Finite Group Databasebeta

← #13161167

Subgroups — #13161167

Order 1728.

888 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 150 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×#88863C211yesmaximal, characteristic
2C3×#64153C211yesmaximal, characteristic
3#11723406C211yesmaximal, characteristic
4C2×#88800C211yesmaximal, characteristic
5C32×#835C211yesmaximal, characteristic
6#11722051C211yesmaximal, characteristic
7#11722039C211yesmaximal, characteristic
8#10592494—31nomaximal
9#10591575—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13#88863—22no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3750—21no—
16C3×#3750—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21Dic3×C3⋊Dic3—21no—
22#88800—22no—
23#64153D311yescharacteristic
24C2×#3756—31no—
25C2×#3756—34no—
26C2×#3756—64no—
27C2×#3746—31no—
28#64096—31no—
29C3×#944—34no—
30#63844—34no—
31C2×#3719—34no—
32C32×#103—31no—
33#63641—31no—
34#63637—31no—
35C3×#835D314yes—
36Dic3⋊(C3⋊C8)—34no—
37#63565—34no—
38C33×Q8—21no—
39C33×Q8—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43#6308—21no—
44#6308D412yes—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3D411yescharacteristic
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×#634—21no—
49C3×#634—21no—
50C2×C32×Dic3—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C32×C3⋊C8—21no—
53#5090—91no—
54#4795—94no—
55C2×C32×Q8—24no—
56C2×C32×Q8C2×D314yes—
57C2×C32×Q8C2×D311yescharacteristic
58C2×#634—31no—
59C2×#634—34no—
60C2×#634—64no—
61C2×#634C2×D311yescharacteristic
62C2×C3×Dic6—34no—
63C2×C32×C8—31no—
64C32×C4⋊C4—31no—
65#3756—62no—
66#3756—68no—
67#3756—616no—
68#3750—21no—
69#3750—21no—
70#3750—61no—
71C4×C3⋊Dic3—61no—
72C4×C3⋊Dic3C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—34no—
75C2×C3⋊(C3⋊C8)—64no—
76#3746—62no—
77C3×C4⋊Dic3C2×D314yes—
78C3×Dic3⋊C4—64no—
79C3×Dic3⋊C4—64no—
80C3×C4×Dic3—34no—
81C2×C3×C3⋊C8C2×D314yes—
82#3724—64no—
83Dic32—64no—
84#3719—68no—
85C22×C33C2×D411yescharacteristic
86C33×C4—41no—
87C33×C4C2×D411yescharacteristic
88C33×C4#4111yescharacteristic
89C3⋊(C3⋊Dic3)—22no—
90C3×C3⋊Dic3—21no—
91C3×C3⋊Dic3—41no—
92C3×C3⋊Dic3Q1612yescomplemented
93C32×Dic3—41no—
94C3×C4⋊Q8—91no—
95#944—34no—
96C2×#216—94no—
97C2×#216—91no—
98C2×#216—91no—
99C2×#216—184no—
100C2×Dic12—94no—
101#887—94no—
102C4×Dic6—91no—
103C3×#103—34no—
104#835—91no—
105#835C3⋊D311yescharacteristic
106Dic3⋊C8—94no—
107#810—94no—
108C4⋊(C3⋊C8)—91no—
109C32×Q8—28no—
110C32×Q8—24no—
111C32×Q8—21no—
112C32×Q8—21no—
113C32×Q8—24no—
114C32×Q8—44no—
115C2×C32×C4—21no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—44no—
119C2×C32×C4—61no—
120C2×C32×C4C22×D314yes—
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—61no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—124no—
126C2×C3⋊Dic3#6711yescharacteristic
127C2×C3⋊Dic3#6711yescharacteristic
128#634—21no—
129#634—21no—
130#634—32no—
131#634—38no—
132#634—68no—
133#634—61no—
134#634—64no—
135#634—124no—
136C2×C3×Dic3—24no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3—64no—
140C3×Dic6—64no—
141C3×Dic6—64no—
142C32×C8—61no—
143C3⋊(C3⋊C8)—68no—
144C3⋊(C3⋊C8)—64no—
145C3⋊(C3⋊C8)—61no—
146C3×C3⋊C8—24no—
147#462—271noSylow
148C2×C33#12111yescharacteristic
149C2×C33C2×Q1611yescharacteristic
150C2×C33#13711yescharacteristic
151C2×C3×Q8—24no—
152C2×C3×Q8—91no—
153C2×C3×Q8D3214yes—
154C2×C3×Q8C2×C3⋊D311yescharacteristic
155C2×Dic6—34no—
156C2×Dic6—94no—
157C2×Dic6—91no—
158C2×Dic6—184no—
159C2×C3×C8—34no—
160C3×C4⋊C4—34no—
161C3×C4⋊C4—181no—
162C3×C4⋊C4—181no—
163C3×C42—91no—
164#216—182no—
165#216—188no—
166#216—182no—
167#216—1816no—
168C4⋊Dic3C2×C3⋊D311yescharacteristic
169Dic3⋊C4—64no—
170Dic3⋊C4—64no—
171Dic3⋊C4—181no—
172Dic3⋊C4—184no—
173C4×Dic3—34no—
174C4×Dic3—181no—
175C4×Dic3—184no—
176C2×C3⋊C8—94no—
177C2×C3⋊C8—91no—
178C2×C3⋊C8—184no—
179C2×C3⋊C8C2×C3⋊D311yescharacteristic
180Dic12—188no—
181C22×C32—24no—
182C22×C32D3×D414yes—
183C22×C32C2×#6711yescharacteristic
184C32×C4—24no—
185C32×C4—24no—
186C32×C4—48no—
187C32×C4—44no—
188C32×C4—41no—
189C32×C4—121no—
190C32×C4#23514yes—
191C32×C4D3×D414yes—
192C32×C4D3×D411yescharacteristic
193C32×C4#23711yescharacteristic
194C3⋊Dic3—21no—
195C3⋊Dic3—41no—
196C3⋊Dic3—62no—
197C3⋊Dic3—68no—
198C3⋊Dic3—128no—
199C3⋊Dic3#21612yescomplemented
200C3×Dic3—38no—
201C3×Dic3—44no—
202C3×Dic3—64no—
203C3×Dic3—124no—
204C2×Q16—271no—
205C2×Q16—271no—
206C4⋊Q8—91no—
207C4×Q8—271no—
208C4⋊C8—271no—
209#103—31no—
210#103—271no—
211C33#46211yescharacteristic, complemented, Sylow
212C3×Q8—28no—
213C3×Q8—24no—
214C3×Q8—24no—
215C3×Q8—21no—
216C3×Q8—21no—
217C3×Q8—44no—
218C3×Q8—181no—
219C3×Q8—181no—
220C2×C3×C4—24no—
221C2×C3×C4—21no—
222C2×C3×C4—24no—
223C2×C3×C4—44no—
224C2×C3×C4—64no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—181no—
228C2×C3×C4C2×D3214yes—
229C2×C3×C4C22×C3⋊D311yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—34no—
232C2×Dic3—34no—
233C2×Dic3—64no—
234C2×Dic3—184no—
235C2×Dic3—181no—
236C2×Dic3—364no—
237Dic6—64no—
238Dic6—64no—
239Dic6—98no—
240Dic6—92no—
241Dic6—188no—
242Dic6—184no—
243Dic6—181no—
244Dic6—364no—
245C3×C8—64no—
246C3⋊C8—21no—
247C3⋊C8—188no—
248C3⋊C8—184no—
249C3⋊C8—181no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32#88414yes—
254C2×C32#91514yes—
255C2×C32D3×Q1614yes—
256C2×C32#93811yescharacteristic
257C2×C32#94211yescharacteristic
258C2×C32C2×#21611yescharacteristic
259C2×Q8—91no—
260C2×Q8—271no—
261C2×Q8D3×C3⋊D311yescharacteristic
262Q16—542no—
263Q16—542no—
264C2×C8—31no—
265C2×C8—271no—
266C4⋊C4—31no—
267C4⋊C4—181no—
268C4⋊C4—181no—
269C4⋊C4—541no—
270C42—91no—
271C42—541no—
272C22×C3—24no—
273C22×C3D3×#6714yes—
274C22×C3D4×C3⋊D311yescharacteristic
275C3×C4—24no—
276C3×C4—24no—
277C3×C4—48no—
278C3×C4—41no—
279C3×C4—44no—
280C3×C4—92no—
281C3×C4—124no—
282C3×C4—181no—
283C3×C4—361no—
284C3×C4#380414yes—
285C3×C4#381114yes—
286C3×C4#382811yescharacteristic
287C3×C4D4×C3⋊D311yescharacteristic
288Dic3—38no—
289Dic3—41no—
290Dic3—64no—
291Dic3—124no—
292Dic3—182no—
293Dic3—188no—
294Dic3—368no—
295C32—24no—
296C32#479514yescomplemented
297C32#509011yescharacteristic, complemented
298Q8—21no—
299Q8—21no—
300Q8—181no—
301Q8—181no—
302Q8—272no—
303Q8—541no—
304C2×C4—21no—
305C2×C4—61no—
306C2×C4—91no—
307C2×C4—91no—
308C2×C4—181no—
309C2×C4—541no—
310C2×C4C2×D3×C3⋊D311yescharacteristic
311C8—61no—
312C8—541no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—24no—
316C2×C3D3×#21614yes—
317C2×C3#6394614yes—
318C2×C3#6395814yes—
319C2×C3#6409311yescharacteristic
320C2×C3#6412411yescharacteristic
321C2×C3Q16×C3⋊D311yescharacteristic
322C22C3⋊D3×#6711yescharacteristic
323C4—41no—
324C4—92no—
325C4—121no—
326C4—181no—
327C4—361no—
328C4—542no—
329C4#8903111yescharacteristic
330C4#8903911yescharacteristic
331C3—24no—
332C3#1059157514yescomplemented
333C3#1059249411yescharacteristic, complemented
334C2C3⋊D3×#21611yescharacteristic
335C2#1172358411yescharacteristic
336C2#1172362311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#462
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
92. C3×C3⋊Dic3Q16semidirect
199. C3⋊Dic3#216semidirect
211. C33#462semidirect
296. C32#4795semidirect
297. C32#5090semidirect
332. C3#10591575semidirect
333. C3#10592494semidirect

Maximal subgroups

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

subgroupindex
C2×#888632
C3×#641532
#117234062
C2×#888002
C32×#8352
#117220512
#117220392
#105924943
#105915753