Finite Group Databasebeta

← #13161077

Subgroups — #13161077

Order 1728.

1284 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1284 conjugacy classes of subgroups — too many to draw. Only its 162 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
1#11723981C211yesmaximal, characteristic
2C2×C3×#3754C211yesmaximal, characteristic, complemented
3C2×C32×#214C211yesmaximal, characteristic, complemented
4#11723451C211yesmaximal, characteristic, complemented
5C2×#88797C211yesmaximal, characteristic, complemented
6C2×#88795C211yesmaximal, characteristic, complemented
7#11722018C211yesmaximal, characteristic, complemented
8#10592874—31nomaximal
9#10591492—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6312—22no—
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3754—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#214—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—22no—
21#88797—22no—
22#88795—22no—
23C2×C32×SD16—31no—
24#64153—31no—
25C2×#3755—31no—
26#64149—31no—
27#64149—34no—
28#64149—64no—
29C2×#3754D311yescharacteristic, complemented
30C2×#3744—31no—
31C2×C3×#214—34no—
32C2×C3×#214D314yescomplemented
33#63873—34no—
34C2×#3717—34no—
35C2×#3717—34no—
36C8×C3⋊Dic3—31no—
37#63553—34no—
38C23×C33—21no—
39C22×C3⋊(C3⋊D3)—21no—
40C33×D4—21no—
41C33×D4—21no—
42C2×C33×C4C2311yescharacteristic
43#6312—44no—
44C2×C3⋊(C3⋊Dic3)D412yes—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—21no—
47C2×C3×C3⋊Dic3—21no—
48C3×#634—21no—
49C3×#634—21no—
50C2×C32×Dic3—21no—
51C32×Dic6—21no—
52C32×Dic6—21no—
53C3×C3⋊(C3⋊C8)D412yescomplemented
54C32×C3⋊C8D412yescomplemented
55#5175—94no—
56#5175—91no—
57C2×C32×Q8—31no—
58C2×C32×D4—24no—
59C2×C32×D4C2×D314yes—
60C2×C32×D4C2×D311yescharacteristic
61C2×#638—62no—
62C2×#638—68no—
63C2×#638—616no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C4⋊(C3⋊D3)—34no—
66C2×C4⋊(C3⋊D3)—64no—
67C2×#634C2×D311yescharacteristic
68C2×C3×Dic6—34no—
69C2×C3×Dic6C2×D314yes—
70C32×SD16—62no—
71C2×C32×C8—31no—
72#3755—62no—
73#3754—22no—
74#3750—62no—
75C4×C3⋊Dic3—31no—
76C4×C3⋊Dic3—34no—
77C4×C3⋊Dic3—64no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79#3744—62no—
80C3×#214—28no—
81C3×#214—68no—
82C2×C3×C3⋊C8—34no—
83C2×C3×C3⋊C8C2×D314yes—
84#3724—68no—
85#3717—68no—
86#3717—68no—
87C22×C33—41no—
88C22×C33—42no—
89C22×C33C2×D411yescharacteristic
90C2×C3⋊(C3⋊D3)—41no—
91C2×C3⋊(C3⋊D3)—42no—
92C33×C4C2×D411yescharacteristic
93C33×C4C2×D411yescharacteristic
94C3⋊(C3⋊Dic3)SD1614yescomplemented
95C3×C3⋊Dic3—41no—
96C32×Dic3—41no—
97C2×C3×SD16—34no—
98C2×C3×SD16—91no—
99#944—94no—
100#944—91no—
101C2×#215—94no—
102C2×#215—91no—
103#940—94no—
104#940—91no—
105#940—184no—
106C2×#214—34no—
107C2×#214C3⋊D311yescharacteristic, complemented
108C2×#204—94no—
109C2×#204—91no—
110C8×Dic3—94no—
111C8×Dic3—91no—
112C23×C32—28no—
113C23×C32—24no—
114C23×C32—21no—
115C22×C3⋊D3—68no—
116C22×C3⋊D3—64no—
117C22×C3⋊D3—61no—
118C32×Q8—61no—
119C32×Q8—61no—
120C32×D4—21no—
121C32×D4—21no—
122C32×D4—24no—
123C32×D4—24no—
124C32×D4—44no—
125C32×D4—44no—
126C2×C32×C4—24no—
127C2×C32×C4—61no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130#638—124no—
131#638—1216no—
132#638—1232no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—32no—
135C2×C3⋊Dic3—38no—
136C2×C3⋊Dic3—68no—
137C4⋊(C3⋊D3)—61no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—64no—
140C4⋊(C3⋊D3)—64no—
141C4⋊(C3⋊D3)—124no—
142C4⋊(C3⋊D3)—124no—
143#634—21no—
144#634—21no—
145C2×C3×Dic3—24no—
146C2×C3×Dic3—64no—
147C3×Dic6—24no—
148C3×Dic6—24no—
149C3×Dic6—64no—
150C3×Dic6—64no—
151C32×C8—32no—
152C3⋊(C3⋊C8)#6712yescomplemented
153C3×C3⋊C8—38no—
154C3×C3⋊C8#6718yescomplemented
155#492—271noSylow
156C2×C33—81no—
157C2×C33#12711yescharacteristic
158C2×C33C2×SD1612yes—
159C3⋊(C3⋊D3)—81no—
160C2×C3×Q8—34no—
161C2×C3×Q8—91no—
162C2×C3×D4—24no—
163C2×C3×D4D3214yes—
164C2×C3×D4C2×C3⋊D311yescharacteristic
165C2×#67—188no—
166C2×#67—182no—
167C2×#67—1816no—
168C2×D12—94no—
169C2×D12—91no—
170C2×D12—184no—
171C2×Dic6—34no—
172C2×Dic6C2×C3⋊D311yescharacteristic
173C3×SD16—68no—
174C3×SD16—182no—
175C2×C3×C8—34no—
176C2×C3×C8—91no—
177#215—188no—
178#215—182no—
179#214—22no—
180#214—68no—
181Dic3⋊C4—182no—
182Dic3⋊C4—188no—
183C4×Dic3—94no—
184C4×Dic3—91no—
185C4×Dic3—184no—
186C2×C3⋊C8—34no—
187C2×C3⋊C8C2×C3⋊D311yescharacteristic
188#204—188no—
189#204—182no—
190C22×C32—24no—
191C22×C32—416no—
192C22×C32—48no—
193C22×C32—48no—
194C22×C32—42no—
195C22×C32—44no—
196C22×C32—41no—
197C22×C32D3×D414yes—
198C22×C32D3×D411yescharacteristic
199C2×C3⋊D3—128no—
200C2×C3⋊D3—1216no—
201C2×C3⋊D3—128no—
202C2×C3⋊D3—122no—
203C2×C3⋊D3—124no—
204C2×C3⋊D3—121no—
205C32×C4—24no—
206C32×C4—24no—
207C32×C4—121no—
208C32×C4D3×D414yes—
209C32×C4D3×D411yescharacteristic
210C32×C4C2×#6714yes—
211C32×C4C2×#6711yescharacteristic
212C3⋊Dic3—34no—
213C3⋊Dic3—316no—
214C3⋊Dic3—41no—
215C3⋊Dic3—616no—
216C3×Dic3—44no—
217C3×Dic3—124no—
218C2×SD16—31no—
219C2×SD16—91no—
220C2×SD16—271no—
221C2×SD16—271no—
222C4⋊Q8—271no—
223#127—271no—
224C4×C8—271no—
225C33#49211yescharacteristic, complemented, Sylow
226C23×C3—28no—
227C23×C3—24no—
228C23×C3—21no—
229C22×D3—188no—
230C22×D3—184no—
231C22×D3—181no—
232C3×Q8—64no—
233C3×Q8—64no—
234C3×Q8—181no—
235C3×Q8—181no—
236C3×D4—24no—
237C3×D4—24no—
238C3×D4—21no—
239C3×D4—21no—
240C3×D4—44no—
241C3×D4—44no—
242C2×C3×C4—24no—
243C2×C3×C4—64no—
244C2×C3×C4—181no—
245C2×C3×C4C2×D3214yes—
246C2×C3×C4C22×C3⋊D311yescharacteristic
247#67—364no—
248#67—3616no—
249#67—3632no—
250C2×Dic3—21no—
251C2×Dic3—64no—
252C2×Dic3—98no—
253C2×Dic3—92no—
254C2×Dic3—188no—
255D12—184no—
256D12—184no—
257D12—181no—
258D12—181no—
259D12—364no—
260D12—364no—
261Dic6—21no—
262Dic6—21no—
263Dic6—64no—
264Dic6—64no—
265C3×C8—38no—
266C3×C8—92no—
267C3⋊C8—38no—
268C3⋊C8#63812yescomplemented
269C2×C32—28no—
270C2×C32—24no—
271C2×C32—88no—
272C2×C32—84no—
273C2×C32—81no—
274C2×C32D3×SD1618yes—
275C2×C32D3×SD1612yes—
276C2×C32#94014yes—
277C2×C32#94011yescharacteristic
278C3⋊D3—248no—
279C3⋊D3—244no—
280C3⋊D3—241no—
281C2×Q8—31no—
282C2×Q8—91no—
283C2×D4—271no—
284C2×D4—542no—
285C2×D4D3×C3⋊D311yescharacteristic
286SD16—62no—
287SD16—182no—
288SD16—542no—
289SD16—542no—
290C2×C8—31no—
291C2×C8—91no—
292C4⋊C4—542no—
293C42—271no—
294C22×C3—24no—
295C22×C3—416no—
296C22×C3—48no—
297C22×C3—42no—
298C22×C3—41no—
299C22×C3—48no—
300C22×C3—44no—
301C22×C3#381114yes—
302C22×C3D4×C3⋊D311yescharacteristic
303C2×D3—368no—
304C2×D3—3616no—
305C2×D3—362no—
306C2×D3—361no—
307C2×D3—368no—
308C2×D3—364no—
309C3×C4—24no—
310C3×C4—24no—
311C3×C4—124no—
312C3×C4—361no—
313C3×C4D3×#6714yes—
314C3×C4D3×#6714yes—
315C3×C4D4×C3⋊D311yescharacteristic
316C3×C4C2×#63811yescharacteristic
317Dic3—41no—
318Dic3—94no—
319Dic3—916no—
320Dic3—124no—
321Dic3—1816no—
322C32—24no—
323C32#517514yescomplemented
324C32#517511yescharacteristic, complemented
325C23—21no—
326C23—541no—
327Q8—61no—
328Q8—61no—
329Q8—181no—
330Q8—181no—
331D4—21no—
332D4—21no—
333D4—541no—
334D4—541no—
335D4—1084no—
336C2×C4—61no—
337C2×C4—181no—
338C2×C4—272no—
339C2×C4C2×D3×C3⋊D311yescharacteristic
340C8—32no—
341C8—92no—
342C2×C3—28no—
343C2×C3—24no—
344C2×C3—88no—
345C2×C3—84no—
346C2×C3—81no—
347C2×C3#6393018yes—
348C2×C3#6397914yes—
349C2×C3SD16×C3⋊D312yes—
350C2×C3#6414911yescharacteristic
351D3—728no—
352D3—724no—
353D3—721no—
354C22—42no—
355C22—41no—
356C22—1082no—
357C22—1081no—
358C22#8903911yescharacteristic
359C4—121no—
360C4—274no—
361C4—361no—
362C4D3×#63811yescharacteristic
363C4C3⋊D3×#6711yescharacteristic
364C3—24no—
365C3#1059149214yescomplemented
366C3#1059287411yescharacteristic, complemented
367C2—81no—
368C2—2161no—
369C2#1172355312yes—
370C2#1172363311yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radical#11723981C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#492
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3754C2semidirect
3. C2×C32×#214C2semidirect
4. #11723451C2semidirect
5. C2×#88797C2semidirect
6. C2×#88795C2semidirect
7. #11722018C2semidirect
29. C2×#3754D3semidirect
32. C2×C3×#214D3semidirect
53. C3×C3⋊(C3⋊C8)D4semidirect
54. C32×C3⋊C8D4semidirect
94. C3⋊(C3⋊Dic3)SD16semidirect
107. C2×#214C3⋊D3semidirect
152. C3⋊(C3⋊C8)#67semidirect
154. C3×C3⋊C8#67semidirect
225. C33#492semidirect
268. C3⋊C8#638semidirect
323. C32#5175semidirect
324. C32#5175semidirect
365. C3#10591492semidirect
366. C3#10592874semidirect

Maximal subgroups

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

subgroupindex
#117239812
C2×C3×#37542
C2×C32×#2142
#117234512
C2×#887972
C2×#887952
#117220182
#105928743
#105914923