Finite Group Databasebeta

← #13159350

Subgroups — #13159350

Order 1728.

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

normal subgroups only

This group has 828 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
1C32×#897C211yesmaximal, characteristic
2#11723426C211yesmaximal, characteristic
3C2×#88799C211yesmaximal, characteristic, complemented
4C2×#88786C211yesmaximal, characteristic
5#11722180C211yesmaximal, characteristic, complemented
6C3×#63635C211yesmaximal, characteristic, complemented
7#11722037C211yesmaximal, characteristic, complemented
8#10592751—31nomaximal
9#10590587—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C32×C4×D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4⋊Dic3—21no—
19#88807—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#88799—22no—
22#88786—22no—
23C32×C22⋊Q8—31no—
24C2×#3756—31no—
25#64133—31no—
26C2×#3748—31no—
27C3×#897D314yes—
28#63857—34no—
29C2×#3719—34no—
30C2×#3711—34no—
31#63662—31no—
32#63637—31no—
33#63637—34no—
34#63637—64no—
35#63635D311yescharacteristic, complemented
36C3×#808—34no—
37#63565—34no—
38C22×C32×D3D411yescharacteristic
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42#6308—21no—
43#6308—21no—
44C2×C3×C3⋊Dic3—21no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3D411yescharacteristic
47C32×C4×D3—21no—
48C32×C4×D3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3×C3⋊(C3⋊C8)D412yes—
53#5052—94no—
54#4867—91no—
55C2×C32×Q8—31no—
56C22×C32×C4—31no—
57C2×#634—31no—
58C2×#634—34no—
59C2×#634—64no—
60C2×C3×C4×D3C2×D314yes—
61C2×C3×Dic6C2×D314yes—
62C32×C4⋊C4—24no—
63C32×C4⋊C4—61no—
64C32×C4⋊C4C2×D314yes—
65C32×C4⋊C4C2×D311yescharacteristic
66C32×C22⋊C4—61no—
67#3757—61no—
68#3756—62no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70#3750—61no—
71#3748—62no—
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—34no—
74C2×C3⋊(C3⋊C8)—64no—
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×(C2×D3)⋊C4—24no—
77C3×C4⋊Dic3—24no—
78C3×C4⋊Dic3—34no—
79C2×C3×C3⋊C8—34no—
80Dic3⋊Dic3—64no—
81(C2×D3)⋊Dic3—64no—
82#3719—68no—
83#3711—68no—
84C22×C33C2×D411yescharacteristic
85C2×C32×D3—21no—
86C2×C32×D3—21no—
87C2×C32×D3—21no—
88C33×C4—41no—
89C33×C4C2×D411yescharacteristic
90C33×C4#4111yescharacteristic
91C3⋊(C3⋊Dic3)—41no—
92C3×C3⋊Dic3—41no—
93C32×Dic3—21no—
94C32×Dic3—41no—
95C3×C22⋊Q8—34no—
96C2×#216—94no—
97#924—94no—
98C2×#208—94no—
99C2×Dic12—91no—
100C2×ΓL(1,𝔽25)—91no—
101#897—91no—
102#897C3⋊D311yescharacteristic
103C3×#106—91no—
104#835—94no—
105#810—91no—
106#810—94no—
107#810—91no—
108#810—184no—
109#808—34no—
110C23×C32—31no—
111C22×C3×D3#6714yes—
112C32×Q8—61no—
113C32×Q8—61no—
114C2×C32×C4—21no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—31no—
118C2×C32×C4—44no—
119C2×C32×C4—61no—
120C2×C32×C4—61no—
121C2×C32×C4—61no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3—64no—
127C2×C3⋊Dic3—124no—
128#634—68no—
129#634—61no—
130#634—64no—
131#634—61no—
132#634—64no—
133#634—124no—
134C2×C3×Dic3—24no—
135C2×C3×Dic3—64no—
136C2×C3×Dic3#6714yes—
137C3×C4×D3—24no—
138C3×C4×D3—24no—
139C3×Dic6—24no—
140C3×Dic6—24no—
141C3⋊(C3⋊C8)—61no—
142C3⋊(C3⋊C8)—64no—
143C3⋊(C3⋊C8)—124no—
144C3⋊(C3⋊C8)D1212yes—
145C3×C3⋊C8—38no—
146#470—271noSylow
147C2×C33#12111yescharacteristic
148C2×C33#13711yescharacteristic
149C2×C33#13711yescharacteristic
150C32×D3—41no—
151C2×C3×Q8—34no—
152C22×C3×C4—34no—
153C2×C4×D3C2×C3⋊D311yescharacteristic
154C2×Dic6—94no—
155C2×Dic6—91no—
156C2×Dic6—184no—
157C2×Dic6C2×C3⋊D311yescharacteristic
158C2×C3×C8—91no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—64no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4D3214yes—
163C3×C4⋊C4C2×C3⋊D311yescharacteristic
164C3×C22⋊C4—64no—
165#217—184no—
166#216—188no—
167(C2×D3)⋊C4—21no—
168(C2×D3)⋊C4—181no—
169C4⋊Dic3—21no—
170C4⋊Dic3—34no—
171C4⋊Dic3—181no—
172Dic3⋊C4—184no—
173#208—188no—
174C2×C3⋊C8—34no—
175C2×C3⋊C8—94no—
176C2×C3⋊C8—91no—
177C2×C3⋊C8—184no—
178Dic12—182no—
179ΓL(1,𝔽25)—182no—
180C22×C32—24no—
181C22×C32—61no—
182C22×C32—61no—
183C22×C32—61no—
184C22×C32D3×D411yescharacteristic
185C22×C32C2×#6714yes—
186C2×C3×D3—24no—
187C2×C3×D3—24no—
188C2×C3×D3—24no—
189C32×C4—24no—
190C32×C4—24no—
191C32×C4—41no—
192C32×C4—44no—
193C32×C4—61no—
194C32×C4—84no—
195C32×C4—121no—
196C32×C4C2×D1211yescharacteristic
197C32×C4D3×D414yes—
198C32×C4#23714yes—
199C32×C4#23911yescharacteristic
200C3⋊Dic3—41no—
201C3⋊Dic3—128no—
202C3⋊Dic3—121no—
203C3⋊Dic3—124no—
204C3×Dic3—24no—
205C3×Dic3—44no—
206C3×Dic3—124no—
207C2×Q16—271no—
208C2×M16—271no—
209C22⋊Q8—31no—
210C22⋊Q8—271no—
211#106—91no—
212#103—271no—
213#103—271no—
214C33#47011yescharacteristic, complemented, Sylow
215C23×C3—34no—
216C22×D3#63811yescharacteristic
217C3×Q8—64no—
218C3×Q8—64no—
219C2×C3×C4—24no—
220C2×C3×C4—21no—
221C2×C3×C4—24no—
222C2×C3×C4—34no—
223C2×C3×C4—44no—
224C2×C3×C4—64no—
225C2×C3×C4—64no—
226C2×C3×C4—64no—
227C2×C3×C4—181no—
228C2×C3×C4C2×D3214yes—
229C2×C3×C4C22×C3⋊D311yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—64no—
232C2×Dic3—184no—
233C2×Dic3—181no—
234C2×Dic3—364no—
235C2×Dic3#63811yescharacteristic
236C4×D3—21no—
237C4×D3—21no—
238Dic6—21no—
239Dic6—21no—
240Dic6—188no—
241Dic6—184no—
242Dic6—184no—
243Dic6—181no—
244Dic6—181no—
245Dic6—364no—
246C3×C8—92no—
247C3⋊C8—38no—
248C3⋊C8—184no—
249C3⋊C8—181no—
250C3⋊C8—364no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32—121no—
255C2×C32#89511yescharacteristic
256C2×C32#91511yescharacteristic
257C2×C32#91511yescharacteristic
258C2×C32#93914yes—
259C2×C32#94214yes—
260C2×C32#95114yes—
261C3×D3—44no—
262C2×Q8—31no—
263C2×Q8—271no—
264C22×C4—31no—
265Q16—542no—
266M16—542no—
267C2×C8—91no—
268C2×C8—271no—
269C4⋊C4—61no—
270C4⋊C4—91no—
271C4⋊C4—541no—
272C4⋊C4D3×C3⋊D311yescharacteristic
273C22⋊C4—61no—
274C22⋊C4—541no—
275C22×C3—24no—
276C22×C3—64no—
277C22×C3—64no—
278C22×C3—64no—
279C22×C3D3×#6714yes—
280C22×C3C2×#63811yescharacteristic
281C2×D3—21no—
282C2×D3—21no—
283C2×D3—21no—
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—41no—
287C3×C4—44no—
288C3×C4—64no—
289C3×C4—84no—
290C3×C4—124no—
291C3×C4—361no—
292C3×C4#379614yes—
293C3×C4D3×D1214yes—
294C3×C4D4×C3⋊D311yescharacteristic
295C3×C4#383011yescharacteristic
296Dic3—21no—
297Dic3—41no—
298Dic3—124no—
299Dic3—368no—
300Dic3—364no—
301Dic3—361no—
302C32—24no—
303C32#486711yescharacteristic, complemented
304C32#505214yescomplemented
305C23—31no—
306Q8—61no—
307Q8—61no—
308Q8—541no—
309Q8—541no—
310C2×C4—21no—
311C2×C4—31no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—181no—
316C2×C4—541no—
317C2×C4C2×D3×C3⋊D311yescharacteristic
318C8—92no—
319C8—541no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—124no—
324C2×C3#6388614yes—
325C2×C3#6392914yes—
326C2×C3#6394614yes—
327C2×C3#6414811yescharacteristic
328C2×C3#6415111yescharacteristic
329C2×C3#6416011yescharacteristic
330D3—41no—
331C22—61no—
332C22—61no—
333C22—61no—
334C22D3×#63811yescharacteristic
335C4—41no—
336C4—61no—
337C4—121no—
338C4—361no—
339C4—1081no—
340C4#8901511yescharacteristic
341C4C3⋊D3×D1211yescharacteristic
342C3—24no—
343C3#1059058714yescomplemented
344C3#1059275111yescharacteristic, complemented
345C2—121no—
346C2#1172347511yescharacteristic
347C2#1172355011yescharacteristic
348C2#1172357911yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#470
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#88799C2semidirect
5. #11722180C2semidirect
6. C3×#63635C2semidirect
7. #11722037C2semidirect
35. #63635D3semidirect
214. C33#470semidirect
303. C32#4867semidirect
304. C32#5052semidirect
343. C3#10590587semidirect
344. C3#10592751semidirect

Maximal subgroups

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

subgroupindex
C32×#8972
#117234262
C2×#887992
C2×#887862
#117221802
C3×#636352
#117220372
#105927513
#105905873