Finite Group Databasebeta

← #13161073

Subgroups — #13161073

Order 1728.

1296 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1296 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
1#11723981C211yesmaximal, characteristic, complemented
2C2×C3×#3753C211yesmaximal, characteristic, complemented
3C2×C32×#214C211yesmaximal, characteristic, complemented
4#11723436C211yesmaximal, characteristic, complemented
5C2×#88795C211yesmaximal, characteristic, complemented
6C2×#88791C211yesmaximal, characteristic, complemented
7#11722021C211yesmaximal, characteristic, complemented
8#10592875—31nomaximal
9#10591475—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6312—22no—
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3753—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#214—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20#88803—22no—
21#88795—22no—
22#88791—22no—
23C2×C32×SD16—31no—
24#64156—31no—
25C2×#3755—31no—
26#64149—31no—
27#64149—34no—
28#64149—64no—
29C2×#3753D311yescharacteristic, complemented
30C2×#3745—31no—
31C2×C3×#214D314yescomplemented
32C2×C3×#213—34no—
33#63863—34no—
34C2×#3717—34no—
35C2×#3714—34no—
36#63642—31no—
37#63555—34no—
38C23×C33—21no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C3×C3⋊D3—21no—
41C33×D4—21no—
42C33×D4—21no—
43C2×C33×C4C2311yescharacteristic
44#6312—44no—
45C2×C3⋊(C3⋊Dic3)D412yes—
46C4⋊(C3⋊(C3⋊D3))—21no—
47C4⋊(C3⋊(C3⋊D3))—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C2×C32×Dic3—21no—
51C32×Dic6—21no—
52C32×Dic6—21no—
53C3×C3⋊(C3⋊C8)D412yescomplemented
54C32×C3⋊C8D412yescomplemented
55#5176—94no—
56#5154—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×C4⋊(C3⋊D3)C2×D311yescharacteristic
68C2×C3×D12—34no—
69C2×C3×Dic6C2×D314yes—
70C32×SD16—62no—
71C2×C32×C8—31no—
72#3755—62no—
73#3753—22no—
74#3752—62no—
75C4×C3⋊Dic3—31no—
76C4×C3⋊Dic3—34no—
77C4×C3⋊Dic3—64no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79#3745—62no—
80C3×#214—28no—
81C3×#213—68no—
82C2×C3×C3⋊C8—34no—
83C2×C3×C3⋊C8C2×D314yes—
84(C2×D3)⋊Dic3—68no—
85#3717—68no—
86#3714—68no—
87C22×C33—41no—
88C22×C33—42no—
89C22×C33C2×D411yescharacteristic
90C2×C3⋊(C3⋊D3)—41no—
91C2×C3⋊(C3⋊D3)—42no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3—42no—
94C33×C4C2×D411yescharacteristic
95C33×C4C2×D411yescharacteristic
96C3⋊(C3⋊Dic3)—22no—
97C32×Dic3—41no—
98C2×C3×SD16—34no—
99C2×C3×D8—91no—
100#947—94no—
101C2×#215—94no—
102#940—94no—
103#940—91no—
104#940—184no—
105#936—91no—
106C2×#214C3⋊D311yescharacteristic, complemented
107C2×#213—34no—
108C2×#213—91no—
109C2×D24—94no—
110C2×#204—91no—
111#815—94no—
112#815—91no—
113C23×C32—28no—
114C23×C32—24no—
115C23×C32—21no—
116C22×C3⋊D3—21no—
117C22×C3⋊D3—68no—
118C22×C3⋊D3—64no—
119C22×C3⋊D3—61no—
120C22×C3×D3—64no—
121C32×Q8—61no—
122C32×Q8—61no—
123C32×D4—21no—
124C32×D4—21no—
125C32×D4—24no—
126C32×D4—24no—
127C32×D4—44no—
128C32×D4—44no—
129C2×C32×C4—24no—
130C2×C32×C4—61no—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C22×D311yescharacteristic
133#638—1232no—
134#638—124no—
135#638—1216no—
136C2×C3⋊Dic3—32no—
137C2×C3⋊Dic3—38no—
138C2×C3⋊Dic3—68no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)—61no—
142C4⋊(C3⋊D3)—61no—
143C4⋊(C3⋊D3)—64no—
144C4⋊(C3⋊D3)—64no—
145C4⋊(C3⋊D3)—124no—
146C4⋊(C3⋊D3)—124no—
147C2×C3×Dic3—24no—
148C3×D12—64no—
149C3×D12—64no—
150C3×Dic6—24no—
151C3×Dic6—24no—
152C32×C8—32no—
153C3⋊(C3⋊C8)#6712yescomplemented
154C3×C3⋊C8—38no—
155C3×C3⋊C8#6718yescomplemented
156#496—271noSylow
157C2×C33—81no—
158C2×C33#12711yescharacteristic
159C2×C33AGL(1,ℤ/8)12yes—
160C3⋊(C3⋊D3)—81no—
161C3×C3⋊D3—81no—
162C2×C3×Q8—34no—
163C2×C3×D4—24no—
164C2×C3×D4—91no—
165C2×C3×D4D3214yes—
166C2×C3×D4C2×C3⋊D311yescharacteristic
167C2×#67—188no—
168C2×#67—182no—
169C2×#67—1816no—
170C2×D12—34no—
171C2×D12—94no—
172C2×D12—91no—
173C2×D12—184no—
174C2×Dic6C2×C3⋊D311yescharacteristic
175C3×SD16—68no—
176C3×D8—182no—
177C2×C3×C8—34no—
178C2×C3×C8—91no—
179#217—182no—
180#215—188no—
181#214—22no—
182#213—68no—
183#213—182no—
184(C2×D3)⋊C4—188no—
185C4×Dic3—94no—
186C4×Dic3—91no—
187C4×Dic3—184no—
188C2×C3⋊C8—34no—
189C2×C3⋊C8C2×C3⋊D311yescharacteristic
190D24—188no—
191#204—182no—
192C22×C32—24no—
193C22×C32—416no—
194C22×C32—48no—
195C22×C32—42no—
196C22×C32—48no—
197C22×C32—44no—
198C22×C32—41no—
199C22×C32D3×D414yes—
200C22×C32D3×D411yescharacteristic
201C2×C3⋊D3—42no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—128no—
204C2×C3⋊D3—1216no—
205C2×C3⋊D3—122no—
206C2×C3⋊D3—128no—
207C2×C3⋊D3—124no—
208C2×C3⋊D3—121no—
209C2×C3×D3—128no—
210C2×C3×D3—124no—
211C32×C4—24no—
212C32×C4—24no—
213C32×C4—121no—
214C32×C4D3×D414yes—
215C32×C4D3×D411yescharacteristic
216C32×C4C2×#6714yes—
217C32×C4C2×#6711yescharacteristic
218C3⋊Dic3—616no—
219C3⋊Dic3—62no—
220C3⋊Dic3—68no—
221C3×Dic3—44no—
222C2×SD16—31no—
223C2×SD16—271no—
224C2×D8—91no—
225C2×D8—271no—
226#127—271no—
227#124—271no—
228#97—271no—
229C33#49611yescharacteristic, complemented, Sylow
230C23×C3—28no—
231C23×C3—24no—
232C23×C3—21no—
233C23×C3—181no—
234C22×D3—64no—
235C22×D3—188no—
236C22×D3—184no—
237C22×D3—181no—
238C3×Q8—64no—
239C3×Q8—64no—
240C3×D4—24no—
241C3×D4—24no—
242C3×D4—21no—
243C3×D4—21no—
244C3×D4—44no—
245C3×D4—44no—
246C3×D4—181no—
247C3×D4—181no—
248C2×C3×C4—24no—
249C2×C3×C4—64no—
250C2×C3×C4C2×D3214yes—
251C2×C3×C4C22×C3⋊D311yescharacteristic
252#67—3632no—
253#67—364no—
254#67—3616no—
255C2×Dic3—21no—
256C2×Dic3—98no—
257C2×Dic3—92no—
258C2×Dic3—188no—
259D12—64no—
260D12—64no—
261D12—184no—
262D12—184no—
263D12—181no—
264D12—181no—
265D12—364no—
266D12—364no—
267Dic6—21no—
268Dic6—21no—
269C3×C8—38no—
270C3×C8—92no—
271C3⋊C8—38no—
272C3⋊C8#63812yescomplemented
273C2×C32—28no—
274C2×C32—24no—
275C2×C32—88no—
276C2×C32—81no—
277C2×C32—84no—
278C2×C32#91112yes—
279C2×C32#91418yes—
280C2×C32#94014yes—
281C2×C32#94011yescharacteristic
282C3⋊D3—81no—
283C3⋊D3—248no—
284C3⋊D3—241no—
285C3⋊D3—244no—
286C3×D3—244no—
287C2×Q8—31no—
288C2×D4—91no—
289C2×D4—271no—
290C2×D4—542no—
291C2×D4D3×C3⋊D311yescharacteristic
292SD16—62no—
293SD16—542no—
294D8—182no—
295D8—542no—
296C2×C8—31no—
297C2×C8—91no—
298C22⋊C4—542no—
299C42—271no—
300C22×C3—24no—
301C22×C3—416no—
302C22×C3—48no—
303C22×C3—42no—
304C22×C3—41no—
305C22×C3—48no—
306C22×C3—44no—
307C22×C3—362no—
308C22×C3—361no—
309C22×C3#381114yes—
310C22×C3D4×C3⋊D311yescharacteristic
311C2×D3—128no—
312C2×D3—124no—
313C2×D3—368no—
314C2×D3—3616no—
315C2×D3—362no—
316C2×D3—361no—
317C2×D3—368no—
318C2×D3—364no—
319C3×C4—24no—
320C3×C4—24no—
321C3×C4—124no—
322C3×C4D3×#6714yes—
323C3×C4D3×#6714yes—
324C3×C4D4×C3⋊D311yescharacteristic
325C3×C4C2×#63811yescharacteristic
326Dic3—41no—
327Dic3—1816no—
328Dic3—182no—
329Dic3—188no—
330C32—24no—
331C32#515411yescharacteristic, complemented
332C32#517614yescomplemented
333C23—21no—
334C23—181no—
335C23—541no—
336Q8—61no—
337Q8—61no—
338D4—21no—
339D4—21no—
340D4—181no—
341D4—181no—
342D4—541no—
343D4—541no—
344D4—1084no—
345C2×C4—61no—
346C2×C4—272no—
347C2×C4C2×D3×C3⋊D311yescharacteristic
348C8—32no—
349C8—92no—
350C2×C3—28no—
351C2×C3—24no—
352C2×C3—88no—
353C2×C3—81no—
354C2×C3—84no—
355C2×C3—721no—
356C2×C3#6393718yes—
357C2×C3#6397914yes—
358C2×C3#6412312yes—
359C2×C3#6414911yescharacteristic
360D3—244no—
361D3—728no—
362D3—721no—
363D3—724no—
364C22—42no—
365C22—41no—
366C22—362no—
367C22—361no—
368C22—1082no—
369C22—1081no—
370C22#8903911yescharacteristic
371C4—121no—
372C4—542no—
373C4D3×#63811yescharacteristic
374C4C3⋊D3×#6711yescharacteristic
375C3—24no—
376C3#1059147514yescomplemented
377C3#1059287511yescharacteristic, complemented
378C2—81no—
379C2—721no—
380C2—2161no—
381C2#1172356612yes—
382C2#1172363311yescharacteristic
subgroupG/H
centerC22#89039
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. #11723981C2semidirect
2. C2×C3×#3753C2semidirect
3. C2×C32×#214C2semidirect
4. #11723436C2semidirect
5. C2×#88795C2semidirect
6. C2×#88791C2semidirect
7. #11722021C2semidirect
29. C2×#3753D3semidirect
31. C2×C3×#214D3semidirect
53. C3×C3⋊(C3⋊C8)D4semidirect
54. C32×C3⋊C8D4semidirect
106. C2×#214C3⋊D3semidirect
153. C3⋊(C3⋊C8)#67semidirect
155. C3×C3⋊C8#67semidirect
229. C33#496semidirect
272. C3⋊C8#638semidirect
331. C32#5154semidirect
332. C32#5176semidirect
376. C3#10591475semidirect
377. C3#10592875semidirect

Maximal subgroups

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

subgroupindex
#117239812
C2×C3×#37532
C2×C32×#2142
#117234362
C2×#887952
C2×#887912
#117220212
#105928753
#105914753