Finite Group Databasebeta

← #13159993

Subgroups — #13159993

Order 1728.

828 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 148 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×#3743C211yesmaximal, characteristic
2#11723435C211yesmaximal, characteristic
3#11723431C211yesmaximal, characteristic
4C2×#88799C211yesmaximal, characteristic, complemented
5#11722186C211yesmaximal, characteristic, complemented
6C32×#816C211yesmaximal, characteristic, complemented
7#11722054C211yesmaximal, characteristic, complemented
8#10592515—31nomaximal
9#10590849—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×#3743—22no—
17C32×C4⋊Dic3C2211yescharacteristic
18#88806—21no—
19Dic3⋊(C3⋊Dic3)—21no—
20#88803—21no—
21#88803—21no—
22#88799—22no—
23#64155—31no—
24C2×#3756—31no—
25C2×#3743D311yescharacteristic
26#64106—31no—
27C2×C3×ΓL(1,𝔽25)—34no—
28#63857—34no—
29#63856—34no—
30C2×#3719—34no—
31C32×#103—31no—
32#63643—31no—
33#63643—34no—
34#63643—64no—
35#63635—31no—
36C3×#816D314yescomplemented
37#63573—34no—
38C22×C3×C3⋊D3D411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)—21no—
42#6308—21no—
43#6308—21no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×C4×C3⋊D3—21no—
46C3×C4×C3⋊D3—21no—
47C2×C32×Dic3—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51C33×C8—21no—
52C3×C3⋊(C3⋊C8)D412yes—
53#5137—91no—
54#4816—94no—
55C2×C32×Q8—31no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×#634—31no—
58C2×#634—34no—
59C2×#634—64no—
60C2×C3×C4×D3—34no—
61C2×C3×Dic6C2×D314yes—
62C2×C32×C8—24no—
63C2×C32×C8C2×D314yes—
64C2×C32×C8C2×D311yescharacteristic
65C32×C4⋊C4—31no—
66#3756—62no—
67#3752—61no—
68#3752—61no—
69C4⋊(C3⋊Dic3)—31no—
70C4⋊(C3⋊Dic3)—34no—
71C4⋊(C3⋊Dic3)—61no—
72C4⋊(C3⋊Dic3)—64no—
73#3750—61no—
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75#3743—22no—
76C3×C4⋊Dic3C2×D314yes—
77C2×C3×C3⋊C8—34no—
78C3×ΓL(1,𝔽25)—68no—
79#3724—64no—
80Dic3⋊Dic3—64no—
81(C2×D3)⋊Dic3—64no—
82(C2×D3)⋊Dic3—64no—
83#3719—68no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—21no—
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—21no—
88C33×C4C2×D411yescharacteristic
89C33×C4#4111yescharacteristic
90C3⋊(C3⋊Dic3)—41no—
91C3⋊(C3⋊Dic3)—41no—
92C3×C3⋊Dic3—21no—
93C32×Dic3—41no—
94C32×Dic3—41no—
95C2×C3×M16—91no—
96#946—94no—
97C2×#216—94no—
98#924—91no—
99#924—91no—
100C2×Dic12—91no—
101C2×ΓL(1,𝔽25)—34no—
102#897—94no—
103C3×#103—34no—
104#817—91no—
105#816—94no—
106#816—91no—
107#816—184no—
108#816C3⋊D311yescharacteristic, complemented
109#808—94no—
110C22×C3⋊D3D1211yescharacteristic
111C22×C3×D3—34no—
112C32×Q8—61no—
113C32×Q8—61no—
114C2×C32×C4—24no—
115C2×C32×C4—61no—
116C2×C32×C4—61no—
117C2×C32×C4C22×D314yes—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—124no—
124C2×C3⋊Dic3—124no—
125C2×C3⋊Dic3D1211yescharacteristic
126C4×C3⋊D3—21no—
127C4×C3⋊D3—21no—
128#634—68no—
129#634—64no—
130#634—61no—
131#634—61no—
132#634—64no—
133#634—124no—
134C2×C3×Dic3—24no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—34no—
137C3×C4×D3—64no—
138C3×C4×D3—64no—
139C3×Dic6—24no—
140C3×Dic6—24no—
141C32×C8—21no—
142C32×C8—24no—
143C32×C8—44no—
144C3⋊(C3⋊C8)#6712yes—
145C3×C3⋊C8—38no—
146#470—271noSylow
147C2×C33#12111yescharacteristic
148C2×C33#13711yescharacteristic
149C2×C33#13711yescharacteristic
150C3×C3⋊D3—41no—
151C2×C3×Q8—34no—
152C22×C3×C4—91no—
153C2×C4×D3—34no—
154C2×Dic6—94no—
155C2×Dic6—91no—
156C2×Dic6—184no—
157C2×Dic6C2×C3⋊D311yescharacteristic
158C3×M16—182no—
159C2×C3×C8—24no—
160C2×C3×C8—91no—
161C2×C3×C8D3214yes—
162C2×C3×C8C2×C3⋊D311yescharacteristic
163C3×C4⋊C4—34no—
164#217—181no—
165#217—181no—
166#216—188no—
167(C2×D3)⋊C4—184no—
168(C2×D3)⋊C4—184no—
169C4⋊Dic3—94no—
170C4⋊Dic3—91no—
171C4⋊Dic3—184no—
172C4⋊Dic3—184no—
173C4⋊Dic3C2×C3⋊D311yescharacteristic
174Dic3⋊C4—181no—
175Dic3⋊C4—181no—
176Dic3⋊C4—184no—
177C2×C3⋊C8—34no—
178Dic12—182no—
179ΓL(1,𝔽25)—68no—
180C22×C32—24no—
181C22×C32C2×D1211yescharacteristic
182C22×C32D3×D414yes—
183C2×C3⋊D3—21no—
184C2×C3⋊D3—21no—
185C2×C3⋊D3—21no—
186C2×C3×D3—64no—
187C2×C3×D3—64no—
188C2×C3×D3—64no—
189C32×C4—24no—
190C32×C4—24no—
191C32×C4—121no—
192C32×C4—121no—
193C32×C4#23514yes—
194C32×C4#23511yescharacteristic
195C32×C4D3×D414yes—
196C32×C4C2×#6711yescharacteristic
197C3⋊Dic3—21no—
198C3⋊Dic3—128no—
199C3⋊Dic3—124no—
200C3⋊Dic3—121no—
201C3⋊Dic3—121no—
202C3⋊Dic3—124no—
203C3⋊Dic3—244no—
204C3×Dic3—44no—
205C3×Dic3—44no—
206C3×Dic3—64no—
207C2×Q16—271no—
208C2×M16—91no—
209C22⋊Q8—271no—
210C22⋊Q8—271no—
211#106—271no—
212#103—31no—
213#103—271no—
214C33#47011yescharacteristic, complemented, Sylow
215C23×C3—91no—
216C22×D3—34no—
217C3×Q8—64no—
218C3×Q8—64no—
219C2×C3×C4—24no—
220C2×C3×C4—64no—
221C2×C3×C4—64no—
222C2×C3×C4—91no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4C2×D3214yes—
226C2×C3×C4C22×C3⋊D311yescharacteristic
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—34no—
230C2×Dic3—184no—
231C2×Dic3—184no—
232C2×Dic3—181no—
233C2×Dic3—181no—
234C2×Dic3—364no—
235C2×Dic3—364no—
236C4×D3—64no—
237C4×D3—64no—
238Dic6—21no—
239Dic6—21no—
240Dic6—188no—
241Dic6—184no—
242Dic6—184no—
243Dic6—181no—
244Dic6—181no—
245Dic6—364no—
246C3×C8—24no—
247C3×C8—21no—
248C3×C8—44no—
249C3×C8—92no—
250C3⋊C8—38no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32#88414yes—
255C2×C32#90911yescharacteristic
256C2×C32#90911yescharacteristic
257C2×C32#91514yes—
258C2×C32#91814yes—
259C2×C32#93011yescharacteristic
260C3⋊D3—41no—
261C3×D3—124no—
262C2×Q8—31no—
263C2×Q8—271no—
264C22×C4—91no—
265Q16—542no—
266M16—182no—
267C2×C8—91no—
268C2×C8D3×C3⋊D311yescharacteristic
269C4⋊C4—31no—
270C4⋊C4—271no—
271C4⋊C4—541no—
272C4⋊C4—541no—
273C22⋊C4—541no—
274C22⋊C4—541no—
275C22×C3—24no—
276C22×C3—181no—
277C22×C3—181no—
278C22×C3—181no—
279C22×C3D3×D1214yes—
280C22×C3D4×C3⋊D311yescharacteristic
281C2×D3—64no—
282C2×D3—64no—
283C2×D3—64no—
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—124no—
287C3×C4—124no—
288C3×C4—181no—
289C3×C4#379814yes—
290C3×C4D3×#6714yes—
291C3×C4#382811yescharacteristic
292C3×C4D4×C3⋊D311yescharacteristic
293Dic3—41no—
294Dic3—41no—
295Dic3—64no—
296Dic3—368no—
297Dic3—364no—
298Dic3—361no—
299Dic3—361no—
300Dic3—364no—
301Dic3—724no—
302C32—24no—
303C32#481614yescomplemented
304C32#513711yescharacteristic, complemented
305C23—91no—
306Q8—61no—
307Q8—61no—
308Q8—541no—
309Q8—541no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—91no—
313C2×C4—181no—
314C2×C4—181no—
315C2×C4—541no—
316C2×C4—541no—
317C2×C4C2×D3×C3⋊D311yescharacteristic
318C8—21no—
319C8—92no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—361no—
324C2×C3#6380014yes—
325C2×C3#6380114yes—
326C2×C3#6390614yes—
327C2×C3#6409311yescharacteristic
328C2×C3#6412411yescharacteristic
329C2×C3#6412711yescharacteristic
330D3—124no—
331C22—181no—
332C22—181no—
333C22—181no—
334C22C3⋊D3×D1211yescharacteristic
335C4—121no—
336C4—121no—
337C4—181no—
338C4—1081no—
339C4—1081no—
340C4#8902011yescharacteristic
341C4C3⋊D3×#6711yescharacteristic
342C3—24no—
343C3#1059084914yescomplemented
344C3#1059251511yescharacteristic, complemented
345C2—361no—
346C2#1172332911yescharacteristic
347C2#1172334011yescharacteristic
348C2#1172352011yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#470
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. C2×#88799C2semidirect
5. #11722186C2semidirect
6. C32×#816C2semidirect
7. #11722054C2semidirect
36. C3×#816D3semidirect
108. #816C3⋊D3semidirect
214. C33#470semidirect
303. C32#4816semidirect
304. C32#5137semidirect
343. C3#10590849semidirect
344. C3#10592515semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37432
#117234352
#117234312
C2×#887992
#117221862
C32×#8162
#117220542
#105925153
#105908493