Finite Group Databasebeta

← #72259397

Subgroups — #72259397

Order 3888.

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

normal subgroups only

This group has 715 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×(#4178⋊C2)⋊C3C212yesmaximal, complemented
2C4⋊#92333C211yesmaximal, characteristic, complemented
3C4⋊(#4179⋊C3)C211yesmaximal, characteristic, complemented
4C4×#92285C211yesmaximal, characteristic, complemented
5#10619565⋊C3C212yesmaximal, complemented
6D3×C4⋊AGL(1,ℤ/9)—31nomaximal
7C3×#88655—31nomaximal
8#13047150C311yesmaximal, characteristic, complemented
9C2×#92333C2212yescomplemented
10C2×#4179⋊C3C2212yescomplemented
11C2×#92285C2211yescharacteristic
12(#4178⋊C2)⋊C3—24no—
13C4×#6911C2211yescharacteristic, complemented
14#11757589C2211yescharacteristic, 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, complemented
19C4×D3×3−1+2—31no—
20C3×C4⋊AGL(1,ℤ/9)—31no—
21C3×C4⋊((C3⋊D3)⋊C3)—31no—
22C3×C4⋊((C3⋊D3)⋊C3)—31no—
23C3×C4⋊((C3⋊D3)⋊C3)—61no—
24#6192⋊C3—32no—
25C3×#6198—32no—
26C2×#4178⋊C2C2×C312yescomplemented
27C4⋊#4185C2×C311yescharacteristic, complemented
28C4⋊#4179—21no—
29C4⋊#4179C2×C311yescharacteristic, complemented
30C4×#4178—21no—
31C4×#4178C2×C311yescharacteristic, complemented
32#10619565C2×C312yescomplemented
33C2×#6911C2311yescharacteristic
34#92333—22no—
35#4179⋊C3—22no—
36#92285D412yescomplemented
37C3×D3×D12—91no—
38C3×D3×D12—91no—
39C2×C4⋊AGL(1,ℤ/9)—91no—
40#88655—31no—
41D3×D36—31no—
42C2×C3⋊AGL(1,ℤ/9)C2×D312yescomplemented
43C2×C3×#272C2×D311yescharacteristic
44C2×D3×3−1+2—31no—
45C2×C3×AGL(1,ℤ/9)—32no—
46C2×C3×(C3⋊D3)⋊C3—32no—
47C2×C3×(C3⋊D3)⋊C3—32no—
48C2×C3×(C3⋊D3)⋊C3—62no—
49D3×AGL(1,ℤ/9)—64no—
50C3×((C3⋊D3)⋊C3)⋊C2—64no—
51C3×C4×3−1+2C2×D311yescharacteristic, complemented
52C3×C4×H(3,𝔽3)—21no—
53C3×C4×H(3,𝔽3)C2×D311yescharacteristic
54C3×#1084C2×D311yescharacteristic
55Dic3×3−1+2—31no—
56C2×#4185C22×C312yescomplemented
57C2×#4179—22no—
58C2×#4179C22×C312yescomplemented
59C2×#4178—21no—
60C2×#4178C22×C311yescharacteristic
61#4178⋊C2—24no—
62C4×#732—21no—
63C4×#732—21no—
64C4×#732—41no—
65C4×#732C22×C311yescharacteristic, complemented
66#66325—21no—
67#66325C22×C311yescharacteristic, complemented
68#6911C2×D411yescharacteristic, complemented, Sylow
69C2×C3×D32—92no—
70C2×C3×D32—92no—
71C3×C4⋊(C3⋊D3)—31no—
72C3×C4⋊(C3⋊D3)—31no—
73C3×C4⋊(C3⋊D3)—61no—
74C32×D12—91no—
75C32×D12—91no—
76C32×C4×D3—31no—
77C32×C4×D3—91no—
78C3×#626—92no—
79C3×#626—92no—
80C22×AGL(1,ℤ/9)—92no—
81C2×((C3⋊D3)⋊C3)⋊C2—32no—
82C2×D3×D9—32no—
83C2×C4×3−1+2—91no—
84C4×#272C3×D311yescharacteristic, complemented
85C4⋊(C3⋊D9)C3×D311yescharacteristic, complemented
86C4⋊AGL(1,ℤ/9)—31no—
87C4⋊AGL(1,ℤ/9)—61no—
88C4⋊AGL(1,ℤ/9)—91no—
89C4⋊AGL(1,ℤ/9)—92no—
90C4⋊((C3⋊D3)⋊C3)—31no—
91C4⋊((C3⋊D3)⋊C3)—31no—
92C4⋊((C3⋊D3)⋊C3)—61no—
93C4⋊((C3⋊D3)⋊C3)—61no—
94C4⋊((C3⋊D3)⋊C3)—61no—
95C4⋊((C3⋊D3)⋊C3)—61no—
96C4⋊((C3⋊D3)⋊C3)—61no—
97C4×D3×C9—31no—
98C4×D3×C9—61no—
99C3×D36—31no—
100#6198—32no—
101#6192—32no—
102C2×C3×3−1+2C22×D311yescharacteristic
103C2×C3×H(3,𝔽3)—21no—
104C2×C3×H(3,𝔽3)C22×D311yescharacteristic
105C3⋊AGL(1,ℤ/9)—22no—
106C3×#272D1212yescomplemented
107D3×3−1+2—32no—
108C3×AGL(1,ℤ/9)—62no—
109C3×(C3⋊D3)⋊C3—62no—
110C3×(C3⋊D3)⋊C3—62no—
111C3×(C3⋊D3)⋊C3—122no—
112C2×#732—21no—
113C2×#732—21no—
114C2×#732—41no—
115C2×#732C23×C311yescharacteristic
116#4185—22no—
117#4179—22no—
118#4179—42no—
119#4178—22no—
120#4178C3×D412yescomplemented
121C3×D3×D4—271no—
122C3×D3×D4—271no—
123C2×C3×D12—271no—
124D3×D12—91no—
125D3×D12—91no—
126C2×D36—91no—
127C2×C3×C3⋊D3—32no—
128C2×C3×C3⋊D3—32no—
129C2×C3×C3⋊D3—62no—
130C2×C32×D3—31no—
131C2×C32×D3—91no—
132C2×C32×D3—92no—
133C2×C32×D3—92no—
134C3×D32—184no—
135C3×D32—184no—
136C33×C4—31no—
137C33×C4—61no—
138C33×C4D3211yescharacteristic
139C32×Dic3—31no—
140C32×Dic3—91no—
141C22×3−1+2—91no—
142C2×#272C2×C3×D311yescharacteristic
143C2×C3⋊D9C2×C3×D312yescomplemented
144C2×AGL(1,ℤ/9)—32no—
145C2×AGL(1,ℤ/9)—62no—
146C2×AGL(1,ℤ/9)—92no—
147C2×AGL(1,ℤ/9)—184no—
148C2×(C3⋊D3)⋊C3—32no—
149C2×(C3⋊D3)⋊C3—32no—
150C2×(C3⋊D3)⋊C3—62no—
151C2×(C3⋊D3)⋊C3—62no—
152C2×(C3⋊D3)⋊C3—62no—
153C2×(C3⋊D3)⋊C3—62no—
154C2×(C3⋊D3)⋊C3—62no—
155C2×D3×C9—31no—
156C2×D3×C9—61no—
157C2×C3×D9—32no—
158((C3⋊D3)⋊C3)⋊C2—64no—
159D3×D9—64no—
160C4×3−1+2—31no—
161C4×3−1+2—91no—
162C4×3−1+2—121no—
163C4×H(3,𝔽3)—21no—
164C4×H(3,𝔽3)—21no—
165C4×H(3,𝔽3)—41no—
166C4×H(3,𝔽3)—61no—
167C4×H(3,𝔽3)—61no—
168C4×H(3,𝔽3)—121no—
169C4×H(3,𝔽3)C2×C3×D311yescharacteristic
170C3×C4×C9—21no—
171C3×C4×C9C2×C3×D311yescharacteristic, complemented
172#1084C2×C3×D311yescharacteristic
173C9×Dic3—31no—
174C9×Dic3—61no—
175C3×3−1+2D3×D411yescharacteristic, complemented
176C3×H(3,𝔽3)—21no—
177C3×H(3,𝔽3)C2×D1211yescharacteristic
178#732—21no—
179#732—21no—
180#732—41no—
181#732C2×C3×D411yescharacteristic, complemented
182C22×C3×D3—272no—
183C22×C3×D3—272no—
184C22×C3×D3—272no—
185C2×D32—92no—
186C2×D32—92no—
187C32×D4—271no—
188C32×D4—271no—
189C2×C32×C4—91no—
190C4⋊(C3⋊D3)—31no—
191C4⋊(C3⋊D3)—31no—
192C4⋊(C3⋊D3)—61no—
193C3×#67—272no—
194C3×#67—272no—
195C3×D12—91no—
196C3×D12—91no—
197C3×D12—91no—
198C3×D12—91no—
199C3×D12—181no—
200C3×D12—181no—
201C3×D12—272no—
202C3×D12—541no—
203C3×D12—541no—
204C3×D12—541no—
205C3×C4×D3—31no—
206C3×C4×D3—181no—
207C3×C4×D3—181no—
208C3×C4×D3—181no—
209C3×C4×D3—181no—
210#626—92no—
211#626—92no—
212C22×D9—92no—
213C2×C4×C9—91no—
214C2×C4×C9—181no—
215D36—31no—
216D36—91no—
217D36—92no—
218C2×C33—31no—
219C2×C33—61no—
220C2×C33C2×D3211yescharacteristic
221C3×C3⋊D3—62no—
222C3×C3⋊D3—62no—
223C3×C3⋊D3—122no—
224C32×D3—32no—
225C32×D3—92no—
226C32×D3—182no—
227C32×D3—182no—
228C2×3−1+2—31no—
229C2×3−1+2—92no—
230C2×3−1+2—121no—
231C2×H(3,𝔽3)—21no—
232C2×H(3,𝔽3)—21no—
233C2×H(3,𝔽3)—41no—
234C2×H(3,𝔽3)—61no—
235C2×H(3,𝔽3)—61no—
236C2×H(3,𝔽3)—121no—
237C2×H(3,𝔽3)C22×C3×D311yescharacteristic
238C2×C3×C9—21no—
239C2×C3×C9C22×C3×D311yescharacteristic
240#272C3×D1212yescomplemented
241C3⋊D9—22no—
242AGL(1,ℤ/9)—62no—
243AGL(1,ℤ/9)—122no—
244AGL(1,ℤ/9)—182no—
245(C3⋊D3)⋊C3—62no—
246(C3⋊D3)⋊C3—62no—
247(C3⋊D3)⋊C3—122no—
248(C3⋊D3)⋊C3—122no—
249(C3⋊D3)⋊C3—122no—
250(C3⋊D3)⋊C3—122no—
251(C3⋊D3)⋊C3—122no—
252D3×C9—32no—
253D3×C9—62no—
254C3×D9—62no—
255C2×C3×D4—811no—
256D3×D4—271no—
257D3×D4—271no—
258C2×D12—271no—
259C22×C32—91no—
260C22×C32—272no—
261C22×C32—272no—
262C2×C3⋊D3—32no—
263C2×C3⋊D3—32no—
264C2×C3⋊D3—62no—
265C2×C3×D3—31no—
266C2×C3×D3—92no—
267C2×C3×D3—92no—
268C2×C3×D3—92no—
269C2×C3×D3—92no—
270C2×C3×D3—181no—
271C2×C3×D3—181no—
272C2×C3×D3—181no—
273C2×C3×D3—181no—
274C2×C3×D3—182no—
275C2×C3×D3—182no—
276C2×C3×D3—542no—
277C2×C3×D3—542no—
278C2×C3×D3—542no—
279C2×C3×D3—544no—
280C2×C3×D3—544no—
281C2×C3×D3—544no—
282D32—184no—
283D32—184no—
284C32×C4—21no—
285C32×C4—61no—
286C32×C4—91no—
287C32×C4—121no—
288C32×C4—121no—
289C32×C4—121no—
290C32×C4—181no—
291C32×C4—361no—
292C32×C4—361no—
293C32×C4—361no—
294C32×C4—361no—
295C32×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
296C32×C4C3×D3211yescharacteristic
297C3×Dic3—31no—
298C3×Dic3—181no—
299C3×Dic3—181no—
300C3×Dic3—181no—
301C3×Dic3—181no—
302C22×C9—91no—
303C22×C9—181no—
304C2×D9—32no—
305C2×D9—92no—
306C2×D9—184no—
307C4×C9—31no—
308C4×C9—91no—
309C4×C9—121no—
310C4×C9—181no—
311C33—31no—
312C33—61no—
313C33D3×D1211yescharacteristic
3143−1+2—31no—
3153−1+2—121no—
316H(3,𝔽3)—21no—
317H(3,𝔽3)—21no—
318H(3,𝔽3)—41no—
319H(3,𝔽3)—61no—
320H(3,𝔽3)—61no—
321H(3,𝔽3)—121no—
322H(3,𝔽3)C2×C3×D1211yescharacteristic
323C3×C9—21no—
324C3×C9C3×D3×D411yescharacteristic, complemented
325C23×C3—812no—
326C22×D3—272no—
327C22×D3—272no—
328C22×D3—272no—
329C3×D4—271no—
330C3×D4—271no—
331C3×D4—271no—
332C3×D4—271no—
333C3×D4—541no—
334C3×D4—541no—
335C3×D4—812no—
336C2×C3×C4—91no—
337C2×C3×C4—541no—
338#67—272no—
339#67—272no—
340D12—91no—
341D12—91no—
342D12—272no—
343D12—541no—
344D12—541no—
345D12—541no—
346C4×D3—31no—
347C4×D3—91no—
348C2×C32—21no—
349C2×C32—61no—
350C2×C32—92no—
351C2×C32—121no—
352C2×C32—121no—
353C2×C32—121no—
354C2×C32—181no—
355C2×C32—361no—
356C2×C32—361no—
357C2×C32—361no—
358C2×C32—361no—
359C2×C32—542no—
360C2×C32—542no—
361C2×C32C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
362C2×C32C2×C3×D3211yescharacteristic
363C3⋊D3—62no—
364C3⋊D3—62no—
365C3⋊D3—122no—
366C3×D3—32no—
367C3×D3—182no—
368C3×D3—182no—
369C3×D3—182no—
370C3×D3—182no—
371C3×D3—182no—
372C3×D3—182no—
373C3×D3—182no—
374C3×D3—182no—
375C3×D3—362no—
376C3×D3—362no—
377C3×D3—1082no—
378C3×D3—1082no—
379C3×D3—1082no—
380C2×C9—31no—
381C2×C9—92no—
382C2×C9—121no—
383C2×C9—182no—
384D9—62no—
385D9—182no—
386C2×D4—811noSylow
387C22×C3—91no—
388C22×C3—272no—
389C22×C3—272no—
390C22×C3—272no—
391C22×C3—272no—
392C22×C3—541no—
393C22×C3—542no—
394C22×C3—542no—
395C22×C3—1624no—
396C2×D3—31no—
397C2×D3—91no—
398C2×D3—92no—
399C2×D3—92no—
400C2×D3—542no—
401C2×D3—542no—
402C2×D3—542no—
403C2×D3—544no—
404C2×D3—544no—
405C2×D3—544no—
406C3×C4—61no—
407C3×C4—61no—
408C3×C4—91no—
409C3×C4—121no—
410C3×C4—181no—
411C3×C4—361no—
412C3×C4—361no—
413C3×C4—361no—
414C3×C4—541no—
415C3×C4C3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
416Dic3—31no—
417Dic3—91no—
418C32—21no—
419C32—61no—
420C32—121no—
421C32—121no—
422C32—121no—
423C32—181no—
424C32—361no—
425C32—361no—
426C32—361no—
427C32—361no—
428C32#8865411yescharacteristic
429C32C3×D3×D1211yescharacteristic
430C9—31no—
431C9—121no—
432C23—812no—
433D4—271no—
434D4—271no—
435D4—812no—
436C2×C4—91no—
437C2×C3—61no—
438C2×C3—61no—
439C2×C3—92no—
440C2×C3—121no—
441C2×C3—181no—
442C2×C3—361no—
443C2×C3—361no—
444C2×C3—361no—
445C2×C3—542no—
446C2×C3—542no—
447C2×C3—542no—
448C2×C3—542no—
449C2×C3—542no—
450C2×C3—1082no—
451C2×C3—1082no—
452C2×C3C2×C3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
453D3—32no—
454D3—92no—
455D3—182no—
456D3—182no—
457D3—1082no—
458D3—1082no—
459D3—1082no—
460C22—91no—
461C22—272no—
462C22—272no—
463C22—1624no—
464C4—91no—
465C4(#4178⋊C2)⋊C311yescharacteristic, complemented
466C3—61no—
467C3—61no—
468C3—121no—
469C3—181no—
470C3—361no—
471C3—361no—
472C3—361no—
473C3C3×#8865411yescharacteristic
474C2—92no—
475C2—542no—
476C2—542no—
477C2C2×(#4178⋊C2)⋊C311yescharacteristic
subgroupG/H
centerC2C2×(#4178⋊C2)⋊C3
commutator, cosocleC2×#732C23×C3
FrattiniC2×C32C2×C3×D32
FittingC4×#6911C22
socleC2×C3C2×C3×((C3⋊D3)⋊C3)⋊C2
hypercenterC4(#4178⋊C2)⋊C3
hypocenter#732C2×C3×D4
LCSG ▷ C2×#732 ▷ #732
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×(#4178⋊C2)⋊C3C2semidirect
2. C4⋊#92333C2semidirect
3. C4⋊(#4179⋊C3)C2semidirect
4. C4×#92285C2semidirect
5. #10619565⋊C3C2semidirect
8. #13047150C3semidirect
9. C2×#92333C22semidirect
10. C2×#4179⋊C3C22semidirect
13. C4×#6911C22semidirect
14. #11757589C22semidirect
17. C4⋊(C3⋊AGL(1,ℤ/9))D3semidirect
18. C3×C4×#272D3semidirect
26. C2×#4178⋊C2C2×C3semidirect
27. C4⋊#4185C2×C3semidirect
29. C4⋊#4179C2×C3semidirect
31. C4×#4178C2×C3semidirect
32. #10619565C2×C3semidirect
36. #92285D4semidirect
42. C2×C3⋊AGL(1,ℤ/9)C2×D3semidirect
51. C3×C4×3−1+2C2×D3semidirect
56. C2×#4185C22×C3semidirect
58. C2×#4179C22×C3semidirect
65. C4×#732C22×C3semidirect
67. #66325C22×C3semidirect
68. #6911C2×D4semidirect
84. C4×#272C3×D3semidirect
85. C4⋊(C3⋊D9)C3×D3semidirect
106. C3×#272D12semidirect
120. #4178C3×D4semidirect
143. C2×C3⋊D9C2×C3×D3semidirect
171. C3×C4×C9C2×C3×D3semidirect
175. C3×3−1+2D3×D4semidirect
181. #732C2×C3×D4semidirect
240. #272C3×D12semidirect
324. C3×C9C3×D3×D4semidirect
465. C4(#4178⋊C2)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×(#4178⋊C2)⋊C32
C4⋊#923332
C4⋊(#4179⋊C3)2
C4×#922852
#10619565⋊C32
D3×C4⋊AGL(1,ℤ/9)3
C3×#886553
#130471503