Finite Group Databasebeta

← #13159991

Subgroups — #13159991

Order 1728.

816 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 168 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×#3746C211yesmaximal, characteristic
2#11723443C211yesmaximal, characteristic
3Dic3×#634C211yesmaximal, characteristic
4C2×#88800C211yesmaximal, characteristic
5#11722186C211yesmaximal, characteristic
6C32×Dic3⋊C8C211yesmaximal, characteristic
7#11722040C211yesmaximal, characteristic
8#10592548—31nomaximal
9#10590800—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×#3746—22no—
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88807—21no—
19Dic3⋊(C3⋊Dic3)—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22#88800—22no—
23C2×#3746—31no—
24C2×#3746D311yescharacteristic
25#64078—31no—
26C4×#634—31no—
27C2×C3×Dic12—34no—
28#63866—34no—
29Dic3×Dic6—34no—
30C2×#3719—34no—
31C32×C4⋊C8—31no—
32#63643—31no—
33#63643—31no—
34#63643—34no—
35#63643—64no—
36C3×Dic3⋊C8D314yes—
37#63566—34no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3⋊(C3⋊Dic3)—21no—
41#6308—21no—
42#6308—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×#634—21no—
46C3×#634—21no—
47C3×#634—21no—
48C3×#634D412yes—
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51C33×C8—21no—
52C32×C3⋊C8—21no—
53#5182—91no—
54#4849—94no—
55C2×#634—31no—
56C2×#634—34no—
57C2×#634—64no—
58C2×#634C2×D311yescharacteristic
59C2×#634C2×D311yescharacteristic
60C2×C3×Dic6—34no—
61C2×C3×Dic6—34no—
62C2×C32×C8—24no—
63C2×C32×C8—31no—
64C2×C32×C8C2×D314yes—
65C2×C32×C8C2×D311yescharacteristic
66C32×C42—31no—
67C4⋊(C3⋊Dic3)—31no—
68C4⋊(C3⋊Dic3)—34no—
69C4⋊(C3⋊Dic3)—61no—
70C4⋊(C3⋊Dic3)—61no—
71C4⋊(C3⋊Dic3)—64no—
72#3750—61no—
73C4×C3⋊Dic3—61no—
74#3746—22no—
75#3746—62no—
76C3×C4×Dic3C2×D314yes—
77C2×C3×C3⋊C8C2×D314yes—
78C3×Dic12—68no—
79Dic3⋊Dic3—64no—
80Dic3⋊Dic3—64no—
81Dic3⋊Dic3—64no—
82Dic32—64no—
83#3719—68no—
84C22×C33C2×D411yescharacteristic
85C33×C4C2×D411yescharacteristic
86C33×C4#4111yescharacteristic
87C3⋊(C3⋊Dic3)—41no—
88C3⋊(C3⋊Dic3)—41no—
89C3×C3⋊Dic3—22no—
90C3×C3⋊Dic3—41no—
91C32×Dic3—21no—
92C32×Dic3Q1612yescomplemented
93C2×C3×Q16—91no—
94Q8×Dic3—91no—
95#944—91no—
96C2×#216—91no—
97C2×Dic12—34no—
98C2×Dic12—94no—
99#869—94no—
100C4×Dic6—94no—
101C3×C4⋊C8—34no—
102#835—91no—
103#816—94no—
104#816—94no—
105#816—91no—
106#816—184no—
107Dic3⋊C8C3⋊D311yescharacteristic
108C2×C32×C4—24no—
109C2×C32×C4—31no—
110C2×C32×C4—31no—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—61no—
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—124no—
120C2×C3⋊Dic3—124no—
121#634—21no—
122#634—21no—
123#634—21no—
124#634—68no—
125#634—61no—
126#634—64no—
127#634—61no—
128#634—64no—
129#634—124no—
130#634#6712yes—
131C2×C3×Dic3—64no—
132C2×C3×Dic3—64no—
133C2×C3×Dic3D1214yes—
134C2×C3×Dic3D1214yes—
135C3×Dic6—38no—
136C3×Dic6—64no—
137C3×Dic6—64no—
138C3×Dic6—64no—
139C32×C8—21no—
140C32×C8—24no—
141C32×C8—44no—
142C32×C8—61no—
143C3×C3⋊C8—24no—
144#462—271noSylow
145C2×C33#12111yescharacteristic
146C2×C33C2×Q1611yescharacteristic
147C2×C33#13711yescharacteristic
148C2×C3×Q8—91no—
149C2×C3×Q8—91no—
150C2×Dic6—34no—
151C2×Dic6—34no—
152C2×Dic6—94no—
153C2×Dic6—91no—
154C2×Dic6—184no—
155C3×Q16—182no—
156C2×C3×C8—24no—
157C2×C3×C8—34no—
158C2×C3×C8D3214yes—
159C2×C3×C8C2×C3⋊D311yescharacteristic
160C3×C42—34no—
161#216—182no—
162C4⋊Dic3—94no—
163C4⋊Dic3—91no—
164C4⋊Dic3—184no—
165C4⋊Dic3—184no—
166C4⋊Dic3—184no—
167C4⋊Dic3—181no—
168Dic3⋊C4—184no—
169Dic3⋊C4—181no—
170Dic3⋊C4—181no—
171C4×Dic3—184no—
172C4×Dic3—181no—
173C4×Dic3C2×C3⋊D311yescharacteristic
174C2×C3⋊C8C2×C3⋊D311yescharacteristic
175Dic12—68no—
176Dic12—188no—
177C22×C32—24no—
178C22×C32C2×D1214yes—
179C22×C32D3×D411yescharacteristic
180C32×C4—24no—
181C32×C4—24no—
182C32×C4—32no—
183C32×C4—61no—
184C32×C4D3×D414yes—
185C32×C4#23711yescharacteristic
186C32×C4#23914yes—
187C32×C4C2×#6711yescharacteristic
188C3⋊Dic3—22no—
189C3⋊Dic3—41no—
190C3⋊Dic3—128no—
191C3⋊Dic3—124no—
192C3⋊Dic3—121no—
193C3⋊Dic3—121no—
194C3⋊Dic3—124no—
195C3⋊Dic3—244no—
196C3×Dic3—24no—
197C3×Dic3—68no—
198C3×Dic3—124no—
199C3×Dic3Dic1218yescomplemented
200C2×Q16—91no—
201C2×Q16—271no—
202C4⋊Q8—271no—
203C4×Q8—271no—
204C4⋊C8—31no—
205#103—271no—
206#103—271no—
207C33#46211yescharacteristic, complemented, Sylow
208C3×Q8—92no—
209C3×Q8—181no—
210C3×Q8—181no—
211C3×Q8—181no—
212C2×C3×C4—24no—
213C2×C3×C4—34no—
214C2×C3×C4—34no—
215C2×C3×C4—181no—
216C2×C3×C4—181no—
217C2×C3×C4C2×D3214yes—
218C2×C3×C4C22×C3⋊D311yescharacteristic
219C2×Dic3—64no—
220C2×Dic3—64no—
221C2×Dic3—184no—
222C2×Dic3—184no—
223C2×Dic3—181no—
224C2×Dic3—181no—
225C2×Dic3—364no—
226C2×Dic3—364no—
227C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
228C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
229Dic6—38no—
230Dic6—64no—
231Dic6—64no—
232Dic6—64no—
233Dic6—188no—
234Dic6—184no—
235Dic6—184no—
236Dic6—181no—
237Dic6—181no—
238Dic6—364no—
239C3×C8—24no—
240C3×C8—21no—
241C3×C8—44no—
242C3×C8—64no—
243C3⋊C8—21no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32#89514yes—
248C2×C32C2×Dic1214yes—
249C2×C32#90914yes—
250C2×C32D3×Q1611yescharacteristic
251C2×C32#91811yescharacteristic
252C2×C32#93911yescharacteristic
253C2×Q8—91no—
254C2×Q8—91no—
255C2×Q8—271no—
256Q16—182no—
257Q16—542no—
258C2×C8—31no—
259C2×C8D3×C3⋊D311yescharacteristic
260C4⋊C4—271no—
261C4⋊C4—541no—
262C4⋊C4—541no—
263C4⋊C4—541no—
264C42—31no—
265C42—541no—
266C22×C3—24no—
267C22×C3D3×D1214yes—
268C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
269C3×C4—24no—
270C3×C4—24no—
271C3×C4—38no—
272C3×C4—64no—
273C3×C4—182no—
274C3×C4—361no—
275C3×C4#379614yes—
276C3×C4D3×#6714yes—
277C3×C4D4×C3⋊D311yescharacteristic
278C3×C4#383211yescharacteristic
279Dic3—21no—
280Dic3—68no—
281Dic3—124no—
282Dic3—368no—
283Dic3—364no—
284Dic3—361no—
285Dic3—361no—
286Dic3—364no—
287Dic3—724no—
288Dic3#374612yescomplemented
289C32—24no—
290C32#484914yescomplemented
291C32#518211yescharacteristic, complemented
292Q8—92no—
293Q8—181no—
294Q8—181no—
295Q8—181no—
296Q8—541no—
297Q8—541no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—181no—
301C2×C4—181no—
302C2×C4—541no—
303C2×C4—541no—
304C2×C4C2×D3×C3⋊D311yescharacteristic
305C8—21no—
306C8—61no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3D3×Dic1214yes—
311C2×C3#6380114yes—
312C2×C3#6388614yes—
313C2×C3#6410411yescharacteristic
314C2×C3C2×#374611yescharacteristic
315C2×C3#6411811yescharacteristic
316C22D3×C4⋊(C3⋊D3)11yescharacteristic
317C4—32no—
318C4—61no—
319C4—182no—
320C4—361no—
321C4—1081no—
322C4—1081no—
323C4#8901711yescharacteristic
324C4C3⋊D3×#6711yescharacteristic
325C3—24no—
326C3#1059080014yescomplemented
327C3#1059254811yescharacteristic, complemented
328C2D3×#374611yescharacteristic
329C2#1172332711yescharacteristic
330C2#1172348211yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#462
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
92. C32×Dic3Q16semidirect
199. C3×Dic3Dic12semidirect
207. C33#462semidirect
288. Dic3#3746semidirect
290. C32#4849semidirect
291. C32#5182semidirect
326. C3#10590800semidirect
327. C3#10592548semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37462
#117234432
Dic3×#6342
C2×#888002
#117221862
C32×Dic3⋊C82
#117220402
#105925483
#105908003