C4D3×D411yescharacteristic
210C32×C4C2×#6714yes—
211C3⋊Dic3—41no—
212C3⋊Dic3#21514yescomplemented
213C3×Dic3—316no—
214C3×Dic3—124no—
215C2×SD16—31no—
216C2×SD16—271no—
217C2×SD16—271no—
218C2×SD16—271no—
219C4⋊Q8—91no—
220#127—271no—
221C4×C8—271no—
222C33#49211yescharacteristic, complemented, Sylow
223C23×C3—64no—
224C22×D3—21no—
225C22×D3—184no—
226C22×D3—181no—
227C22×D3—364no—
228C3×Q8—28no—
229C3×Q8—24no—
230C3×Q8—24no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—44no—
234C3×Q8—181no—
235C3×Q8—181no—
236C3×D4—64no—
237C3×D4—64no—
238C2×C3×C4—24no—
239C2×C3×C4—21no—
240C2×C3×C4—24no—
241C2×C3×C4—44no—
242C2×C3×C4—92no—
243C2×C3×C4—181no—
244C2×C3×C4C2×D3214yes—
245C2×C3×C4C22×C3⋊D311yescharacteristic
246#67—3616no—
247C2×Dic3—38no—
248C2×Dic3—64no—
249D12—21no—
250D12—21no—
251D12—188no—
252D12—184no—
253D12—184no—
254D12—181no—
255D12—181no—
256D12—364no—
257D12—364no—
258Dic6—64no—
259Dic6—64no—
260C3×C8—38no—
261C3⋊C8—98no—
262C3⋊C8—92no—
263C3⋊C8—188no—
264C3⋊C8#63812yescomplemented
265C2×C32—28no—
266C2×C32—24no—
267C2×C32—241no—
268C2×C32D3×SD1618yes—
269C2×C32#94014yes—
270C2×C32#94011yescharacteristic
271C2×C32C2×#21512yes—
272C3⋊D3—248no—
273C3⋊D3—244no—
274C3⋊D3—241no—
275C3×D3—84no—
276C2×Q8—91no—
277C2×Q8D3×C3⋊D311yescharacteristic
278C2×D4—31no—
279C2×D4—271no—
280C2×D4—542no—
281SD16—62no—
282SD16—542no—
283SD16—542no—
284SD16—542no—
285C2×C8—31no—
286C2×C8—271no—
287C4⋊C4—182no—
288C42—91no—
289C22×C3—24no—
290C22×C3—128no—
291C22×C3—124no—
292C22×C3D3×#6714yes—
293C22×C3D4×C3⋊D311yescharacteristic
294C2×D3—42no—
295C2×D3—41no—
296C2×D3—368no—
297C2×D3—362no—
298C2×D3—361no—
299C2×D3—368no—
300C2×D3—364no—
301C2×D3—728no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—48no—
305C3×C4—41no—
306C3×C4—44no—
307C3×C4—94no—
308C3×C4—361no—
309C3×C4D3×#6714yes—
310C3×C4#381114yes—
311C3×C4D4×C3⋊D311yescharacteristic
312C3×C4C2×#63811yescharacteristic
313Dic3—316no—
314Dic3—124no—
315C32—24no—
316C32#508511yescharacteristic, complemented
317C32#517514yescomplemented
318C23—61no—
319C23—541no—
320Q8—21no—
321Q8—21no—
322Q8—181no—
323Q8—181no—
324D4—61no—
325D4—61no—
326D4—541no—
327D4—541no—
328D4—1084no—
329C2×C4—21no—
330C2×C4—92no—
331C2×C4—181no—
332C2×C4C2×D3×C3⋊D311yescharacteristic
333C8—32no—
334C8—272no—
335C2×C3—28no—
336C2×C3—24no—
337C2×C3—244no—
338C2×C3D3×#21518yes—
339C2×C3#6397914yes—
340C2×C3SD16×C3⋊D312yes—
341C2×C3#6414911yescharacteristic
342D3—81no—
343D3—728no—
344D3—721no—
345D3—724no—
346C22—122no—
347C22—121no—
348C22—1082no—
349C22—1081no—
350C22C3⋊D3×#6711yescharacteristic
351C4—41no—
352C4—94no—
353C4—361no—
354C4D3×#63811yescharacteristic
355C4#8903911yescharacteristic
356C3—24no—
357C3#1059161814yescomplemented
358C3#1059287411yescharacteristic, complemented
359C2—241no—
360C2—2161no—
361C2C3⋊D3×#21512yes—
362C2#1172363311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radical#11723524C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#492
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88862C2semidirect
2. C3×#64153C2semidirect
3. C2×C32×#215C2semidirect
5. C2×#88797C2semidirect
6. C2×#88792C2semidirect
7. C3⋊C8×C3⋊Dic3C2semidirect
24. #64153D3semidirect
32. C2×C3×#215D3semidirect
53. C3⋊(C3⋊(C3⋊C8))D4semidirect
54. C32×C3⋊C8D4semidirect
96. C3×C3⋊Dic3SD16semidirect
105. C2×#215C3⋊D3semidirect
153. C3×C3⋊C8#67semidirect
212. C3⋊Dic3#215semidirect
222. C33#492semidirect
264. C3⋊C8#638semidirect
316. C32#5085semidirect
317. C32#5175semidirect
357. C3#10591618semidirect
358. C3#10592874semidirect

Maximal subgroups

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

subgroupindex
C2×#888622
C3×#641532
C2×C32×#2152
#117235242
C2×#887972
C2×#887922
C3⋊C8×C3⋊Dic32
#105928743
#105916183