Finite Group Databasebeta

← #13161044

Subgroups — #13161044

Order 1728.

920 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 156 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×#3754C211yesmaximal, characteristic, complemented
2C32×#938C211yesmaximal, characteristic
3#11723425C211yesmaximal, characteristic, complemented
4C2×#88786C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6#11722057C211yesmaximal, characteristic, complemented
7#11722039C211yesmaximal, characteristic, complemented
8#10592741—31nomaximal
9#10591502—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×#67—21no—
13C2×C32×C4×D3C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3754—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#217—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20#88806—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88786—22no—
23C32×#121—31no—
24C2×#3754D311yescharacteristic, complemented
25#64133—31no—
26C2×#3748—31no—
27C3×#938D314yes—
28C2×C3×#214—34no—
29#63856—34no—
30C2×#3711—34no—
31#63659—31no—
32#63659—34no—
33#63659—64no—
34#63637—31no—
35#63634—31no—
36#63575—34no—
37#63565—34no—
38C23×C33—21no—
39C22×C32×D3D411yescharacteristic
40C33×D4—21no—
41C33×D4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×#634—21no—
46C3×#634—21no—
47C32×#67—42no—
48C2×C32×Dic3—21no—
49C2×C32×Dic3D411yescharacteristic
50C32×C4×D3—21no—
51C32×C4×D3—21no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C3×C3⋊(C3⋊C8)D412yescomplemented
54#5174—91no—
55#5042—94no—
56C2×C32×D4—24no—
57C2×C32×D4—61no—
58C2×C32×D4C2×D314yes—
59C2×C32×D4C2×D311yescharacteristic
60C22×C32×C4—31no—
61C2×#634C2×D311yescharacteristic
62C2×C3×#67—24no—
63C2×C3×C4×D3C2×D314yes—
64C2×C3×Dic6—34no—
65C32×C4⋊C4—31no—
66C32×C22⋊C4—61no—
67#3757—61no—
68#3754—22no—
69C4⋊(C3⋊Dic3)—31no—
70C4⋊(C3⋊Dic3)—34no—
71C4⋊(C3⋊Dic3)—64no—
72#3750—61no—
73#3748—62no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×#217—24no—
79C3×#214—68no—
80C3×C4⋊Dic3C2×D314yes—
81C2×C3×C3⋊C8—34no—
82#3724—64no—
83(C2×D3)⋊Dic3—64no—
84#3711—68no—
85C22×C33—41no—
86C22×C33—41no—
87C22×C33—41no—
88C22×C33C2×D411yescharacteristic
89C2×C32×D3—21no—
90C2×C32×D3—21no—
91C2×C32×D3—21no—
92C33×C4C2×D411yescharacteristic
93C33×C4#4111yescharacteristic
94C3⋊(C3⋊Dic3)—41no—
95C3×C3⋊Dic3—41no—
96C32×Dic3—21no—
97C32×Dic3—41no—
98C2×C3×SD16—91no—
99C3×#121—34no—
100#946—91no—
101#938C3⋊D311yescharacteristic
102C2×#214—34no—
103#924—94no—
104C2×#208—94no—
105C2×ΓL(1,𝔽25)—91no—
106#835—91no—
107#832—94no—
108#832—91no—
109#832—184no—
110#818—91no—
111#810—94no—
112#807—94no—
113C23×C32—21no—
114C23×C32—24no—
115C23×C32—31no—
116C23×C32—44no—
117C22×C3×D3#6714yes—
118C32×D4—28no—
119C32×D4—24no—
120C32×D4—21no—
121C32×D4—21no—
122C32×D4—24no—
123C32×D4—44no—
124C32×D4—122no—
125C2×C32×C4—24no—
126C2×C32×C4—31no—
127C2×C32×C4—61no—
128C2×C32×C4—61no—
129C2×C32×C4—61no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—61no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—124no—
136#634—21no—
137#634—21no—
138C3×#67—48no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—64no—
141C2×C3×Dic3#6714yes—
142C3×C4×D3—24no—
143C3×C4×D3—24no—
144C3×Dic6—64no—
145C3×Dic6—64no—
146C3⋊(C3⋊C8)—61no—
147C3⋊(C3⋊C8)—64no—
148C3⋊(C3⋊C8)—124no—
149C3⋊(C3⋊C8)#6712yescomplemented
150C3×C3⋊C8—38no—
151#468—271noSylow
152C2×C33—81no—
153C2×C33#12111yescharacteristic
154C2×C33AGL(1,ℤ/8)11yescharacteristic
155C2×C33#13711yescharacteristic
156C32×D3—41no—
157C2×C3×Q8—91no—
158C2×C3×D4—24no—
159C2×C3×D4—64no—
160C2×C3×D4D3214yes—
161C2×C3×D4C2×C3⋊D311yescharacteristic
162C22×C3×C4—34no—
163C2×#67—21no—
164C2×C4×D3C2×C3⋊D311yescharacteristic
165C2×Dic6—34no—
166C3×SD16—182no—
167C2×C3×C8—91no—
168C3×C4⋊C4—34no—
169C3×C22⋊C4—64no—
170#217—21no—
171#217—184no—
172#214—68no—
173(C2×D3)⋊C4—181no—
174C4⋊Dic3—94no—
175C4⋊Dic3—91no—
176C4⋊Dic3—184no—
177C4⋊Dic3C2×C3⋊D311yescharacteristic
178Dic3⋊C4—184no—
179Dic3⋊C4—181no—
180#208—188no—
181C2×C3⋊C8—34no—
182C2×C3⋊C8—94no—
183C2×C3⋊C8—91no—
184C2×C3⋊C8—184no—
185ΓL(1,𝔽25)—182no—
186C22×C32—24no—
187C22×C32—48no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—41no—
191C22×C32—44no—
192C22×C32—44no—
193C22×C32—44no—
194C22×C32—61no—
195C22×C32—61no—
196C22×C32—61no—
197C22×C32—84no—
198C22×C32—84no—
199C22×C32D3×D411yescharacteristic
200C22×C32C2×#6714yes—
201C2×C3×D3—24no—
202C2×C3×D3—24no—
203C2×C3×D3—24no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—61no—
207C32×C4—121no—
208C32×C4D3×D414yes—
209C32×C4#23714yes—
210C32×C4#23711yescharacteristic
211C32×C4C2×#6711yescharacteristic
212C3⋊Dic3—41no—
213C3⋊Dic3—121no—
214C3⋊Dic3—124no—
215C3⋊Dic3—244no—
216C3×Dic3—24no—
217C3×Dic3—44no—
218C3×Dic3—124no—
219C2×SD16—91no—
220C2×M16—271no—
221C22⋊Q8—271no—
222#121—31no—
223#107—271no—
224#103—271no—
225#102—271no—
226C33#46811yescharacteristic, complemented, Sylow
227C23×C3—24no—
228C23×C3—21no—
229C23×C3—34no—
230C23×C3—44no—
231C22×D3#63811yescharacteristic
232C3×Q8—181no—
233C3×Q8—181no—
234C3×D4—28no—
235C3×D4—24no—
236C3×D4—24no—
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—44no—
240C3×D4—128no—
241C2×C3×C4—24no—
242C2×C3×C4—34no—
243C2×C3×C4—64no—
244C2×C3×C4—64no—
245C2×C3×C4—64no—
246C2×C3×C4—181no—
247C2×C3×C4C2×D3214yes—
248C2×C3×C4C22×C3⋊D311yescharacteristic
249#67—42no—
250C2×Dic3—21no—
251C2×Dic3—64no—
252C2×Dic3—184no—
253C2×Dic3—181no—
254C2×Dic3—364no—
255C2×Dic3#63811yescharacteristic
256C4×D3—21no—
257C4×D3—21no—
258Dic6—64no—
259Dic6—64no—
260C3×C8—92no—
261C3⋊C8—38no—
262C3⋊C8—184no—
263C3⋊C8—181no—
264C3⋊C8—364no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—88no—
269C2×C32—81no—
270C2×C32—84no—
271C2×C32—121no—
272C2×C32#91411yescharacteristic
273C2×C32#91511yescharacteristic
274C2×C32#93214yes—
275C2×C32#93811yescharacteristic
276C2×C32#93914yes—
277C2×C32#95114yes—
278C3×D3—44no—
279C2×Q8—91no—
280C2×D4—61no—
281C2×D4D3×C3⋊D311yescharacteristic
282C22×C4—31no—
283SD16—182no—
284M16—542no—
285C2×C8—91no—
286C2×C8—271no—
287C4⋊C4—31no—
288C4⋊C4—271no—
289C4⋊C4—541no—
290C22⋊C4—61no—
291C22⋊C4—541no—
292C22×C3—24no—
293C22×C3—48no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—44no—
298C22×C3—44no—
299C22×C3—44no—
300C22×C3—64no—
301C22×C3—64no—
302C22×C3—64no—
303C22×C3—84no—
304C22×C3—84no—
305C22×C3D3×#6714yes—
306C22×C3C2×#63811yescharacteristic
307C2×D3—21no—
308C2×D3—21no—
309C2×D3—21no—
310C3×C4—24no—
311C3×C4—24no—
312C3×C4—64no—
313C3×C4—124no—
314C3×C4—361no—
315C3×C4#380514yes—
316C3×C4D3×#6714yes—
317C3×C4D4×C3⋊D311yescharacteristic
318C3×C4#383011yescharacteristic
319Dic3—21no—
320Dic3—41no—
321Dic3—124no—
322Dic3—361no—
323Dic3—364no—
324Dic3—724no—
325C32—24no—
326C32#504214yescomplemented
327C32#517411yescharacteristic, complemented
328C23—21no—
329C23—31no—
330Q8—181no—
331Q8—181no—
332D4—21no—
333D4—21no—
334D4—122no—
335C2×C4—31no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—181no—
340C2×C4—541no—
341C2×C4C2×D3×C3⋊D311yescharacteristic
342C8—92no—
343C8—541no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—88no—
348C2×C3—84no—
349C2×C3—81no—
350C2×C3—124no—
351C2×C3#6392914yes—
352C2×C3#6393414yes—
353C2×C3#6397014yes—
354C2×C3#6414111yescharacteristic
355C2×C3#6414811yescharacteristic
356C2×C3#6416011yescharacteristic
357D3—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—61no—
362C22—61no—
363C22—61no—
364C22D3×#63811yescharacteristic
365C4—61no—
366C4—121no—
367C4—361no—
368C4—1081no—
369C4#8902911yescharacteristic
370C4C3⋊D3×#6711yescharacteristic
371C3—24no—
372C3#1059150214yescomplemented
373C3#1059274111yescharacteristic, complemented
374C2—81no—
375C2—121no—
376C2#1172355011yescharacteristic
377C2#1172355711yescharacteristic
378C2#1172361511yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC32×#938C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#468
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3754C2semidirect
3. #11723425C2semidirect
4. C2×#88786C2semidirect
5. #11722202C2semidirect
6. #11722057C2semidirect
7. #11722039C2semidirect
24. C2×#3754D3semidirect
53. C3×C3⋊(C3⋊C8)D4semidirect
149. C3⋊(C3⋊C8)#67semidirect
226. C33#468semidirect
326. C32#5042semidirect
327. C32#5174semidirect
372. C3#10591502semidirect
373. C3#10592741semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37542
C32×#9382
#117234252
C2×#887862
#117222022
#117220572
#117220392
#105927413
#105915023