Finite Group Databasebeta

← #13161126

Subgroups — #13161126

Order 1728.

1140 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1140 conjugacy classes of subgroups — too many to draw. Only its 196 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×C3×#3832C211yesmaximal, characteristic
2C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#88787C211yesmaximal, characteristic, complemented
4#11722205C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic, complemented
8#10592502—31nomaximal
9#10591550—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×#3832—22no—
12C3×#3832C414yescomplemented
13C2×C3×C4⋊(C3⋊D3)—21no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3—21no—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×Dic3×C3⋊D3—21no—
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C32×C4⋊Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88807—21no—
23#88787—22no—
24C2×#3832D311yescharacteristic
25C2×C3×#239—34no—
26C2×#3743—31no—
27C4⋊C4×C3⋊D3—31no—
28D3×C4⋊Dic3—34no—
29C2×#3711—34no—
30C32×#103—31no—
31#63662—31no—
32#63662—34no—
33#63662—64no—
34#63648—31no—
35#63636—31no—
36C3×#835D314yescomplemented
37#63563—34no—
38#63562—34no—
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3D411yescharacteristic
41C33×Q8—21no—
42C33×Q8C2×C412yescomplemented
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)—21no—
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×C4⋊(C3⋊D3)—24no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)C2×C412yescomplemented
50C3×C4×C3⋊D3—24no—
51C3×C4×C3⋊D3C2×C412yes—
52C3×C4×C3⋊D3D412yes—
53C2×C32×Dic3—21no—
54Dic3×C3⋊D3—42no—
55C3⋊(C3⋊(C3⋊C8))—21no—
56C32×C3⋊C8—21no—
57#5232—91no—
58#4803—94no—
59C2×C32×Q8—24no—
60C2×C32×Q8C2×D314yes—
61C2×C32×Q8C2×D311yescharacteristic
62#3832—22no—
63#3832Dic314yescomplemented
64C2×C4⋊(C3⋊D3)—21no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C4×C3⋊D3—21no—
67C2×C4×C3⋊D3—61no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C3×#239—316no—
70C3×#239—68no—
71C2×C3×D12—34no—
72C2×C3×D12—64no—
73C2×C3×C4×D3—34no—
74C2×C3×C4×D3—64no—
75C2×D3×Dic3—64no—
76C2×C32×C8—31no—
77C32×C4⋊C4—31no—
78C4⋊(C3⋊Dic3)—31no—
79C4⋊(C3⋊Dic3)—34no—
80C4⋊(C3⋊Dic3)—64no—
81#3750—61no—
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—34no—
84C2×C3⋊(C3⋊C8)—64no—
85#3743—62no—
86C3×C4⋊Dic3C2×D314yes—
87C2×C3×C3⋊C8C2×D314yes—
88Dic3⋊Dic3—64no—
89#3711—68no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—21no—
93C2×C3×C3⋊D3—22no—
94C2×C3×C3⋊D3—41no—
95C2×C3×C3⋊D3—41no—
96C2×C3×C3⋊D3C22⋊C412yes—
97C33×C4—22no—
98C33×C4C22×C411yescharacteristic
99C33×C4C2×D411yescharacteristic
100C3⋊(C3⋊Dic3)—41no—
101C3×C3⋊Dic3C22⋊C412yes—
102C32×Dic3—41no—
103C2×C3×#41—91no—
104C2×#239—34no—
105C2×C4⋊Dic3—91no—
106C2×#208—91no—
107C2×ΓL(1,𝔽25)—94no—
108D3×C4⋊C4—94no—
109C3×#103—34no—
110#835—94no—
111#835—91no—
112#835—184no—
113#835C3⋊D311yescharacteristic, complemented
114#832—91no—
115#832—91no—
116#821—94no—
117#809—94no—
118C22×C3⋊D3—21no—
119C22×C3⋊D3#6711yescharacteristic
120C22×C3×D3—34no—
121C22×C3×D3—64no—
122C32×Q8—21no—
123C32×Q8—28no—
124C32×Q8—24no—
125C32×Q8—44no—
126C32×Q8C4×D318yescomplemented
127C32×Q8C2×Dic312yescomplemented
128C2×C32×C4—21no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4—44no—
132C2×C32×C4—61no—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3—61no—
136C2×C3⋊Dic3—64no—
137C2×C3⋊Dic3—124no—
138C2×C3⋊Dic3#6711yescharacteristic
139C4⋊(C3⋊D3)—24no—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)C2×Dic312yescomplemented
142C4×C3⋊D3—24no—
143C4×C3⋊D3—122no—
144C4×C3⋊D3C2×Dic312yes—
145C4×C3⋊D3#6712yes—
146C2×C3×Dic3—24no—
147C2×C3×Dic3—34no—
148C3×D12—38no—
149C3×D12—616no—
150C3×D12—64no—
151C3×C4×D3—38no—
152C3×C4×D3—38no—
153C3×C4×D3—616no—
154D3×Dic3—128no—
155C32×C8—61no—
156C3⋊(C3⋊C8)—61no—
157C3⋊(C3⋊C8)—64no—
158C3⋊(C3⋊C8)—124no—
159C3×C3⋊C8—24no—
160#417—271noSylow
161C2×C33C2×C22⋊C411yescharacteristic
162C2×C33AGL(1,ℤ/8)11yescharacteristic
163C2×C33#13711yescharacteristic
164C3×C3⋊D3—22no—
165C3×C3⋊D3—42no—
166C3×#41—94no—
167C3×#41—182no—
168C2×C3×Q8—24no—
169C2×C3×Q8D3214yes—
170C2×C3×Q8C2×C3⋊D311yescharacteristic
171C2×C3×D4—91no—
172C2×C3×D4—181no—
173C22×C3×C4—91no—
174C22×C3×C4—181no—
175C22×Dic3—181no—
176#239—316no—
177#239—68no—
178C2×D12—34no—
179C2×D12—64no—
180C2×C4×D3—34no—
181C2×C4×D3—64no—
182C2×C4×D3—184no—
183C2×C3×C8—34no—
184C3×C4⋊C4—34no—
185C4⋊Dic3—94no—
186C4⋊Dic3—91no—
187C4⋊Dic3—184no—
188C4⋊Dic3—181no—
189C4⋊Dic3C2×C3⋊D311yescharacteristic
190Dic3⋊C4—184no—
191#208—182no—
192C2×C3⋊C8—94no—
193C2×C3⋊C8—91no—
194C2×C3⋊C8—184no—
195C2×C3⋊C8C2×C3⋊D311yescharacteristic
196ΓL(1,𝔽25)—188no—
197C22×C32—24no—
198C22×C32D3×D414yes—
199C22×C32C2×#6711yescharacteristic
200C2×C3⋊D3—21no—
201C2×C3⋊D3—21no—
202C2×C3⋊D3—22no—
203C2×C3⋊D3—41no—
204C2×C3⋊D3—41no—
205C2×C3⋊D3#21712yes—
206C2×C3×D3—38no—
207C2×C3×D3—64no—
208C2×C3×D3—64no—
209C2×C3×D3—68no—
210C2×C3×D3—124no—
211C2×C3×D3—124no—
212C32×C4—22no—
213C32×C4—24no—
214C32×C4—24no—
215C32×C4—28no—
216C32×C4—48no—
217C32×C4—121no—
218C32×C4C2×C4×D314yes—
219C32×C4D3×D414yes—
220C32×C4C22×Dic311yescharacteristic
221C32×C4C2×#6711yescharacteristic
222C3⋊Dic3—121no—
223C3⋊Dic3—124no—
224C3⋊Dic3—244no—
225C3⋊Dic3#21712yes—
226C3×Dic3—38no—
227C3×Dic3—44no—
228C2×#41—91no—
229C2×M16—271no—
230C2×C4⋊C4—271no—
231#103—31no—
232#103—271no—
233#102—271no—
234#102—271no—
235C33#41711yescharacteristic, complemented, Sylow
236C23×C3—91no—
237C23×C3—181no—
238C22×D3—34no—
239C22×D3—64no—
240C3×Q8—24no—
241C3×Q8—21no—
242C3×Q8—28no—
243C3×Q8—44no—
244C3×Q8D3×Dic318yescomplemented
245C3×Q8C4×C3⋊D312yescomplemented
246C3×D4—92no—
247C3×D4—184no—
248C3×D4—181no—
249C2×C3×C4—24no—
250C2×C3×C4—21no—
251C2×C3×C4—24no—
252C2×C3×C4—44no—
253C2×C3×C4—64no—
254C2×C3×C4—91no—
255C2×C3×C4—92no—
256C2×C3×C4—92no—
257C2×C3×C4—184no—
258C2×C3×C4C2×D3214yes—
259C2×C3×C4C22×C3⋊D311yescharacteristic
260C2×Dic3—21no—
261C2×Dic3—34no—
262C2×Dic3—184no—
263C2×Dic3—181no—
264C2×Dic3—364no—
265C2×Dic3—362no—
266D12—38no—
267D12—616no—
268D12—64no—
269C4×D3—38no—
270C4×D3—38no—
271C4×D3—616no—
272C4×D3—368no—
273C3×C8—64no—
274C3⋊C8—21no—
275C3⋊C8—184no—
276C3⋊C8—181no—
277C3⋊C8—364no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—24no—
281C2×C32D3×C22⋊C414yes—
282C2×C32#91414yes—
283C2×C32#91814yes—
284C2×C32#94911yescharacteristic
285C2×C32#95111yescharacteristic
286C2×C32C2×#21711yescharacteristic
287C3⋊D3—22no—
288C3⋊D3—42no—
289C3×D3—68no—
290C3×D3—128no—
291#41—94no—
292#41—182no—
293C2×Q8D3×C3⋊D311yescharacteristic
294C2×D4—91no—
295C2×D4—181no—
296C22×C4—91no—
297C22×C4—181no—
298C22×C4—541no—
299M16—542no—
300C2×C8—31no—
301C2×C8—271no—
302C4⋊C4—31no—
303C4⋊C4—271no—
304C4⋊C4—541no—
305C22×C3—24no—
306C22×C3—92no—
307C22×C3—181no—
308C22×C3—181no—
309C22×C3—182no—
310C22×C3—361no—
311C22×C3—361no—
312C22×C3D3×#6714yes—
313C22×C3D4×C3⋊D311yescharacteristic
314C2×D3—38no—
315C2×D3—64no—
316C2×D3—64no—
317C2×D3—68no—
318C2×D3—124no—
319C2×D3—124no—
320C3×C4—22no—
321C3×C4—28no—
322C3×C4—24no—
323C3×C4—24no—
324C3×C4—48no—
325C3×C4—92no—
326C3×C4—124no—
327C3×C4C2×D3×Dic314yes—
328C3×C4D3×#6714yes—
329C3×C4C2×C4×C3⋊D311yescharacteristic
330C3×C4D4×C3⋊D311yescharacteristic
331Dic3—38no—
332Dic3—41no—
333Dic3—361no—
334Dic3—364no—
335Dic3—724no—
336C32—24no—
337C32#480314yescomplemented
338C32#523211yescharacteristic, complemented
339C23—91no—
340C23—181no—
341Q8—21no—
342Q8Dic3×C3⋊D312yescomplemented
343D4—92no—
344D4—184no—
345D4—181no—
346C2×C4—21no—
347C2×C4—61no—
348C2×C4—91no—
349C2×C4—92no—
350C2×C4—92no—
351C2×C4—184no—
352C2×C4—541no—
353C2×C4—1082no—
354C2×C4C2×D3×C3⋊D311yescharacteristic
355C8—61no—
356C8—541no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—24no—
360C2×C3—182no—
361C2×C3—362no—
362C2×C3#6393914yes—
363C2×C3#6394314yes—
364C2×C3D3×#21714yes—
365C2×C3C22⋊C4×C3⋊D311yescharacteristic
366C2×C3#6412311yescharacteristic
367C2×C3#6412711yescharacteristic
368D3—68no—
369D3—128no—
370C22—92no—
371C22—181no—
372C22—181no—
373C22—182no—
374C22—361no—
375C22—361no—
376C22C3⋊D3×#6711yescharacteristic
377C4—22no—
378C4—92no—
379C4—121no—
380C4—1081no—
381C4C2×Dic3×C3⋊D311yescharacteristic
382C4C3⋊D3×#6711yescharacteristic
383C3—24no—
384C3#1059155014yescomplemented
385C3#1059250211yescharacteristic, complemented
386C2—182no—
387C2—362no—
388C2#1172357011yescharacteristic
389C2#1172357611yescharacteristic
390C2C3⋊D3×#21711yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×#3832C2
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#417
LCSG ▷ 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
2. C3⋊D3×C4⋊Dic3C2semidirect
3. C2×#88787C2semidirect
4. #11722205C2semidirect
5. C32×#835C2semidirect
6. #11722034C2semidirect
7. #11722033C2semidirect
12. C3×#3832C4semidirect
36. C3×#835D3semidirect
42. C33×Q8C2×C4semidirect
49. C3×C4⋊(C3⋊D3)C2×C4semidirect
63. #3832Dic3semidirect
113. #835C3⋊D3semidirect
126. C32×Q8C4×D3semidirect
127. C32×Q8C2×Dic3semidirect
141. C4⋊(C3⋊D3)C2×Dic3semidirect
235. C33#417semidirect
244. C3×Q8D3×Dic3semidirect
245. C3×Q8C4×C3⋊D3semidirect
337. C32#4803semidirect
338. C32#5232semidirect
342. Q8Dic3×C3⋊D3semidirect
384. C3#10591550semidirect
385. C3#10592502semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C3⋊D3×C4⋊Dic32
C2×#887872
#117222052
C32×#8352
#117220342
#117220332
#105925023
#105915503