Finite Group Databasebeta

← #13167027

Subgroups — #13167027

Order 1728.

616 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C3×#64306C211yesmaximal, characteristic
2C2×C4⋊(C3⋊#157)C211yesmaximal, characteristic, complemented
3#11723656C211yesmaximal, characteristic, complemented
4#11722760C212yesmaximal, complemented
5#11722758C212yesmaximal, complemented
6#10593319—31nomaximal
7C22×C3⋊#157—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C32×#235—21no—
10C3×#3811—21no—
11C3×#3805C412yescomplemented
12C3×#3802C2211yescharacteristic
13C3×C4×D32—21no—
14C3×#3798C412yescomplemented
15C3×#3797C2211yescharacteristic
16C3×#3796—21no—
17C4⋊(C3⋊#157)—21no—
18C4⋊(C3⋊#157)C2212yes—
19C3×#3748—21no—
20#88809—21no—
21#88808C2212yes—
22C3×D3×#41—61no—
23#64306D311yescharacteristic
24C2×C4⋊#157—31no—
25#63818—31no—
26D3×#208—61no—
27#63730—32no—
28#63729—32no—
29C22×C3×C3⋊D3D411yescharacteristic
30C2×C3×D32—21no—
31C2×C3⋊#157—22no—
32C2×C3⋊#157—42no—
33C2×C33×C4D411yescharacteristic
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C3×C4×C3⋊D3C2×C411yescharacteristic
36C3×C4×C3⋊D3D411yescharacteristic
37C3×C4×C3⋊D3D411yescharacteristic
38C3×C4×C3⋊D3C2311yescharacteristic
39C32×#67—22no—
40C32×D12—21no—
41C32×C4×D3—22no—
42C32×Dic6—21no—
43C3×#627C2×C412yescomplemented
44C3×#626—22no—
45C3×#625C2×C412yescomplemented
46C3×#624—21no—
47C3×D3×Dic3—22no—
48C3×C3⋊(C3⋊C8)—22no—
49#5232—91nomaximal
50C22×#157—61no—
51C32×#41—61no—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C3×#239—61no—
54C3×D3×Q8—61no—
55C3×#237—61no—
56C3×#237—62no—
57C3×D3×D4—61no—
58C3×D3×D4—62no—
59C3×#235—21no—
60C3×#235—21no—
61C3×#235—41no—
62C3×#235—62no—
63C2×C3×C4×D3—62no—
64C2×C3×C4×D3—61no—
65C2×C3×C4×D3—61no—
66#3811—21no—
67#3805Dic312yescomplemented
68#3802C2×D311yescharacteristic
69C4×D32—21no—
70#3798Dic312yescomplemented
71#3797C2×D311yescharacteristic
72#3796—21no—
73C4⋊#157—32no—
74C4⋊#157—61no—
75#3748—21no—
76C3×#208—61no—
77C3×#208—61no—
78C3×#208—121no—
79#3712—61no—
80#3711—62no—
81#3710—32no—
82D3×C3⋊C8—62no—
83C22×C33C22⋊C411yescharacteristic
84C2×C3×C3⋊D3—21no—
85C2×C3×C3⋊D3—21no—
86C2×C3×C3⋊D3C22⋊C411yescharacteristic
87C2×C3×C3⋊D3C2×D411yescharacteristic
88C2×C32×D3—22no—
89C3×D32—41no—
90C3×D32—41no—
91C3⋊#157—42no—
92C33×C4C22⋊C411yescharacteristic
93C33×C4C2×D411yescharacteristic
94C3×C3⋊Dic3C22⋊C411yescharacteristic
95C3×C3⋊Dic3C22×C411yescharacteristic
96C32×Dic3—22no—
97C2×C3×#41—91no—
98D3×#41—61no—
99C2×C4⋊Dic3—91no—
100C2×#208—91no—
101D3×M16—181no—
102#835—92no—
103#832—92no—
104C22×C3⋊D3#6711yescharacteristic
105C22×C3×D3—61no—
106C22×C3×D3—61no—
107C22×C3×D3—62no—
108C2×D32—21no—
109C2×#157—62no—
110C2×#157—122no—
111C32×Q8—61no—
112C32×D4—61no—
113C32×D4—62no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4—62no—
119C2×C32×C4#6711yescharacteristic
120C2×C3⋊Dic3C2×Dic311yescharacteristic
121C4×C3⋊D3C2×Dic311yescharacteristic
122C4×C3⋊D3#6711yescharacteristic
123C4×C3⋊D3#6711yescharacteristic
124C4×C3⋊D3C22×D311yescharacteristic
125C3×#67—22no—
126C3×#67—22no—
127C3×#67—42no—
128C3×#67—62no—
129C3×#67—62no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—62no—
133C3×D12—21no—
134C3×D12—21no—
135C3×D12—41no—
136C3×D12—62no—
137C3×C4×D3—22no—
138C3×C4×D3—22no—
139C3×C4×D3—42no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144C3×C4×D3—61no—
145C3×C4×D3—61no—
146C3×C4×D3—61no—
147C3×C4×D3—61no—
148C3×C4×D3—62no—
149C3×C4×D3—62no—
150C3×Dic6—21no—
151C3×Dic6—21no—
152C3×Dic6—41no—
153C3×Dic6—62no—
154#627C2×Dic312yescomplemented
155#626—22no—
156#625C2×Dic312yescomplemented
157#624—21no—
158D3×Dic3—22no—
159C3⋊(C3⋊C8)—22no—
160C3×C3⋊C8—62no—
161C3×C3⋊C8—62no—
162C3×C3⋊C8—122no—
163#417—271noSylow
164C2×C33—21no—
165C2×C33C2×C22⋊C411yescharacteristic
166C3×C3⋊D3—21no—
167C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
168C3×C3⋊D3#13711yescharacteristic
169C32×D3—42no—
170C3×#41—61no—
171C3×#41—61no—
172C3×#41—92no—
173C3×#41—92no—
174C3×#41—121no—
175C3×#41—181no—
176C2×C3×Q8—91no—
177C2×C3×D4—91no—
178C2×C3×D4—181no—
179C22×C3×C4—91no—
180C22×C3×C4—181no—
181C22×Dic3—181no—
182#239—61no—
183D3×Q8—61no—
184#237—62no—
185#237—61no—
186D3×D4—61no—
187D3×D4—62no—
188#235—21no—
189#235—62no—
190C2×C4×D3—61no—
191C2×C4×D3—61no—
192C2×C4×D3—62no—
193C3×M16—181no—
194C4⋊Dic3—92no—
195C4⋊Dic3—181no—
196#208—61no—
197#208—61no—
198#208—121no—
199#208—181no—
200C2×C3⋊C8—92no—
201ΓL(1,𝔽25)—182no—
202D3×C8—182no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32—62no—
208C22×C32#21711yescharacteristic
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3#21711yescharacteristic
212C2×C3⋊D3C2×#6711yescharacteristic
213C2×C3×D3—22no—
214C2×C3×D3—22no—
215C2×C3×D3—42no—
216C2×C3×D3—61no—
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223C2×C3×D3—121no—
224C2×C3×D3—122no—
225C2×C3×D3—122no—
226D32—41no—
227D32—41no—
228#157—122no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—41no—
234C32×C4—41no—
235C32×C4—41no—
236C32×C4—41no—
237C32×C4—62no—
238C32×C4#21711yescharacteristic
239C32×C4C2×#6711yescharacteristic
240C3⋊Dic3#21711yescharacteristic
241C3⋊Dic3C22×Dic311yescharacteristic
242C3×Dic3—22no—
243C3×Dic3—22no—
244C3×Dic3—42no—
245C3×Dic3—61no—
246C3×Dic3—61no—
247C3×Dic3—61no—
248C3×Dic3—61no—
249C2×#41—91no—
250C2×M16—271no—
251C2×C4⋊C4—271no—
252#103—272no—
253#102—272no—
254C33#41711yescharacteristic, complemented, Sylow
255C23×C3—91no—
256C23×C3—181no—
257C22×D3—61no—
258C22×D3—61no—
259C22×D3—62no—
260C3×Q8—61no—
261C3×Q8—61no—
262C3×Q8—92no—
263C3×Q8—121no—
264C3×D4—61no—
265C3×D4—62no—
266C3×D4—62no—
267C3×D4—61no—
268C3×D4—92no—
269C3×D4—121no—
270C3×D4—122no—
271C3×D4—182no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—62no—
277C2×C3×C4—62no—
278C2×C3×C4—91no—
279C2×C3×C4—91no—
280C2×C3×C4—91no—
281C2×C3×C4—91no—
282C2×C3×C4—91no—
283C2×C3×C4—122no—
284C2×C3×C4—181no—
285C2×C3×C4—182no—
286C2×C3×C4GO+(4,𝔽2)11yescharacteristic
287#67—22no—
288#67—62no—
289#67—62no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—62no—
293C2×Dic3—182no—
294C2×Dic3—362no—
295D12—21no—
296D12—62no—
297C4×D3—22no—
298C4×D3—61no—
299C4×D3—61no—
300C4×D3—61no—
301C4×D3—61no—
302C4×D3—61no—
303C4×D3—61no—
304C4×D3—61no—
305C4×D3—61no—
306C4×D3—62no—
307C4×D3—62no—
308Dic6—21no—
309Dic6—62no—
310C3×C8—182no—
311C3⋊C8—62no—
312C3⋊C8—62no—
313C3⋊C8—122no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32—41no—
321C2×C32—81no—
322C2×C32—81no—
323C2×C32—122no—
324C2×C32C2×#21711yescharacteristic
325C3⋊D3—21no—
326C3⋊D3#94911yescharacteristic
327C3⋊D3#95111yescharacteristic
328C3×D3—42no—
329C3×D3—42no—
330C3×D3—61no—
331C3×D3—61no—
332C3×D3—61no—
333C3×D3—61no—
334C3×D3—82no—
335C3×D3—121no—
336C3×D3—121no—
337#41—61no—
338#41—92no—
339#41—92no—
340#41—181no—
341C2×Q8—91no—
342C2×D4—91no—
343C2×D4—181no—
344C22×C4—91no—
345C22×C4—181no—
346C22×C4—541no—
347M16—181no—
348M16—541no—
349C2×C8—272no—
350C4⋊C4—272no—
351C4⋊C4—541no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—62no—
357C22×C3—62no—
358C22×C3—91no—
359C22×C3—91no—
360C22×C3—122no—
361C22×C3—181no—
362C22×C3—181no—
363C22×C3—361no—
364C22×C3—361no—
365C22×C3D32⋊C411yescharacteristic
366C2×D3—22no—
367C2×D3—61no—
368C2×D3—61no—
369C2×D3—61no—
370C2×D3—61no—
371C2×D3—121no—
372C2×D3—121no—
373C2×D3—121no—
374C2×D3—121no—
375C2×D3—122no—
376C2×D3—122no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—41no—
382C3×C4—41no—
383C3×C4—41no—
384C3×C4—41no—
385C3×C4—62no—
386C3×C4—62no—
387C3×C4—91no—
388C3×C4—91no—
389C3×C4—122no—
390C3×C4D32⋊C411yescharacteristic
391C3×C4C2×GO+(4,𝔽2)11yescharacteristic
392Dic3—22no—
393Dic3—61no—
394Dic3—61no—
395Dic3—61no—
396Dic3—61no—
397Dic3—362no—
398C32—21no—
399C32—21no—
400C32—41no—
401C32—41no—
402C32#523211yescharacteristic, complemented
403C23—91no—
404C23—181no—
405Q8—61no—
406Q8—92no—
407D4—62no—
408D4—61no—
409D4—92no—
410D4—182no—
411C2×C4—62no—
412C2×C4—91no—
413C2×C4—91no—
414C2×C4—91no—
415C2×C4—91no—
416C2×C4—91no—
417C2×C4—181no—
418C2×C4—182no—
419C2×C4—542no—
420C2×C4—1082no—
421C2×C4D32⋊D311yescharacteristic
422C8—182no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—41no—
430C2×C3—81no—
431C2×C3—81no—
432C2×C3—91no—
433C2×C3—91no—
434C2×C3—122no—
435C2×C3—122no—
436C2×C3—181no—
437C2×C3—242no—
438C2×C3C2×D32⋊C411yescharacteristic
439D3—42no—
440D3—61no—
441D3—61no—
442D3—61no—
443D3—61no—
444D3—121no—
445D3—121no—
446C22—62no—
447C22—91no—
448C22—91no—
449C22—181no—
450C22—181no—
451C22—361no—
452C22—361no—
453C22D32⋊Dic311yescharacteristic
454C4—62no—
455C4—91no—
456C4—91no—
457C4—1082no—
458C4D32⋊Dic311yescharacteristic
459C4C2×D32⋊D311yescharacteristic
460C3—21no—
461C3—21no—
462C3—41no—
463C3—41no—
464C3#1059331911yescharacteristic, complemented
465C2—21no—
466C2—91no—
467C2—91no—
468C2—122no—
469C2—181no—
470C2C2×D32⋊Dic311yescharacteristic
subgroupG/H
centerC2C2×D32⋊Dic3
commutatorC3×C3⋊Dic3C22×C4
FrattiniC4C2×D32⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radicalC3×#64306C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D32⋊D3
hypocenterC33#417
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(C3⋊#157)C2semidirect
3. #11723656C2semidirect
4. #11722760C2semidirect
5. #11722758C2semidirect
11. C3×#3805C4semidirect
14. C3×#3798C4semidirect
43. C3×#627C2×C4semidirect
45. C3×#625C2×C4semidirect
67. #3805Dic3semidirect
70. #3798Dic3semidirect
154. #627C2×Dic3semidirect
156. #625C2×Dic3semidirect
254. C33#417semidirect
402. C32#5232semidirect
464. C3#10593319semidirect

Maximal subgroups

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

subgroupindex
C3×#643062
C2×C4⋊(C3⋊#157)2
#117236562
#117227602
#117227582
#105933193
#52329