Finite Group Databasebeta

← #72259725

Subgroups — #72259725

Order 3888.

693 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 693 conjugacy classes of subgroups — too many to draw. Only its 81 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×(#4176⋊C2)⋊C3C212yesmaximal, complemented
2C4⋊(#4184⋊C3)C211yesmaximal, characteristic, complemented
3C4×#4177⋊C3C211yesmaximal, characteristic, complemented
4C4⋊#92283C211yesmaximal, characteristic, complemented
5#10619556⋊C3C212yesmaximal, complemented
6#88646⋊C3—31nomaximal
7C3×#88654—31nomaximal
8#13047135C311yesmaximal, characteristic, complemented
9C2×#4184⋊C3C2212yescomplemented
10C2×#4177⋊C3C2211yescharacteristic
11C2×#92283C2212yescomplemented
12(#4176⋊C2)⋊C3—24no—
13C4×#6910C2211yescharacteristic, complemented
14#66324⋊C3C2211yescharacteristic, complemented
15C2×D3×AGL(1,ℤ/9)—32no—
16C2×C3×((C3⋊D3)⋊C3)⋊C2—32no—
17C4×C3⋊AGL(1,ℤ/9)D311yescharacteristic, complemented
18C3×C4⋊#272D311yescharacteristic
19D12×3−1+2—31no—
20C3×C4⋊AGL(1,ℤ/9)—31no—
21C3×C4⋊AGL(1,ℤ/9)—61no—
22C3×C4⋊((C3⋊D3)⋊C3)—31no—
23C3×C4×(C3⋊D3)⋊C3—31no—
24#6194⋊C3—32no—
25C3×#6200—32no—
26C2×#4176⋊C2C2×C312yescomplemented
27C4⋊#4184C2×C311yescharacteristic, complemented
28C4×#4177—21no—
29C4×#4177C2×C311yescharacteristic, complemented
30C4⋊#4176—21no—
31C4⋊#4176C2×C311yescharacteristic, complemented
32#10619556C2×C312yescomplemented
33C2×#6910C2311yescharacteristic
34#4184⋊C3—22no—
35#4177⋊C3D412yescomplemented
36#92283—22no—
37C3×#3802—91no—
38C3×D3×D12—91no—
39D4×AGL(1,ℤ/9)—91no—
40#88654—31no—
41#88646—31no—
42C2×C3⋊AGL(1,ℤ/9)C2×D311yescharacteristic
43C2×C3×#272C2×D312yes—
44C2×D3×3−1+2—32no—
45C2×C3×AGL(1,ℤ/9)—32no—
46C2×C3×AGL(1,ℤ/9)—62no—
47C2×C3×(C3⋊D3)⋊C3—31no—
48C2×C3×(C3⋊D3)⋊C3—32no—
49D3×AGL(1,ℤ/9)—64no—
50C3×((C3⋊D3)⋊C3)⋊C2—64no—
51C3×C4×3−1+2—21no—
52C3×C4×3−1+2C2×D311yescharacteristic, complemented
53C3×C4×H(3,𝔽3)C2×D311yescharacteristic
54C3⋊(3−1+2⋊C4)C2×D311yescharacteristic, complemented
55C3×(C3⋊Dic3)⋊C3—31no—
56C2×#4184C22×C312yescomplemented
57C2×#4177—21no—
58C2×#4177C22×C311yescharacteristic
59C2×#4176—22no—
60C2×#4176C22×C312yescomplemented
61#4176⋊C2—24no—
62C4×#731—21no—
63C4×#731—21no—
64C4×#731—41no—
65C4×#731C22×C311yescharacteristic, complemented
66#66324—21no—
67#66324C22×C311yescharacteristic, complemented
68#6910C2×D411yescharacteristic, complemented, Sylow
69C2×C3×D32—92no—
70C2×C3×D32—92no—
71C3×C4⋊(C3⋊D3)—31no—
72C3×C4×C3⋊D3—31no—
73C32×D12—31no—
74C32×D12—91no—
75C32×D12—181no—
76C32×C4×D3—91no—
77C3×#626—92no—
78C3×#625—92no—
79C22×AGL(1,ℤ/9)—92no—
80C2×((C3⋊D3)⋊C3)⋊C2—32no—
81C2×D3×D9—32no—
82D4×3−1+2—91no—
83C4⋊#272C3×D311yescharacteristic
84C4×C3⋊D9C3×D311yescharacteristic, complemented
85#611⋊C3—92no—
86C4⋊AGL(1,ℤ/9)—31no—
87C4⋊AGL(1,ℤ/9)—61no—
88C4⋊AGL(1,ℤ/9)—61no—
89C4⋊AGL(1,ℤ/9)—61no—
90C4⋊AGL(1,ℤ/9)—61no—
91C4×AGL(1,ℤ/9)—181no—
92C4⋊((C3⋊D3)⋊C3)—31no—
93C4⋊((C3⋊D3)⋊C3)—61no—
94C4×(C3⋊D3)⋊C3—31no—
95C4×(C3⋊D3)⋊C3—61no—
96C9×D12—31no—
97C9×D12—61no—
98C3×D36—31no—
99C3×D36—61no—
100#6200—32no—
101#6194—32no—
102C2×C3×3−1+2—21no—
103C2×C3×3−1+2C22×D311yescharacteristic
104C2×C3×H(3,𝔽3)C22×D311yescharacteristic
105C3⋊AGL(1,ℤ/9)D1212yescomplemented
106C3×#272—22no—
107D3×3−1+2—62no—
108C3×AGL(1,ℤ/9)—62no—
109C3×AGL(1,ℤ/9)—122no—
110C3×(C3⋊D3)⋊C3—32no—
111C3×(C3⋊D3)⋊C3—62no—
112C2×#731—21no—
113C2×#731—21no—
114C2×#731—41no—
115C2×#731C23×C311yescharacteristic
116#4184—22no—
117#4177—22no—
118#4177C3×D412yescomplemented
119#4176—22no—
120#4176—42no—
121C3×D3×D4—271no—
122C3×D3×D4—271no—
123C2×C3×D12—271no—
124#3802—91no—
125D3×D12—91no—
126D4×D9—91no—
127C2×C3×C3⋊D3—31no—
128C2×C3×C3⋊D3—32no—
129C2×C32×D3—32no—
130C2×C32×D3—91no—
131C2×C32×D3—92no—
132C2×C32×D3—182no—
133C3×D32—184no—
134C3×D32—184no—
135C33×C4—61no—
136C33×C4D3211yescharacteristic
137C3×C3⋊Dic3—31no—
138C32×Dic3—91no—
139C22×3−1+2—92no—
140C2×#272C2×C3×D312yes—
141C2×C3⋊D9C2×C3×D311yescharacteristic
142C2×AGL(1,ℤ/9)—32no—
143C2×AGL(1,ℤ/9)—62no—
144C2×AGL(1,ℤ/9)—62no—
145C2×AGL(1,ℤ/9)—62no—
146C2×AGL(1,ℤ/9)—62no—
147C2×AGL(1,ℤ/9)—184no—
148C2×AGL(1,ℤ/9)—181no—
149C2×(C3⋊D3)⋊C3—31no—
150C2×(C3⋊D3)⋊C3—32no—
151C2×(C3⋊D3)⋊C3—61no—
152C2×(C3⋊D3)⋊C3—62no—
153C2×D3×C9—32no—
154C2×D3×C9—62no—
155C2×C3×D9—32no—
156C2×C3×D9—62no—
157((C3⋊D3)⋊C3)⋊C2—64no—
158D3×D9—64no—
159C4×3−1+2—21no—
160C4×3−1+2—41no—
161C4×3−1+2—61no—
162C4×3−1+2—61no—
163C4×3−1+2—61no—
164C4×3−1+2—121no—
165C4×H(3,𝔽3)—21no—
166C4×H(3,𝔽3)—121no—
167C4×H(3,𝔽3)C2×C3×D311yescharacteristic
168C3×C4×C9—21no—
169C3×C4×C9—121no—
170C3×C4×C9C2×C3×D311yescharacteristic, complemented
171C3⋊Dic9C2×C3×D311yescharacteristic, complemented
1723−1+2⋊C4—181no—
173(C3⋊Dic3)⋊C3—31no—
174(C3⋊Dic3)⋊C3—61no—
175C3×3−1+2—21no—
176C3×3−1+2C2×D1211yescharacteristic, complemented
177C3×H(3,𝔽3)D3×D411yescharacteristic
178#731—21no—
179#731—21no—
180#731—41no—
181#731C2×C3×D411yescharacteristic, complemented
182C22×C3×D3—272no—
183C22×C3×D3—272no—
184C22×C3×D3—272no—
185C2×D32—92no—
186C2×D32—92no—
187C32×D4—91no—
188C32×D4—271no—
189C2×C32×C4—271no—
190C4⋊(C3⋊D3)—31no—
191C4×C3⋊D3—31no—
192C3×#67—272no—
193C3×#67—272no—
194C3×D12—31no—
195C3×D12—91no—
196C3×D12—91no—
197C3×D12—91no—
198C3×D12—181no—
199C3×D12—181no—
200C3×D12—181no—
201C3×D12—181no—
202C3×D12—272no—
203C3×D12—541no—
204C3×C4×D3—91no—
205C3×C4×D3—91no—
206C3×C4×D3—181no—
207C3×C4×D3—541no—
208#626—92no—
209#625—92no—
210C22×D9—92no—
211D4×C9—91no—
212D4×C9—181no—
213#611—92no—
214D36—31no—
215D36—61no—
216C4×D9—181no—
217C2×C33—61no—
218C2×C33C2×D3211yescharacteristic
219C3×C3⋊D3—32no—
220C3×C3⋊D3—62no—
221C32×D3—62no—
222C32×D3—92no—
223C32×D3—182no—
224C32×D3—362no—
225C2×3−1+2—21no—
226C2×3−1+2—41no—
227C2×3−1+2—61no—
228C2×3−1+2—61no—
229C2×3−1+2—61no—
230C2×3−1+2—121no—
231C2×3−1+2—182no—
232C2×H(3,𝔽3)—21no—
233C2×H(3,𝔽3)—121no—
234C2×H(3,𝔽3)C22×C3×D311yescharacteristic
235C2×C3×C9—21no—
236C2×C3×C9—121no—
237C2×C3×C9C22×C3×D311yescharacteristic
238#272—22no—
239C3⋊D9C3×D1212yescomplemented
240AGL(1,ℤ/9)—62no—
241AGL(1,ℤ/9)—122no—
242AGL(1,ℤ/9)—122no—
243AGL(1,ℤ/9)—122no—
244AGL(1,ℤ/9)—122no—
245AGL(1,ℤ/9)—182no—
246(C3⋊D3)⋊C3—32no—
247(C3⋊D3)⋊C3—62no—
248(C3⋊D3)⋊C3—62no—
249(C3⋊D3)⋊C3—122no—
250D3×C9—62no—
251D3×C9—122no—
252C3×D9—62no—
253C3×D9—122no—
254C2×C3×D4—811no—
255D3×D4—271no—
256D3×D4—271no—
257C2×D12—271no—
258C22×C32—92no—
259C22×C32—272no—
260C22×C32—271no—
261C2×C3⋊D3—31no—
262C2×C3⋊D3—32no—
263C2×C3×D3—32no—
264C2×C3×D3—92no—
265C2×C3×D3—91no—
266C2×C3×D3—91no—
267C2×C3×D3—92no—
268C2×C3×D3—92no—
269C2×C3×D3—182no—
270C2×C3×D3—181no—
271C2×C3×D3—182no—
272C2×C3×D3—182no—
273C2×C3×D3—182no—
274C2×C3×D3—541no—
275C2×C3×D3—542no—
276C2×C3×D3—544no—
277C2×C3×D3—544no—
278C2×C3×D3—544no—
279D32—184no—
280D32—184no—
281C32×C4—21no—
282C32×C4—31no—
283C32×C4—121no—
284C32×C4—181no—
285C32×C4—271no—
286C32×C4—361no—
287C32×C4—361no—
288C32×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
289C32×C4C3×D3211yescharacteristic
290C3⋊Dic3—31no—
291C3×Dic3—91no—
292C3×Dic3—91no—
293C3×Dic3—181no—
294C3×Dic3—541no—
295C22×C9—92no—
296C22×C9—182no—
297C2×D9—32no—
298C2×D9—62no—
299C2×D9—181no—
300C2×D9—184no—
301C4×C9—61no—
302C4×C9—121no—
303C4×C9—121no—
304C4×C9—121no—
305Dic9—181no—
306C33—61no—
307C33D3×D1211yescharacteristic
3083−1+2—21no—
3093−1+2—41no—
3103−1+2—61no—
3113−1+2—61no—
3123−1+2—61no—
3133−1+2—121no—
314H(3,𝔽3)—21no—
315H(3,𝔽3)—121no—
316H(3,𝔽3)C3×D3×D411yescharacteristic
317C3×C9—21no—
318C3×C9—121no—
319C3×C9C2×C3×D1211yescharacteristic, complemented
320C23×C3—812no—
321C22×D3—272no—
322C22×D3—272no—
323C22×D3—272no—
324C3×D4—91no—
325C3×D4—271no—
326C3×D4—271no—
327C3×D4—541no—
328C3×D4—541no—
329C3×D4—812no—
330C2×C3×C4—271no—
331C2×C3×C4—271no—
332C2×C3×C4—541no—
333#67—272no—
334#67—272no—
335D12—31no—
336D12—91no—
337D12—91no—
338D12—272no—
339D12—541no—
340C4×D3—91no—
341C4×D3—541no—
342C2×C32—21no—
343C2×C32—31no—
344C2×C32—121no—
345C2×C32—181no—
346C2×C32—182no—
347C2×C32—272no—
348C2×C32—361no—
349C2×C32—361no—
350C2×C32—542no—
351C2×C32C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
352C2×C32C2×C3×D3211yescharacteristic
353C3⋊D3—32no—
354C3⋊D3—62no—
355C3×D3—62no—
356C3×D3—92no—
357C3×D3—92no—
358C3×D3—182no—
359C3×D3—182no—
360C3×D3—182no—
361C3×D3—182no—
362C3×D3—362no—
363C3×D3—362no—
364C3×D3—362no—
365C3×D3—362no—
366C3×D3—542no—
367C3×D3—1082no—
368C2×C9—61no—
369C2×C9—121no—
370C2×C9—121no—
371C2×C9—121no—
372C2×C9—182no—
373C2×C9—362no—
374D9—62no—
375D9—122no—
376D9—182no—
377C2×D4—811noSylow
378C22×C3—92no—
379C22×C3—271no—
380C22×C3—272no—
381C22×C3—272no—
382C22×C3—271no—
383C22×C3—542no—
384C22×C3—542no—
385C22×C3—541no—
386C22×C3—1624no—
387C2×D3—32no—
388C2×D3—92no—
389C2×D3—91no—
390C2×D3—92no—
391C2×D3—541no—
392C2×D3—542no—
393C2×D3—544no—
394C2×D3—544no—
395C2×D3—544no—
396C3×C4—61no—
397C3×C4—61no—
398C3×C4—121no—
399C3×C4—181no—
400C3×C4—271no—
401C3×C4—271no—
402C3×C4—361no—
403C3×C4—541no—
404C3×C4C3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
405Dic3—91no—
406Dic3—541no—
407C32—21no—
408C32—31no—
409C32—121no—
410C32—181no—
411C32—361no—
412C32—361no—
413C32#8865411yescharacteristic
414C32C3×D3×D1211yescharacteristic
415C9—61no—
416C9—121no—
417C9—121no—
418C9—121no—
419C23—812no—
420D4—91no—
421D4—271no—
422D4—812no—
423C2×C4—271no—
424C2×C3—61no—
425C2×C3—61no—
426C2×C3—121no—
427C2×C3—182no—
428C2×C3—181no—
429C2×C3—272no—
430C2×C3—272no—
431C2×C3—361no—
432C2×C3—542no—
433C2×C3—542no—
434C2×C3—542no—
435C2×C3—1082no—
436C2×C3—1082no—
437C2×C3C2×C3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
438D3—62no—
439D3—92no—
440D3—182no—
441D3—182no—
442D3—542no—
443D3—1082no—
444C22—92no—
445C22—271no—
446C22—272no—
447C22—1624no—
448C4—271no—
449C4(#4176⋊C2)⋊C311yescharacteristic, complemented
450C3—61no—
451C3—61no—
452C3—121no—
453C3—181no—
454C3—361no—
455C3C3×#8865411yescharacteristic
456C2—182no—
457C2—272no—
458C2—542no—
459C2C2×(#4176⋊C2)⋊C311yescharacteristic
subgroupG/H
centerC2C2×(#4176⋊C2)⋊C3
commutator, cosocleC2×#731C23×C3
FrattiniC2×C32C2×C3×D32
FittingC4×#6910C22
socleC2×C3C2×C3×((C3⋊D3)⋊C3)⋊C2
hypercenterC4(#4176⋊C2)⋊C3
hypocenter#731C2×C3×D4
LCSG ▷ C2×#731 ▷ #731
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×(#4176⋊C2)⋊C3C2semidirect
2. C4⋊(#4184⋊C3)C2semidirect
3. C4×#4177⋊C3C2semidirect
4. C4⋊#92283C2semidirect
5. #10619556⋊C3C2semidirect
8. #13047135C3semidirect
9. C2×#4184⋊C3C22semidirect
11. C2×#92283C22semidirect
13. C4×#6910C22semidirect
14. #66324⋊C3C22semidirect
17. C4×C3⋊AGL(1,ℤ/9)D3semidirect
26. C2×#4176⋊C2C2×C3semidirect
27. C4⋊#4184C2×C3semidirect
29. C4×#4177C2×C3semidirect
31. C4⋊#4176C2×C3semidirect
32. #10619556C2×C3semidirect
35. #4177⋊C3D4semidirect
52. C3×C4×3−1+2C2×D3semidirect
54. C3⋊(3−1+2⋊C4)C2×D3semidirect
56. C2×#4184C22×C3semidirect
60. C2×#4176C22×C3semidirect
65. C4×#731C22×C3semidirect
67. #66324C22×C3semidirect
68. #6910C2×D4semidirect
84. C4×C3⋊D9C3×D3semidirect
105. C3⋊AGL(1,ℤ/9)D12semidirect
118. #4177C3×D4semidirect
170. C3×C4×C9C2×C3×D3semidirect
171. C3⋊Dic9C2×C3×D3semidirect
176. C3×3−1+2C2×D12semidirect
181. #731C2×C3×D4semidirect
239. C3⋊D9C3×D12semidirect
319. C3×C9C2×C3×D12semidirect
449. C4(#4176⋊C2)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×(#4176⋊C2)⋊C32
C4⋊(#4184⋊C3)2
C4×#4177⋊C32
C4⋊#922832
#10619556⋊C32
#88646⋊C33
C3×#886543
#130471353