Finite Group Databasebeta

← #13161260

Subgroups — #13161260

Order 1728.

1116 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1116 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
2#11723404C211yesmaximal, characteristic, complemented
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
4#11722205C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic, complemented
8#10592503—31nomaximal
9#10591556—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
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88803—21no—
22Dic3×C3⋊Dic3—21no—
23C3⋊D3×C3⋊C8—22no—
24C2×#3832D311yescharacteristic
25C2×C3×#239—34no—
26C2×C8×C3⋊D3—31no—
27#64101—31no—
28#63841—34no—
29C2×D3×C3⋊C8—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×C412yes—
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×C412yes—
50C3×C4×C3⋊D3—24no—
51C3×C4×C3⋊D3C2×C412yes—
52C3×C4×C3⋊D3D412yes—
53C2×C32×Dic3—21no—
54C3⋊(C3⋊(C3⋊C8))—21no—
55C32×C3⋊C8—21no—
56#5233—91no—
57#4804—94no—
58C2×C32×Q8—24no—
59C2×C32×Q8C2×D314yes—
60C2×C32×Q8C2×D311yescharacteristic
61#3832—22no—
62#3832Dic314yescomplemented
63C2×C4⋊(C3⋊D3)—21no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C4×C3⋊D3—21no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C3×#239—316no—
68C3×#239—68no—
69C2×C3×D12—34no—
70C2×C3×D12—64no—
71C2×C3×C4×D3—34no—
72C2×C3×C4×D3—64no—
73C2×C32×C8—31no—
74C32×C4⋊C4—31no—
75#3752—61no—
76C4⋊(C3⋊Dic3)—31no—
77C4⋊(C3⋊Dic3)—34no—
78C4⋊(C3⋊Dic3)—64no—
79C4×C3⋊Dic3—61no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—34no—
82C2×C3⋊(C3⋊C8)—64no—
83C8×C3⋊D3—62no—
84C3×C4⋊Dic3C2×D314yes—
85C2×C3×C3⋊C8C2×D314yes—
86(C2×D3)⋊Dic3—64no—
87Dic32—64no—
88D3×C3⋊C8—68no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—22no—
93C2×C3×C3⋊D3—41no—
94C2×C3×C3⋊D3—41no—
95C2×C3×C3⋊D3C22⋊C412yes—
96C33×C4—22no—
97C33×C4C22×C411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3⋊(C3⋊Dic3)—41no—
100C3×C3⋊Dic3C22⋊C412yes—
101C32×Dic3—41no—
102C2×C3×#41—91no—
103C2×#239—34no—
104#926—91no—
105C22×C3⋊C8—91no—
106C2×D3×C8—94no—
107#892—94no—
108C3×#103—34no—
109#835—94no—
110#835—91no—
111#835—184no—
112#835C3⋊D311yescharacteristic, complemented
113#832—91no—
114#832—91no—
115#821—94no—
116#809—94no—
117C22×C3⋊D3—21no—
118C22×C3⋊D3#6711yescharacteristic
119C22×C3×D3—34no—
120C22×C3×D3—64no—
121C32×Q8—21no—
122C32×Q8—28no—
123C32×Q8—24no—
124C32×Q8—44no—
125C32×Q8C4×D318yes—
126C32×Q8C2×Dic312yes—
127C2×C32×C4—21no—
128C2×C32×C4—24no—
129C2×C32×C4—24no—
130C2×C32×C4—44no—
131C2×C32×C4—61no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134C2×C3⋊Dic3—61no—
135C2×C3⋊Dic3—64no—
136C2×C3⋊Dic3—124no—
137C2×C3⋊Dic3#6711yescharacteristic
138C4⋊(C3⋊D3)—24no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)C2×Dic312yes—
141C4×C3⋊D3—24no—
142C4×C3⋊D3C2×Dic312yes—
143C4×C3⋊D3#6712yes—
144C2×C3×Dic3—24no—
145C2×C3×Dic3—34no—
146C3×D12—38no—
147C3×D12—616no—
148C3×D12—64no—
149C3×C4×D3—38no—
150C3×C4×D3—38no—
151C3×C4×D3—616no—
152C32×C8—61no—
153C3⋊(C3⋊C8)—61no—
154C3⋊(C3⋊C8)—64no—
155C3⋊(C3⋊C8)—124no—
156C3×C3⋊C8—24no—
157#416—271noSylow
158C2×C33C2×C22⋊C411yescharacteristic
159C2×C33#13511yescharacteristic
160C2×C33#13511yescharacteristic
161C3×C3⋊D3—22no—
162C3×C3⋊D3—42no—
163C3×#41—94no—
164C3×#41—182no—
165C2×C3×Q8—24no—
166C2×C3×Q8D3214yes—
167C2×C3×Q8C2×C3⋊D311yescharacteristic
168C2×C3×D4—91no—
169C2×C3×D4—181no—
170C22×C3×C4—91no—
171C22×C3×C4—181no—
172#239—316no—
173#239—68no—
174C2×D12—34no—
175C2×D12—64no—
176C2×C4×D3—34no—
177C2×C4×D3—64no—
178C2×C3×C8—34no—
179C3×C4⋊C4—34no—
180#217—181no—
181(C2×D3)⋊C4—184no—
182C4⋊Dic3—94no—
183C4⋊Dic3—91no—
184C4⋊Dic3—184no—
185C4⋊Dic3C2×C3⋊D311yescharacteristic
186C4×Dic3—181no—
187C4×Dic3—184no—
188C2×C3⋊C8—94no—
189C2×C3⋊C8—91no—
190C2×C3⋊C8—184no—
191C2×C3⋊C8—182no—
192C2×C3⋊C8C2×C3⋊D311yescharacteristic
193D3×C8—188no—
194C22×C32—24no—
195C22×C32D3×D414yes—
196C22×C32C2×#6711yescharacteristic
197C2×C3⋊D3—21no—
198C2×C3⋊D3—21no—
199C2×C3⋊D3—22no—
200C2×C3⋊D3—41no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3#21712yes—
203C2×C3×D3—38no—
204C2×C3×D3—64no—
205C2×C3×D3—64no—
206C2×C3×D3—68no—
207C2×C3×D3—124no—
208C2×C3×D3—124no—
209C32×C4—22no—
210C32×C4—24no—
211C32×C4—24no—
212C32×C4—28no—
213C32×C4—48no—
214C32×C4—121no—
215C32×C4C2×C4×D314yes—
216C32×C4D3×D414yes—
217C32×C4C22×Dic311yescharacteristic
218C32×C4C2×#6711yescharacteristic
219C3⋊Dic3—121no—
220C3⋊Dic3—124no—
221C3⋊Dic3—244no—
222C3⋊Dic3#21712yes—
223C3×Dic3—38no—
224C3×Dic3—44no—
225C2×#41—91no—
226C22×C8—271no—
227#117—271no—
228#103—31no—
229#103—271no—
230#102—271no—
231#102—271no—
232C33#41611yescharacteristic, complemented, Sylow
233C23×C3—91no—
234C23×C3—181no—
235C22×D3—34no—
236C22×D3—64no—
237C3×Q8—24no—
238C3×Q8—21no—
239C3×Q8—28no—
240C3×Q8—44no—
241C3×Q8D3×Dic318yes—
242C3×Q8C4×C3⋊D312yes—
243C3×D4—92no—
244C3×D4—184no—
245C3×D4—181no—
246C2×C3×C4—24no—
247C2×C3×C4—21no—
248C2×C3×C4—24no—
249C2×C3×C4—44no—
250C2×C3×C4—64no—
251C2×C3×C4—91no—
252C2×C3×C4—92no—
253C2×C3×C4—92no—
254C2×C3×C4—184no—
255C2×C3×C4C2×D3214yes—
256C2×C3×C4C22×C3⋊D311yescharacteristic
257C2×Dic3—21no—
258C2×Dic3—34no—
259C2×Dic3—184no—
260C2×Dic3—181no—
261C2×Dic3—364no—
262D12—38no—
263D12—616no—
264D12—64no—
265C4×D3—38no—
266C4×D3—38no—
267C4×D3—616no—
268C3×C8—64no—
269C3⋊C8—21no—
270C3⋊C8—184no—
271C3⋊C8—181no—
272C3⋊C8—364no—
273C2×C32—24no—
274C2×C32—24no—
275C2×C32—24no—
276C2×C32D3×C22⋊C414yes—
277C2×C32#91614yes—
278C2×C32#91914yes—
279C2×C32#95011yescharacteristic
280C2×C32#95011yescharacteristic
281C2×C32C2×#21711yescharacteristic
282C3⋊D3—22no—
283C3⋊D3—42no—
284C3×D3—68no—
285C3×D3—128no—
286#41—94no—
287#41—182no—
288C2×Q8D3×C3⋊D311yescharacteristic
289C2×D4—91no—
290C2×D4—181no—
291C22×C4—91no—
292C22×C4—181no—
293C2×C8—31no—
294C2×C8—271no—
295C2×C8—542no—
296C4⋊C4—31no—
297C4⋊C4—271no—
298C22⋊C4—541no—
299C42—541no—
300C22×C3—24no—
301C22×C3—92no—
302C22×C3—181no—
303C22×C3—181no—
304C22×C3—182no—
305C22×C3—361no—
306C22×C3—361no—
307C22×C3D3×#6714yes—
308C22×C3D4×C3⋊D311yescharacteristic
309C2×D3—38no—
310C2×D3—64no—
311C2×D3—64no—
312C2×D3—68no—
313C2×D3—124no—
314C2×D3—124no—
315C3×C4—22no—
316C3×C4—28no—
317C3×C4—24no—
318C3×C4—24no—
319C3×C4—48no—
320C3×C4—92no—
321C3×C4—124no—
322C3×C4C2×D3×Dic314yes—
323C3×C4D3×#6714yes—
324C3×C4C2×C4×C3⋊D311yescharacteristic
325C3×C4D4×C3⋊D311yescharacteristic
326Dic3—38no—
327Dic3—41no—
328Dic3—361no—
329Dic3—364no—
330Dic3—724no—
331C32—24no—
332C32#480414yescomplemented
333C32#523311yescharacteristic, complemented
334C23—91no—
335C23—181no—
336Q8—21no—
337Q8Dic3×C3⋊D312yes—
338D4—92no—
339D4—184no—
340D4—181no—
341C2×C4—21no—
342C2×C4—61no—
343C2×C4—91no—
344C2×C4—92no—
345C2×C4—92no—
346C2×C4—184no—
347C2×C4—541no—
348C2×C4C2×D3×C3⋊D311yescharacteristic
349C8—61no—
350C8—541no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—24no—
354C2×C3—182no—
355C2×C3—362no—
356C2×C3#6394714yes—
357C2×C3#6394914yes—
358C2×C3D3×#21714yes—
359C2×C3C22⋊C4×C3⋊D311yescharacteristic
360C2×C3#6412511yescharacteristic
361C2×C3#6412811yescharacteristic
362D3—68no—
363D3—128no—
364C22—92no—
365C22—181no—
366C22—181no—
367C22—182no—
368C22—361no—
369C22—361no—
370C22C3⋊D3×#6711yescharacteristic
371C4—22no—
372C4—92no—
373C4—121no—
374C4—1081no—
375C4C2×Dic3×C3⋊D311yescharacteristic
376C4C3⋊D3×#6711yescharacteristic
377C3—24no—
378C3#1059155614yescomplemented
379C3#1059250311yescharacteristic, complemented
380C2—182no—
381C2—362no—
382C2#1172358311yescharacteristic
383C2#1172358511yescharacteristic
384C2C3⋊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#416
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. #11723404C2semidirect
3. C2×C3⋊D3×C3⋊C8C2semidirect
4. #11722205C2semidirect
5. C32×#835C2semidirect
6. #11722034C2semidirect
7. #11722033C2semidirect
12. C3×#3832C4semidirect
36. C3×#835D3semidirect
62. #3832Dic3semidirect
112. #835C3⋊D3semidirect
232. C33#416semidirect
332. C32#4804semidirect
333. C32#5233semidirect
378. C3#10591556semidirect
379. C3#10592503semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
#117234042
C2×C3⋊D3×C3⋊C82
#117222052
C32×#8352
#117220342
#117220332
#105925033
#105915563