Finite Group Databasebeta

← #13160405

Subgroups — #13160405

Order 1728.

1012 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1012 conjugacy classes of subgroups — too many to draw. Only its 164 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#11723971C211yesmaximal, characteristic
2C3×#64135C211yesmaximal, characteristic, complemented
3C32×#830C211yesmaximal, characteristic, complemented
4#11722055C211yesmaximal, characteristic, complemented
5#11722052C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic, complemented
7#11722036C211yesmaximal, characteristic, complemented
8#10592436—31nomaximal
9#10591136—34nomaximal
10C22×C33×C4C2211yescharacteristic
11C2×#6312—21no—
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
13#88859—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×#3757—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4×C3⋊Dic3—21no—
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#64139—31no—
22#64139—34no—
23#64139—64no—
24#64135D311yescharacteristic, complemented
25C3×#926—34no—
26C32×C22⋊C8—31no—
27#63648—31no—
28#63648—31no—
29#63645—31no—
30#63644—31no—
31C3×#830D314yescomplemented
32#63575—34no—
33#63573—34no—
34#63564—34no—
35#63564—34no—
36C23×C33D411yescharacteristic
37C22×C3⋊(C3⋊D3)—21no—
38C2×C33×C4—21no—
39C2×C33×C4—21no—
40C2×C33×C4D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42#6312—42no—
43C2×C3⋊(C3⋊Dic3)—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—21no—
48C32×C3⋊C8—21no—
49C32×C3⋊C8—21no—
50#5003—91no—
51#4737—94no—
52C22×C32×C4—24no—
53C22×C32×C4C2×D314yes—
54C22×C32×C4C2×D311yescharacteristic
55C2×#638—68no—
56C2×#638—64no—
57C2×#638—61no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4⋊(C3⋊D3)—34no—
60C2×C4⋊(C3⋊D3)—64no—
61C2×C32×C8—31no—
62C2×C32×C8—31no—
63#3757—21no—
64#3752—68no—
65#3752—64no—
66#3752—61no—
67C4⋊(C3⋊Dic3)—31no—
68C4⋊(C3⋊Dic3)—34no—
69C4⋊(C3⋊Dic3)—64no—
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72C4×C3⋊Dic3—21no—
73C3×#217—64no—
74C3×C4⋊Dic3—34no—
75C3×C4⋊Dic3—34no—
76C3×C4×Dic3—64no—
77C2×C3×C3⋊C8C2×D314yes—
78C2×C3×C3⋊C8C2×D314yes—
79C22×C33—21no—
80C22×C33—21no—
81C22×C33—21no—
82C22×C33C2×D411yescharacteristic
83C2×C3⋊(C3⋊D3)—41no—
84C2×C3⋊(C3⋊D3)—41no—
85C2×C3⋊(C3⋊D3)—41no—
86C33×C4—21no—
87C33×C4#4111yescharacteristic
88C33×C4#4111yescharacteristic
89C3⋊(C3⋊Dic3)—41no—
90C3×C3⋊Dic3—41no—
91C3×C3⋊Dic3—41no—
92C3×#117—91no—
93#930—94no—
94#930—91no—
95#930—184no—
96#926—34no—
97C3×C22⋊C8—34no—
98#830C3⋊D311yescharacteristic, complemented
99#821—94no—
100#821—94no—
101#818—94no—
102#817—94no—
103#809—91no—
104#809—91no—
105#808—91no—
106#807—91no—
107C23×C32—24no—
108C23×C32D1214yes—
109C23×C32#6711yescharacteristic
110C22×C3⋊D3—68no—
111C22×C3⋊D3—64no—
112C22×C3⋊D3—61no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—44no—
120C2×C32×C4—44no—
121C2×C32×C4D1214yes—
122C2×C32×C4#6711yescharacteristic
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125#638—1216no—
126#638—128no—
127#638—122no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—68no—
131C2×C3⋊Dic3—64no—
132C2×C3⋊Dic3—61no—
133C4⋊(C3⋊D3)—61no—
134C4⋊(C3⋊D3)—61no—
135C4⋊(C3⋊D3)—64no—
136C4⋊(C3⋊D3)—64no—
137C4⋊(C3⋊D3)—124no—
138C4⋊(C3⋊D3)—124no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3—64no—
141C32×C8—61no—
142C32×C8—61no—
143C3×C3⋊C8—24no—
144C3×C3⋊C8—24no—
145#482—271noSylow
146C2×C33—41no—
147C2×C33#12311yescharacteristic
148C2×C33#13511yescharacteristic
149C2×C33AGL(1,ℤ/8)11yescharacteristic
150C3⋊(C3⋊D3)—81no—
151C22×C3×C4—24no—
152C22×C3×C4D3214yes—
153C22×C3×C4C2×C3⋊D311yescharacteristic
154C2×#67—188no—
155C2×#67—184no—
156C2×#67—181no—
157C2×D12—94no—
158C2×D12—91no—
159C2×D12—184no—
160C2×C3×C8—34no—
161C2×C3×C8—34no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—91no—
164C3×C22⋊C4—181no—
165C3×C42—181no—
166#217—64no—
167(C2×D3)⋊C4—188no—
168(C2×D3)⋊C4—184no—
169(C2×D3)⋊C4—181no—
170C4⋊Dic3—34no—
171C4⋊Dic3—34no—
172C4⋊Dic3—94no—
173C4⋊Dic3—91no—
174C4⋊Dic3—184no—
175C4×Dic3—64no—
176C2×C3⋊C8C2×C3⋊D311yescharacteristic
177C2×C3⋊C8C2×C3⋊D311yescharacteristic
178C22×C32—28no—
179C22×C32—24no—
180C22×C32—21no—
181C22×C32—21no—
182C22×C32—21no—
183C22×C32—24no—
184C22×C32—24no—
185C22×C32—24no—
186C22×C32—44no—
187C22×C32—44no—
188C22×C32C2×D1214yes—
189C22×C32C2×#6711yescharacteristic
190C2×C3⋊D3—128no—
191C2×C3⋊D3—128no—
192C2×C3⋊D3—128no—
193C2×C3⋊D3—124no—
194C2×C3⋊D3—124no—
195C2×C3⋊D3—124no—
196C2×C3⋊D3—121no—
197C2×C3⋊D3—121no—
198C2×C3⋊D3—121no—
199C32×C4—21no—
200C32×C4—24no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—44no—
204C32×C4#23511yescharacteristic
205C32×C4#23511yescharacteristic
206C32×C4#23714yes—
207C32×C4#23714yes—
208C3⋊Dic3—41no—
209C3⋊Dic3—41no—
210C3⋊Dic3—128no—
211C3⋊Dic3—124no—
212C3⋊Dic3—121no—
213C3×Dic3—124no—
214C3×Dic3—124no—
215#121—271no—
216#117—91no—
217#107—271no—
218#106—271no—
219#102—271no—
220#102—271no—
221C22⋊C8—31no—
222C33#48211yescharacteristic, complemented, Sylow
223C23×C3—24no—
224C23×C3#62614yes—
225C23×C3C4⋊(C3⋊D3)11yescharacteristic
226C22×D3—188no—
227C22×D3—184no—
228C22×D3—181no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—24no—
234C2×C3×C4—24no—
235C2×C3×C4—44no—
236C2×C3×C4—44no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4#62614yes—
240C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
241C2×C3×C4C2×D3214yes—
242C2×C3×C4C22×C3⋊D311yescharacteristic
243#67—3616no—
244#67—368no—
245#67—362no—
246C2×Dic3—64no—
247C2×Dic3—64no—
248C2×Dic3—188no—
249C2×Dic3—184no—
250C2×Dic3—181no—
251D12—184no—
252D12—184no—
253D12—181no—
254D12—181no—
255D12—364no—
256D12—364no—
257C3×C8—64no—
258C3×C8—64no—
259C3⋊C8—21no—
260C3⋊C8—21no—
261C2×C32—24no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32—48no—
265C2×C32—44no—
266C2×C32—41no—
267C2×C32#88614yes—
268C2×C32#90414yes—
269C2×C32#90814yes—
270C2×C32#92911yescharacteristic
271C2×C32#94911yescharacteristic
272C2×C32#95011yescharacteristic
273C3⋊D3—248no—
274C3⋊D3—244no—
275C3⋊D3—241no—
276C2×D4—271no—
277C2×D4—541no—
278C22×C4D3×C3⋊D311yescharacteristic
279C2×C8—31no—
280C2×C8—31no—
281C4⋊C4—91no—
282C4⋊C4—91no—
283C4⋊C4—271no—
284C22⋊C4—181no—
285C22⋊C4—541no—
286C42—181no—
287C22×C3—28no—
288C22×C3—24no—
289C22×C3—24no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—24no—
294C22×C3—24no—
295C22×C3—44no—
296C22×C3—44no—
297C22×C3C2×#62614yes—
298C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
299C2×D3—368no—
300C2×D3—368no—
301C2×D3—368no—
302C2×D3—364no—
303C2×D3—364no—
304C2×D3—361no—
305C2×D3—361no—
306C2×D3—361no—
307C2×D3—364no—
308C3×C4—21no—
309C3×C4—24no—
310C3×C4—24no—
311C3×C4—24no—
312C3×C4—44no—
313C3×C4—361no—
314C3×C4—361no—
315C3×C4#380414yes—
316C3×C4#380414yes—
317C3×C4#383011yescharacteristic
318C3×C4#383011yescharacteristic
319Dic3—124no—
320Dic3—124no—
321Dic3—368no—
322Dic3—364no—
323Dic3—361no—
324C32—24no—
325C32#473714yescomplemented
326C32#500311yescharacteristic, complemented
327C23—541no—
328C23#629211yescharacteristic
329D4—541no—
330D4—541no—
331D4—1082no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4—541no—
337C2×C4#629211yescharacteristic
338C2×C4C2×D3×C3⋊D311yescharacteristic
339C8—61no—
340C8—61no—
341C2×C3—24no—
342C2×C3—24no—
343C2×C3—24no—
344C2×C3—48no—
345C2×C3—44no—
346C2×C3—41no—
347C2×C3#6382514yes—
348C2×C3#6382914yes—
349C2×C3#6396614yes—
350C2×C3#6409511yescharacteristic
351C2×C3#6411311yescharacteristic
352C2×C3#6411711yescharacteristic
353D3—728no—
354D3—724no—
355D3—721no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—1081no—
360C22—1081no—
361C22—1081no—
362C22C2×#629211yescharacteristic
363C4—21no—
364C4—361no—
365C4—361no—
366C4—1081no—
367C4#8902811yescharacteristic
368C4#8902811yescharacteristic
369C3—24no—
370C3#1059113614yescomplemented
371C3#1059243611yescharacteristic, complemented
372C2—41no—
373C2—2161no—
374C2#1172336911yescharacteristic
375C2#1172337111yescharacteristic
376C2#1172360811yescharacteristic
subgroupG/H
centerC22C2×#6292
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radical#11723971C2
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#482
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#64135C2semidirect
3. C32×#830C2semidirect
4. #11722055C2semidirect
5. #11722052C2semidirect
6. #11722036C2semidirect
7. #11722036C2semidirect
24. #64135D3semidirect
31. C3×#830D3semidirect
98. #830C3⋊D3semidirect
222. C33#482semidirect
325. C32#4737semidirect
326. C32#5003semidirect
370. C3#10591136semidirect
371. C3#10592436semidirect

Maximal subgroups

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

subgroupindex
#117239712
C3×#641352
C32×#8302
#117220552
#117220522
#117220362
#105924363
#105911363