Finite Group Databasebeta

← #13161192

Subgroups — #13161192

Order 1728.

1016 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 166 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×#88862C211yesmaximal, characteristic, complemented
2C32×#946C211yesmaximal, characteristic, complemented
3#11723516C211yesmaximal, characteristic
4C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722053C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic, complemented
8#10592747—31nomaximal
9#10591585—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6291—21no—
14#88862—22no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×(C2×D3)⋊C4—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20C32×Dic3⋊C4—21no—
21#88804—21no—
22D3×C3⋊(C3⋊C8)—22no—
23C32×C22⋊Q8—31no—
24C2×#3755—31no—
25C2×#3755—34no—
26C2×#3755—64no—
27#64139—31no—
28C22×C3⋊(C3⋊C8)—31no—
29C3×#946D314yescomplemented
30#63909—34no—
31C2×D3×C3⋊C8—34no—
32#63662D311yescharacteristic, complemented
33#63636—31no—
34#63635—31no—
35C3×#835—34no—
36#63574—34no—
37#63564—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C32×D3D411yescharacteristic
40C33×Q8—21no—
41C33×Q8—21no—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C32×Dic3—21no—
48C2×C32×Dic3D411yescharacteristic
49C32×C4×D3—21no—
50C32×C4×D3—21no—
51#6291—42no—
52C3⋊(C3⋊(C3⋊C8))D412yescomplemented
53C3×C3⋊(C3⋊C8)—21no—
54#5048—94no—
55#4807—91no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D311yescharacteristic
59C22×C32×C4—31no—
60C2×#638—61no—
61C2×C4⋊(C3⋊D3)—31no—
62C2×C4⋊(C3⋊D3)—34no—
63C2×C4⋊(C3⋊D3)—64no—
64C2×C3×C4×D3C2×D314yes—
65C2×#626—64no—
66C32×C4⋊C4—31no—
67C32×C4⋊C4—61no—
68C32×C22⋊C4—61no—
69#3755—62no—
70#3755—68no—
71#3755—616no—
72#3752—61no—
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—62no—
77C2×C3⋊(C3⋊C8)—64no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×(C2×D3)⋊C4—24no—
80C3×C4⋊Dic3—34no—
81C3×C4⋊Dic3C2×D314yes—
82C3×Dic3⋊C4—24no—
83C2×C3×C3⋊C8—34no—
84#3722—64no—
85D3×C3⋊C8—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C2×C32×D3—21no—
91C2×C32×D3—21no—
92C2×C32×D3SD1612yescomplemented
93C33×C4—41no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3×C3⋊Dic3—41no—
97C32×Dic3—21no—
98C32×Dic3—41no—
99C3×C22⋊Q8—34no—
100#946C3⋊D311yescharacteristic, complemented
101C2×#215—94no—
102C2×#215—91no—
103C2×#215—184no—
104#930—94no—
105C22×C3⋊C8—94no—
106C2×D3×C8—91no—
107#895—91no—
108C3×#103—91no—
109#835—34no—
110#821—91no—
111#809—94no—
112#808—94no—
113#808—91no—
114C23×C32—31no—
115C22×C3⋊D3—61no—
116C22×C3⋊D3—64no—
117C22×C3⋊D3—124no—
118C22×C3×D3#6714yes—
119C32×Q8—28no—
120C32×Q8—21no—
121C32×Q8—24no—
122C32×Q8—21no—
123C32×Q8—24no—
124C32×Q8—44no—
125C2×C32×C4—21no—
126C2×C32×C4—24no—
127C2×C32×C4—24no—
128C2×C32×C4—31no—
129C2×C32×C4—44no—
130C2×C32×C4—61no—
131C2×C32×C4—61no—
132C2×C32×C4—61no—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C22×D311yescharacteristic
135#638—122no—
136C2×C3⋊Dic3—21no—
137C4⋊(C3⋊D3)—68no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—64no—
140C4⋊(C3⋊D3)—61no—
141C4⋊(C3⋊D3)—64no—
142C4⋊(C3⋊D3)—124no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—64no—
145C2×C3×Dic3#6714yes—
146C3×C4×D3—24no—
147C3×C4×D3—24no—
148#626—128no—
149C3⋊(C3⋊C8)—21no—
150C3⋊(C3⋊C8)—32no—
151C3⋊(C3⋊C8)—38no—
152C3⋊(C3⋊C8)—68no—
153C3×C3⋊C8—64no—
154#465—271noSylow
155C2×C33#12111yescharacteristic
156C2×C33C2×SD1611yescharacteristic
157C2×C33#13511yescharacteristic
158C3⋊(C3⋊D3)—81no—
159C32×D3—22no—
160C2×C3×Q8—24no—
161C2×C3×Q8D3214yes—
162C2×C3×Q8C2×C3⋊D311yescharacteristic
163C22×C3×C4—34no—
164C2×#67—184no—
165C2×D12—94no—
166C2×D12—91no—
167C2×D12—184no—
168C2×D12—181no—
169C2×C4×D3C2×C3⋊D311yescharacteristic
170C2×C3×C8—91no—
171C3×C4⋊C4—34no—
172C3×C4⋊C4—64no—
173C3×C4⋊C4—91no—
174C3×C22⋊C4—64no—
175#215—188no—
176#215—182no—
177#215—1816no—
178(C2×D3)⋊C4—21no—
179(C2×D3)⋊C4—184no—
180(C2×D3)⋊C4—181no—
181C4⋊Dic3—34no—
182C4⋊Dic3C2×C3⋊D311yescharacteristic
183Dic3⋊C4—21no—
184C2×C3⋊C8—34no—
185C2×C3⋊C8—94no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—184no—
188C2×C3⋊C8—188no—
189D3×C8—182no—
190C22×C32—24no—
191C22×C32—32no—
192C22×C32—61no—
193C22×C32—61no—
194C22×C32D3×D411yescharacteristic
195C22×C32C2×#6714yes—
196C2×C3⋊D3—128no—
197C2×C3⋊D3—121no—
198C2×C3⋊D3—121no—
199C2×C3⋊D3—121no—
200C2×C3⋊D3—124no—
201C2×C3⋊D3—124no—
202C2×C3⋊D3—124no—
203C2×C3⋊D3—244no—
204C2×C3⋊D3—244no—
205C2×C3×D3—24no—
206C2×C3×D3—24no—
207C2×C3×D3#21518yescomplemented
208C32×C4—24no—
209C32×C4—24no—
210C32×C4—48no—
211C32×C4—41no—
212C32×C4—44no—
213C32×C4—61no—
214C32×C4—121no—
215C32×C4#23511yescharacteristic
216C32×C4D3×D414yes—
217C32×C4D3×D411yescharacteristic
218C32×C4#23714yes—
219C3⋊Dic3—41no—
220C3×Dic3—24no—
221C3×Dic3—44no—
222C3×Dic3—124no—
223C2×SD16—271no—
224C22×C8—271no—
225C22⋊Q8—31no—
226#121—271no—
227#106—271no—
228#103—91no—
229#102—271no—
230C33#46511yescharacteristic, complemented, Sylow
231C23×C3—34no—
232C22×D3—184no—
233C22×D3—181no—
234C22×D3—364no—
235C22×D3#63811yescharacteristic
236C3×Q8—28no—
237C3×Q8—24no—
238C3×Q8—24no—
239C3×Q8—21no—
240C3×Q8—21no—
241C3×Q8—44no—
242C2×C3×C4—24no—
243C2×C3×C4—21no—
244C2×C3×C4—24no—
245C2×C3×C4—34no—
246C2×C3×C4—44no—
247C2×C3×C4—64no—
248C2×C3×C4—64no—
249C2×C3×C4—64no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253#67—368no—
254C2×Dic3—21no—
255C2×Dic3—64no—
256C2×Dic3#63811yescharacteristic
257D12—188no—
258D12—184no—
259D12—184no—
260D12—181no—
261D12—181no—
262D12—364no—
263D12—362no—
264C4×D3—21no—
265C4×D3—21no—
266C3×C8—181no—
267C3⋊C8—64no—
268C3⋊C8—98no—
269C3⋊C8—92no—
270C3⋊C8—188no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32—24no—
274C2×C32—62no—
275C2×C32#88411yescharacteristic
276C2×C32D3×SD1611yescharacteristic
277C2×C32#91911yescharacteristic
278C2×C32#93914yes—
279C2×C32C2×#21514yes—
280C2×C32#95014yes—
281C3⋊D3—248no—
282C3⋊D3—244no—
283C3⋊D3—241no—
284C3×D3—28no—
285C2×Q8D3×C3⋊D311yescharacteristic
286C2×D4—271no—
287C2×D4—541no—
288C22×C4—31no—
289SD16—542no—
290C2×C8—91no—
291C2×C8—271no—
292C2×C8—542no—
293C4⋊C4—31no—
294C4⋊C4—61no—
295C4⋊C4—91no—
296C22⋊C4—61no—
297C22⋊C4—541no—
298C22×C3—24no—
299C22×C3—38no—
300C22×C3—64no—
301C22×C3—64no—
302C22×C3D3×#6714yes—
303C22×C3C2×#63811yescharacteristic
304C2×D3—21no—
305C2×D3—21no—
306C2×D3—368no—
307C2×D3—361no—
308C2×D3—361no—
309C2×D3—361no—
310C2×D3—364no—
311C2×D3—364no—
312C2×D3—364no—
313C2×D3—724no—
314C2×D3—724no—
315C2×D3#375512yescomplemented
316C3×C4—24no—
317C3×C4—24no—
318C3×C4—48no—
319C3×C4—41no—
320C3×C4—44no—
321C3×C4—64no—
322C3×C4—124no—
323C3×C4—361no—
324C3×C4#380414yes—
325C3×C4#381114yes—
326C3×C4D4×C3⋊D311yescharacteristic
327C3×C4#383011yescharacteristic
328Dic3—21no—
329Dic3—41no—
330Dic3—124no—
331C32—24no—
332C32#480711yescharacteristic, complemented
333C32#504814yescomplemented
334C23—31no—
335C23—541no—
336Q8—21no—
337Q8—21no—
338D4—541no—
339D4—541no—
340D4—1082no—
341C2×C4—21no—
342C2×C4—31no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—181no—
347C2×C4C2×D3×C3⋊D311yescharacteristic
348C8—181no—
349C8—272no—
350C2×C3—24no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—68no—
354C2×C3D3×#21514yes—
355C2×C3#6394914yes—
356C2×C3#6397114yes—
357C2×C3#6414811yescharacteristic
358C2×C3C2×#375511yescharacteristic
359C2×C3#6415911yescharacteristic
360D3—22no—
361D3—728no—
362D3—724no—
363D3—721no—
364C22—32no—
365C22—61no—
366C22—61no—
367C22—1081no—
368C22—1081no—
369C22—1081no—
370C22D3×#63811yescharacteristic
371C4—41no—
372C4—61no—
373C4—121no—
374C4—361no—
375C4#8902811yescharacteristic
376C4#8903911yescharacteristic
377C3—24no—
378C3#1059158514yescomplemented
379C3#1059274711yescharacteristic, complemented
380C2—62no—
381C2—2161no—
382C2D3×#375511yescharacteristic
383C2#1172358211yescharacteristic
384C2#1172361611yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radical#11723516C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#465
LCSG ▷ C2×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
1. C2×#88862C2semidirect
2. C32×#946C2semidirect
4. C2×D3×C3⋊(C3⋊C8)C2semidirect
5. C3×#63662C2semidirect
6. #11722053C2semidirect
7. #11722035C2semidirect
29. C3×#946D3semidirect
32. #63662D3semidirect
52. C3⋊(C3⋊(C3⋊C8))D4semidirect
92. C2×C32×D3SD16semidirect
100. #946C3⋊D3semidirect
207. C2×C3×D3#215semidirect
230. C33#465semidirect
315. C2×D3#3755semidirect
332. C32#4807semidirect
333. C32#5048semidirect
378. C3#10591585semidirect
379. C3#10592747semidirect

Maximal subgroups

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

subgroupindex
C2×#888622
C32×#9462
#117235162
C2×D3×C3⋊(C3⋊C8)2
C3×#636622
#117220532
#117220352
#105927473
#105915853