Finite Group Databasebeta

← #13161013

Subgroups — #13161013

Order 1728.

1168 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 158 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×#88860C211yesmaximal, characteristic, complemented
2C3×#64145C211yesmaximal, characteristic, complemented
3C2×C32×#213C211yesmaximal, characteristic, complemented
4#11723524C211yesmaximal, characteristic, complemented
5C2×#88797C211yesmaximal, characteristic, complemented
6C2×#88792C211yesmaximal, characteristic, complemented
7#11722020C211yesmaximal, characteristic, complemented
8#10592853—31nomaximal
9#10591483—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6291—22no—
15#88860—22no—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3757—22no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C32×#213—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21#88797—22no—
22#88792—22no—
23C2×C32×D8—31no—
24#64149—31no—
25#64145D311yescharacteristic, complemented
26C2×#3754—31no—
27C2×#3753—31no—
28C2×#3753—34no—
29C2×#3753—64no—
30C2×#3744—31no—
31C3×#936—34no—
32C2×C3×#213D314yescomplemented
33#63911—34no—
34C2×#3717—34no—
35C2×#3715—34no—
36#63642—31no—
37#63554—34no—
38C23×C33—21no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C32×D3—21no—
41C33×D4—21no—
42C33×D4—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#5154—94no—
56#5058—91no—
57C2×C32×D4—24no—
58C2×C32×D4—31no—
59C2×C32×D4C2×D314yes—
60C2×C32×D4C2×D311yescharacteristic
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×D8—62no—
70C2×C32×C8—31no—
71#3757—22no—
72#3754—62no—
73#3753—62no—
74#3753—68no—
75#3753—616no—
76C4×C3⋊Dic3C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—34no—
79C2×C3⋊(C3⋊C8)—64no—
80#3744—62no—
81C3×#217—68no—
82C3×#213—28no—
83C3×C4×Dic3—34no—
84C2×C3×C3⋊C8C2×D314yes—
85#3717—68no—
86#3715—68no—
87C22×C33—41no—
88C22×C33—42no—
89C22×C33C2×D411yescharacteristic
90C2×C3⋊(C3⋊D3)—41no—
91C2×C3⋊(C3⋊D3)—42no—
92C2×C32×D3—41no—
93C2×C32×D3—42no—
94C33×C4C2×D411yescharacteristic
95C33×C4C2×D411yescharacteristic
96C3×C3⋊Dic3—22no—
97C3×C3⋊Dic3—41no—
98C2×C3×D8—34no—
99C3×#124—91no—
100C2×#215—91no—
101C2×#215—91no—
102#940—94no—
103#936—34no—
104C2×#214—94no—
105C2×#213—94no—
106C2×#213—91no—
107C2×#213—184no—
108C2×#213C3⋊D311yescharacteristic, complemented
109C2×#204—94no—
110#874—91no—
111#815—94no—
112#803—91no—
113C23×C32—21no—
114C23×C32—24no—
115C23×C32—44no—
116C23×C32—61no—
117C22×C3⋊D3—61no—
118C22×C3⋊D3—64no—
119C22×C3⋊D3—124no—
120C22×C3×D3—24no—
121C32×D4—28no—
122C32×D4—24no—
123C32×D4—21no—
124C32×D4—21no—
125C32×D4—24no—
126C32×D4—44no—
127C32×D4—61no—
128C32×D4—61no—
129C2×C32×C4—24no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D311yescharacteristic
132#638—124no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3#6712yes—
135C4⋊(C3⋊D3)—68no—
136C4⋊(C3⋊D3)—61no—
137C4⋊(C3⋊D3)—64no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—64no—
140C4⋊(C3⋊D3)—124no—
141#634—21no—
142#634—21no—
143C2×C3×Dic3—38no—
144C2×C3×Dic3—64no—
145C3×D12—24no—
146C3×D12—24no—
147C3×Dic6—64no—
148C3×Dic6—64no—
149#626—1216no—
150C32×C8—32no—
151C3⋊(C3⋊C8)—32no—
152C3⋊(C3⋊C8)—38no—
153C3⋊(C3⋊C8)—68no—
154C3×C3⋊C8#6718yescomplemented
155#496—271noSylow
156C2×C33—81no—
157C2×C33#12711yescharacteristic
158C2×C33AGL(1,ℤ/8)12yes—
159C3⋊(C3⋊D3)—81no—
160C32×D3—81no—
161C2×C3×Q8—91no—
162C2×C3×D4—24no—
163C2×C3×D4—34no—
164C2×C3×D4D3214yes—
165C2×C3×D4C2×C3⋊D311yescharacteristic
166C2×#67—188no—
167C2×D12—94no—
168C2×D12—91no—
169C2×D12—184no—
170C2×D12—182no—
171C2×D12C2×C3⋊D311yescharacteristic
172C2×Dic6—34no—
173C3×D8—68no—
174C2×C3×C8—34no—
175C3×C22⋊C4—182no—
176C3×C42—91no—
177#217—68no—
178#215—182no—
179#215—182no—
180#214—188no—
181#213—22no—
182#213—188no—
183#213—182no—
184#213—1816no—
185C4×Dic3—34no—
186C2×C3⋊C8—94no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—184no—
189C2×C3⋊C8C2×C3⋊D311yescharacteristic
190#204—188no—
191C22×C32—24no—
192C22×C32—48no—
193C22×C32—44no—
194C22×C32—41no—
195C22×C32—42no—
196C22×C32—48no—
197C22×C32—88no—
198C22×C32—121no—
199C22×C32—122no—
200C22×C32D3×D414yes—
201C22×C32C2×#6711yescharacteristic
202C2×C3⋊D3—128no—
203C2×C3⋊D3—124no—
204C2×C3⋊D3—121no—
205C2×C3⋊D3—122no—
206C2×C3⋊D3—128no—
207C2×C3⋊D3—248no—
208C2×C3×D3—48no—
209C2×C3×D3—44no—
210C32×C4—24no—
211C32×C4—24no—
212C32×C4D3×D414yes—
213C32×C4D3×D411yescharacteristic
214C32×C4D3×D411yescharacteristic
215C32×C4C2×#6714yes—
216C3⋊Dic3—22no—
217C3⋊Dic3—41no—
218C3×Dic3—68no—
219C3×Dic3—124no—
220C2×SD16—271no—
221C2×SD16—271no—
222C2×D8—31no—
223C2×D8—271no—
224#127—271no—
225#124—91no—
226#97—271no—
227C33#49611yescharacteristic, complemented, Sylow
228C23×C3—24no—
229C23×C3—21no—
230C23×C3—44no—
231C23×C3—64no—
232C22×D3—21no—
233C22×D3—184no—
234C22×D3—181no—
235C22×D3—364no—
236C3×Q8—181no—
237C3×Q8—181no—
238C3×D4—28no—
239C3×D4—24no—
240C3×D4—24no—
241C3×D4—21no—
242C3×D4—21no—
243C3×D4—44no—
244C3×D4—64no—
245C3×D4—64no—
246C2×C3×C4—24no—
247C2×C3×C4—92no—
248C2×C3×C4—181no—
249C2×C3×C4C2×D3214yes—
250C2×C3×C4C22×C3⋊D311yescharacteristic
251#67—3616no—
252C2×Dic3—38no—
253C2×Dic3—64no—
254D12—21no—
255D12—21no—
256D12—188no—
257D12—184no—
258D12—184no—
259D12—181no—
260D12—181no—
261D12—364no—
262D12—364no—
263Dic6—64no—
264Dic6—64no—
265C3×C8—38no—
266C3⋊C8—98no—
267C3⋊C8—92no—
268C3⋊C8—188no—
269C3⋊C8#63812yescomplemented
270C2×C32—28no—
271C2×C32—24no—
272C2×C32—88no—
273C2×C32—84no—
274C2×C32—81no—
275C2×C32—241no—
276C2×C32#91118yes—
277C2×C32#94014yes—
278C2×C32#94011yescharacteristic
279C2×C32#94912yes—
280C3⋊D3—248no—
281C3⋊D3—244no—
282C3⋊D3—241no—
283C3×D3—84no—
284C2×Q8—91no—
285C2×D4—31no—
286C2×D4—271no—
287C2×D4—542no—
288C2×D4D3×C3⋊D311yescharacteristic
289SD16—542no—
290SD16—542no—
291D8—62no—
292D8—542no—
293C2×C8—31no—
294C2×C8—271no—
295C22⋊C4—182no—
296C42—91no—
297C22×C3—24no—
298C22×C3—48no—
299C22×C3—42no—
300C22×C3—41no—
301C22×C3—48no—
302C22×C3—44no—
303C22×C3—88no—
304C22×C3—128no—
305C22×C3—124no—
306C22×C3D3×#6714yes—
307C22×C3D4×C3⋊D311yescharacteristic
308C2×D3—42no—
309C2×D3—41no—
310C2×D3—368no—
311C2×D3—362no—
312C2×D3—361no—
313C2×D3—368no—
314C2×D3—364no—
315C2×D3—728no—
316C3×C4—24no—
317C3×C4—24no—
318C3×C4—182no—
319C3×C4—361no—
320C3×C4D3×#6714yes—
321C3×C4#381114yes—
322C3×C4D4×C3⋊D311yescharacteristic
323C3×C4C2×#63811yescharacteristic
324Dic3—68no—
325Dic3—124no—
326C32—24no—
327C32#505811yescharacteristic, complemented
328C32#515414yescomplemented
329C23—21no—
330C23—61no—
331C23—541no—
332Q8—181no—
333Q8—181no—
334D4—21no—
335D4—21no—
336D4—61no—
337D4—61no—
338D4—541no—
339D4—541no—
340D4—1084no—
341C2×C4—92no—
342C2×C4—181no—
343C2×C4C2×D3×C3⋊D311yescharacteristic
344C8—32no—
345C8—272no—
346C2×C3—28no—
347C2×C3—24no—
348C2×C3—88no—
349C2×C3—81no—
350C2×C3—84no—
351C2×C3—244no—
352C2×C3#6392518yes—
353C2×C3#6397914yes—
354C2×C3#6412012yes—
355C2×C3#6414911yescharacteristic
356D3—81no—
357D3—728no—
358D3—721no—
359D3—724no—
360C22—42no—
361C22—41no—
362C22—122no—
363C22—121no—
364C22—1082no—
365C22—1081no—
366C22C3⋊D3×#6711yescharacteristic
367C4—182no—
368C4—361no—
369C4D3×#63811yescharacteristic
370C4#8903911yescharacteristic
371C3—24no—
372C3#1059148314yescomplemented
373C3#1059285311yescharacteristic, complemented
374C2—81no—
375C2—241no—
376C2—2161no—
377C2#1172354612yes—
378C2#1172363311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#496
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
1. C2×#88860C2semidirect
2. C3×#64145C2semidirect
3. C2×C32×#213C2semidirect
4. #11723524C2semidirect
5. C2×#88797C2semidirect
6. C2×#88792C2semidirect
7. #11722020C2semidirect
25. #64145D3semidirect
32. C2×C3×#213D3semidirect
53. C3⋊(C3⋊(C3⋊C8))D4semidirect
54. C32×C3⋊C8D4semidirect
108. C2×#213C3⋊D3semidirect
154. C3×C3⋊C8#67semidirect
227. C33#496semidirect
269. C3⋊C8#638semidirect
327. C32#5058semidirect
328. C32#5154semidirect
372. C3#10591483semidirect
373. C3#10592853semidirect

Maximal subgroups

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

subgroupindex
C2×#888602
C3×#641452
C2×C32×#2132
#117235242
C2×#887972
C2×#887922
#117220202
#105928533
#105914833