Finite Group Databasebeta

← #13167026

Subgroups — #13167026

Order 1728.

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

normal subgroups only

This group has 758 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
1C2×C3×#3802C211yesmaximal, characteristic
2#11724514C211yesmaximal, characteristic, complemented
3#11723656C211yesmaximal, characteristic, complemented
4#11722758C214yesmaximal, complemented
5#10593318—31nomaximal
6C22×C3×D32—21no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×D12—21no—
9C2×C3×#625C412yescomplemented
10C3×#3802—21no—
11C3×#3802—21no—
12C3×#3802C412yescomplemented
13C3×#3802C2212yes—
14#88994—21no—
15C4⋊(C3⋊#157)C2212yescomplemented
16C4×C3⋊#157—21no—
17C3×#3748—21no—
18#88809—21no—
19#88808C2212yescomplemented
20C2×C3×D3×D4—61no—
21C2×#3802D311yescharacteristic
22#64286—31no—
23#63818—31no—
24D3×#208—61no—
25#63729—34no—
26C22×C3×C3⋊D3D411yescharacteristic
27C22×C32×D3—22no—
28C2×C3×D32—22no—
29C2×C3×D32—22no—
30C2×C3×D32—44no—
31C2×C3⋊#157—22no—
32C2×C33×C4D411yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C3×C4×C3⋊D3C2×C411yescharacteristic
35C3×C4×C3⋊D3D411yescharacteristic
36C3×C4×C3⋊D3D411yescharacteristic
37C3×C4×C3⋊D3C2311yescharacteristic
38C32×D12—22no—
39C32×D12—22no—
40C3×#625—22no—
41C3×#625C2×C414yes—
42C3×C3⋊(C3⋊C8)—22no—
43#5215—91nomaximal
44C23×C3×D3—62no—
45C22×D32—21no—
46C2×C32×D4—61no—
47C2×C4×C3⋊D3C2×D311yescharacteristic
48C2×C3×#67—62no—
49C3×D3×D4—62no—
50C3×D3×D4—62no—
51C3×D3×D4—62no—
52C3×D3×D4—62no—
53C2×C3×D12—21no—
54C2×C3×D12—21no—
55C2×C3×D12—41no—
56C2×C3×C4×D3—61no—
57C2×C3×C4×D3—61no—
58C2×#625Dic312yescomplemented
59#3802—21no—
60#3802—21no—
61#3802Dic312yescomplemented
62#3802C2×D312yes—
63C22⋊#157—61no—
64C4⋊#157—32no—
65C4×#157—61no—
66#3748—21no—
67C3×#208—61no—
68C3×#208—61no—
69C3×#208—121no—
70#3712—61no—
71#3711—62no—
72#3710—32no—
73D3×C3⋊C8—62no—
74C22×C33C22⋊C411yescharacteristic
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3C22⋊C411yescharacteristic
77C2×C3×C3⋊D3C2×D411yescharacteristic
78C2×C32×D3—24no—
79C2×C32×D3—44no—
80C3×D32—44no—
81C3×D32—44no—
82C3⋊#157—42no—
83C33×C4C22⋊C411yescharacteristic
84C33×C4C2×D411yescharacteristic
85C3×C3⋊Dic3C22⋊C411yescharacteristic
86C3×C3⋊Dic3C22×C411yescharacteristic
87C22×C3×D4—91no—
88C2×D3×D4—61no—
89#926—91no—
90C2×#208—91no—
91D3×M16—181no—
92#832—94no—
93C23×C32—62no—
94C22×C3⋊D3#6711yescharacteristic
95C22×C3×D3—22no—
96C22×C3×D3—22no—
97C22×C3×D3—42no—
98C22×C3×D3—61no—
99C22×C3×D3—61no—
100C22×C3×D3—64no—
101C22×C3×D3—64no—
102C22×C3×D3—128no—
103C2×D32—22no—
104C2×D32—22no—
105C2×D32—44no—
106C2×#157—62no—
107C32×D4—62no—
108C32×D4—62no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—41no—
112C2×C32×C4—41no—
113C2×C32×C4#6711yescharacteristic
114C2×C3⋊Dic3C2×Dic311yescharacteristic
115C4×C3⋊D3C2×Dic311yescharacteristic
116C4×C3⋊D3#6711yescharacteristic
117C4×C3⋊D3#6711yescharacteristic
118C4×C3⋊D3C22×D311yescharacteristic
119C3×#67—64no—
120C3×#67—64no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C3×D12—22no—
124C3×D12—22no—
125C3×D12—22no—
126C3×D12—22no—
127C3×D12—42no—
128C3×D12—42no—
129C3×C4×D3—61no—
130C3×C4×D3—61no—
131C3×C4×D3—61no—
132C3×C4×D3—61no—
133C3×C4×D3—61no—
134C3×C4×D3—61no—
135C3×C4×D3—61no—
136C3×C4×D3—61no—
137#625—22no—
138#625C2×Dic314yes—
139C3⋊(C3⋊C8)—22no—
140C3×C3⋊C8—62no—
141C3×C3⋊C8—62no—
142C3×C3⋊C8—122no—
143#418—271noSylow
144C2×C33—21no—
145C2×C33C2×C22⋊C411yescharacteristic
146C3×C3⋊D3—21no—
147C3×C3⋊D3AGL(1,ℤ/8)12yes—
148C32×D3—44no—
149C24×C3—181no—
150C23×D3—62no—
151C2×C3×D4—61no—
152C2×C3×D4—61no—
153C2×C3×D4—92no—
154C2×C3×D4—92no—
155C2×C3×D4—92no—
156C2×C3×D4—121no—
157C2×C3×D4—181no—
158C2×C3×D4—181no—
159C22×C3×C4—91no—
160C2×#67—62no—
161D3×D4—62no—
162D3×D4—62no—
163D3×D4—62no—
164D3×D4—62no—
165C2×D12—21no—
166C2×C4×D3—61no—
167C2×C4×D3—61no—
168C3×M16—181no—
169#217—181no—
170C4⋊Dic3—92no—
171C4×Dic3—181no—
172#208—61no—
173#208—61no—
174#208—121no—
175#208—181no—
176C2×C3⋊C8—92no—
177ΓL(1,𝔽25)—182no—
178D3×C8—182no—
179C22×C32—21no—
180C22×C32—21no—
181C22×C32—41no—
182C22×C32—41no—
183C22×C32—64no—
184C22×C32—124no—
185C22×C32#21711yescharacteristic
186C2×C3⋊D3—22no—
187C2×C3⋊D3#21711yescharacteristic
188C2×C3⋊D3C2×#6711yescharacteristic
189C2×C3×D3—24no—
190C2×C3×D3—24no—
191C2×C3×D3—44no—
192C2×C3×D3—44no—
193C2×C3×D3—44no—
194C2×C3×D3—61no—
195C2×C3×D3—61no—
196C2×C3×D3—61no—
197C2×C3×D3—61no—
198C2×C3×D3—64no—
199C2×C3×D3—84no—
200C2×C3×D3—122no—
201C2×C3×D3—128no—
202C2×C3×D3—128no—
203D32—44no—
204D32—44no—
205#157—122no—
206C32×C4—21no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—41no—
211C32×C4—41no—
212C32×C4—41no—
213C32×C4—41no—
214C32×C4#21711yescharacteristic
215C32×C4C2×#6711yescharacteristic
216C3⋊Dic3#21711yescharacteristic
217C3⋊Dic3C22×Dic311yescharacteristic
218C3×Dic3—61no—
219C3×Dic3—61no—
220C3×Dic3—61no—
221C3×Dic3—61no—
222C22×D4—91no—
223C2×M16—271no—
224#117—271no—
225#102—274no—
226C33#41811yescharacteristic, complemented, Sylow
227C23×C3—62no—
228C23×C3—62no—
229C23×C3—91no—
230C23×C3—122no—
231C23×C3—182no—
232C23×C3—182no—
233C23×C3—364no—
234C22×D3—22no—
235C22×D3—61no—
236C22×D3—61no—
237C22×D3—64no—
238C22×D3—64no—
239C22×D3—128no—
240C3×D4—62no—
241C3×D4—62no—
242C3×D4—62no—
243C3×D4—62no—
244C3×D4—94no—
245C3×D4—122no—
246C3×D4—122no—
247C3×D4—182no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—91no—
253C2×C3×C4—91no—
254C2×C3×C4—91no—
255C2×C3×C4—91no—
256C2×C3×C4—91no—
257C2×C3×C4GO+(4,𝔽2)11yescharacteristic
258#67—64no—
259#67—64no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—182no—
263D12—22no—
264D12—22no—
265C4×D3—61no—
266C4×D3—61no—
267C4×D3—61no—
268C4×D3—61no—
269C4×D3—61no—
270C4×D3—61no—
271C4×D3—61no—
272C4×D3—61no—
273C3×C8—182no—
274C3⋊C8—62no—
275C3⋊C8—62no—
276C3⋊C8—122no—
277C2×C32—22no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—41no—
282C2×C32—41no—
283C2×C32—41no—
284C2×C32—42no—
285C2×C32—81no—
286C2×C32—124no—
287C2×C32C2×#21711yescharacteristic
288C3⋊D3—21no—
289C3⋊D3#93212yes—
290C3×D3—44no—
291C3×D3—44no—
292C3×D3—62no—
293C3×D3—62no—
294C3×D3—62no—
295C3×D3—84no—
296C3×D3—121no—
297C24—181no—
298C2×D4—61no—
299C2×D4—92no—
300C2×D4—92no—
301C2×D4—92no—
302C2×D4—181no—
303C2×D4—181no—
304C22×C4—91no—
305M16—181no—
306M16—541no—
307C2×C8—272no—
308C4⋊C4—272no—
309C22⋊C4—541no—
310C42—541no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—64no—
316C22×C3—64no—
317C22×C3—91no—
318C22×C3—91no—
319C22×C3—124no—
320C22×C3—124no—
321C22×C3—124no—
322C22×C3—182no—
323C22×C3—244no—
324C22×C3—364no—
325C22×C3—364no—
326C22×C3D32⋊C411yescharacteristic
327C2×D3—24no—
328C2×D3—44no—
329C2×D3—61no—
330C2×D3—61no—
331C2×D3—61no—
332C2×D3—61no—
333C2×D3—64no—
334C2×D3—122no—
335C2×D3—128no—
336C2×D3—128no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—41no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4—41no—
345C3×C4—91no—
346C3×C4—91no—
347C3×C4D32⋊C411yescharacteristic
348C3×C4C2×GO+(4,𝔽2)11yescharacteristic
349Dic3—61no—
350Dic3—61no—
351Dic3—61no—
352Dic3—61no—
353Dic3—362no—
354C32—21no—
355C32—21no—
356C32—41no—
357C32—41no—
358C32#521511yescharacteristic, complemented
359C23—62no—
360C23—91no—
361C23—182no—
362C23—182no—
363C23—364no—
364D4—62no—
365D4—62no—
366D4—94no—
367D4—182no—
368C2×C4—91no—
369C2×C4—91no—
370C2×C4—91no—
371C2×C4—91no—
372C2×C4—91no—
373C2×C4—542no—
374C2×C4D32⋊D311yescharacteristic
375C8—182no—
376C2×C3—22no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—41no—
381C2×C3—42no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—81no—
385C2×C3—92no—
386C2×C3—124no—
387C2×C3—124no—
388C2×C3—181no—
389C2×C3—244no—
390C2×C3C2×D32⋊C411yescharacteristic
391D3—44no—
392D3—62no—
393D3—62no—
394D3—62no—
395D3—121no—
396C22—64no—
397C22—91no—
398C22—91no—
399C22—124no—
400C22—182no—
401C22—364no—
402C22—364no—
403C22D32⋊Dic311yescharacteristic
404C4—91no—
405C4—91no—
406C4—1082no—
407C4D32⋊Dic311yescharacteristic
408C4C2×D32⋊D311yescharacteristic
409C3—21no—
410C3—21no—
411C3—41no—
412C3—41no—
413C3#1059331811yescharacteristic, complemented
414C2—21no—
415C2—92no—
416C2—124no—
417C2—181no—
418C2C2×D32⋊Dic311yescharacteristic
subgroupG/H
centerC2C2×D32⋊Dic3
commutatorC3×C3⋊Dic3C22×C4
FrattiniC4C2×D32⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3802C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D32⋊D3
hypocenterC33#418
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. #11724514C2semidirect
3. #11723656C2semidirect
4. #11722758C2semidirect
9. C2×C3×#625C4semidirect
12. C3×#3802C4semidirect
15. C4⋊(C3⋊#157)C22semidirect
19. #88808C22semidirect
58. C2×#625Dic3semidirect
61. #3802Dic3semidirect
226. C33#418semidirect
358. C32#5215semidirect
413. C3#10593318semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38022
#117245142
#117236562
#117227582
#105933183
#52159