Finite Group Databasebeta

← #13161248

Subgroups — #13161248

Order 1728.

972 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 972 conjugacy classes of subgroups — too many to draw. Only its 166 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
3C2×C32×#216C211yesmaximal, characteristic
4#11723455C211yesmaximal, characteristic
5C2×#88800C211yesmaximal, characteristic
6C2×#88798C211yesmaximal, characteristic
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic
8#10592882—31nomaximal
9#10591626—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14#88863—22no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3750—22no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×#216—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—22no—
21#88800—22no—
22#88798—22no—
23C2×C32×Q16—31no—
24#64153—31no—
25#64153D311yescharacteristic
26C2×#3756—31no—
27C2×#3756—31no—
28C2×#3756—34no—
29C2×#3756—64no—
30C2×#3746—31no—
31C3×#944—34no—
32C2×C3×#216D314yes—
33#63866—34no—
34C2×#3719—34no—
35C2×#3718—34no—
36C8×C3⋊Dic3—31no—
37Dic3×C3⋊C8—34no—
38C33×Q8—21no—
39C33×Q8—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43#6308—21no—
44#6308—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3D412yes—
47C3×#634—21no—
48C3×#634—21no—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6—21no—
52C3⋊(C3⋊(C3⋊C8))D412yes—
53C32×C3⋊C8D412yes—
54#5183—94no—
55#5092—91no—
56C2×C32×Q8—24no—
57C2×C32×Q8—31no—
58C2×C32×Q8C2×D314yes—
59C2×C32×Q8C2×D311yescharacteristic
60C2×#634—31no—
61C2×#634—34no—
62C2×#634—64no—
63C2×#634C2×D311yescharacteristic
64C2×C3×Dic6—34no—
65C2×C3×Dic6C2×D314yes—
66C32×Q16—62no—
67C2×C32×C8—31no—
68#3756—62no—
69#3756—62no—
70#3756—68no—
71#3756—616no—
72#3750—22no—
73#3750—62no—
74C4×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78#3746—62no—
79C3×#216—28no—
80C3×Dic3⋊C4—68no—
81C3×C4×Dic3—34no—
82C2×C3×C3⋊C8C2×D314yes—
83Dic3⋊Dic3—68no—
84#3719—68no—
85#3718—68no—
86C22×C33C2×D411yescharacteristic
87C33×C4—41no—
88C33×C4C2×D411yescharacteristic
89C33×C4C2×D411yescharacteristic
90C3⋊(C3⋊Dic3)—41no—
91C3×C3⋊Dic3—41no—
92C3×C3⋊Dic3Q1614yescomplemented
93C32×Dic3—41no—
94C2×C3×Q16—34no—
95C3×C4⋊Q8—91no—
96#944—34no—
97#944—94no—
98C2×#216—94no—
99C2×#216—94no—
100C2×#216—91no—
101C2×#216—91no—
102C2×#216—91no—
103C2×#216—184no—
104C2×#216C3⋊D311yescharacteristic
105C2×Dic12—94no—
106#869—91no—
107C8×Dic3—94no—
108C4×C3⋊C8—91no—
109C32×Q8—28no—
110C32×Q8—24no—
111C32×Q8—21no—
112C32×Q8—21no—
113C32×Q8—24no—
114C32×Q8—44no—
115C32×Q8—61no—
116C32×Q8—61no—
117C2×C32×C4—21no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4—44no—
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—
128C2×C3⋊Dic3#6712yes—
129#634—21no—
130#634—21no—
131#634—68no—
132#634—61no—
133#634—64no—
134#634—61no—
135#634—64no—
136#634—124no—
137C2×C3×Dic3—24no—
138C2×C3×Dic3—38no—
139C2×C3×Dic3—64no—
140C3×Dic6—24no—
141C3×Dic6—24no—
142C3×Dic6—64no—
143C3×Dic6—64no—
144C32×C8—32no—
145C3⋊(C3⋊C8)—32no—
146C3⋊(C3⋊C8)—38no—
147C3⋊(C3⋊C8)—68no—
148C3×C3⋊C8#6718yes—
149#494—271noSylow
150C2×C33#12711yescharacteristic
151C2×C33C2×Q1612yes—
152C2×C3×Q8—24no—
153C2×C3×Q8—34no—
154C2×C3×Q8—91no—
155C2×C3×Q8D3214yes—
156C2×C3×Q8C2×C3⋊D311yescharacteristic
157C2×Dic6—34no—
158C2×Dic6—94no—
159C2×Dic6—91no—
160C2×Dic6—184no—
161C2×Dic6C2×C3⋊D311yescharacteristic
162C3×Q16—68no—
163C2×C3×C8—34no—
164C3×C4⋊C4—182no—
165C3×C42—91no—
166#216—22no—
167#216—182no—
168#216—188no—
169#216—188no—
170#216—182no—
171#216—1816no—
172#216—182no—
173C4⋊Dic3—182no—
174Dic3⋊C4—68no—
175Dic3⋊C4—188no—
176C4×Dic3—34no—
177C2×C3⋊C8—94no—
178C2×C3⋊C8—91no—
179C2×C3⋊C8—184no—
180C2×C3⋊C8C2×C3⋊D311yescharacteristic
181Dic12—188no—
182C22×C32—24no—
183C22×C32D3×D414yes—
184C22×C32C2×#6711yescharacteristic
185C32×C4—24no—
186C32×C4—24no—
187C32×C4—48no—
188C32×C4—44no—
189C32×C4—41no—
190C32×C4—121no—
191C32×C4D3×D414yes—
192C32×C4D3×D411yescharacteristic
193C32×C4D3×D411yescharacteristic
194C32×C4C2×#6714yes—
195C3⋊Dic3—41no—
196C3⋊Dic3—128no—
197C3⋊Dic3—124no—
198C3⋊Dic3—121no—
199C3⋊Dic3#21614yescomplemented
200C3×Dic3—316no—
201C3×Dic3—44no—
202C3×Dic3—124no—
203C2×Q16—31no—
204C2×Q16—271no—
205C2×Q16—271no—
206C2×Q16—271no—
207C4⋊Q8—91no—
208C4⋊Q8—271no—
209C4×C8—271no—
210C33#49411yescharacteristic, complemented, Sylow
211C3×Q8—28no—
212C3×Q8—24no—
213C3×Q8—24no—
214C3×Q8—21no—
215C3×Q8—21no—
216C3×Q8—44no—
217C3×Q8—64no—
218C3×Q8—64no—
219C3×Q8—181no—
220C3×Q8—181no—
221C2×C3×C4—24no—
222C2×C3×C4—21no—
223C2×C3×C4—24no—
224C2×C3×C4—44no—
225C2×C3×C4—64no—
226C2×C3×C4—92no—
227C2×C3×C4—181no—
228C2×C3×C4C2×D3214yes—
229C2×C3×C4C22×C3⋊D311yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—38no—
232C2×Dic3—64no—
233C2×Dic3—184no—
234C2×Dic3—181no—
235C2×Dic3—364no—
236Dic6—21no—
237Dic6—21no—
238Dic6—64no—
239Dic6—64no—
240Dic6—188no—
241Dic6—184no—
242Dic6—184no—
243Dic6—181no—
244Dic6—181no—
245Dic6—364no—
246C3×C8—38no—
247C3⋊C8—98no—
248C3⋊C8—92no—
249C3⋊C8—188no—
250C3⋊C8#63812yes—
251C2×C32—28no—
252C2×C32—24no—
253C2×C32D3×Q1618yes—
254C2×C32#94014yes—
255C2×C32#94011yescharacteristic
256C2×C32C2×#21612yes—
257C2×Q8—31no—
258C2×Q8—91no—
259C2×Q8—271no—
260C2×Q8D3×C3⋊D311yescharacteristic
261Q16—62no—
262Q16—542no—
263Q16—542no—
264Q16—542no—
265C2×C8—31no—
266C2×C8—271no—
267C4⋊C4—182no—
268C4⋊C4—542no—
269C42—91no—
270C22×C3—24no—
271C22×C3D3×#6714yes—
272C22×C3D4×C3⋊D311yescharacteristic
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—48no—
276C3×C4—41no—
277C3×C4—44no—
278C3×C4—94no—
279C3×C4—124no—
280C3×C4—361no—
281C3×C4D3×#6714yes—
282C3×C4#381114yes—
283C3×C4D4×C3⋊D311yescharacteristic
284C3×C4C2×#63811yescharacteristic
285Dic3—316no—
286Dic3—41no—
287Dic3—124no—
288Dic3—368no—
289Dic3—361no—
290Dic3—364no—
291C32—24no—
292C32#509211yescharacteristic, complemented
293C32#518314yescomplemented
294Q8—21no—
295Q8—21no—
296Q8—61no—
297Q8—61no—
298Q8—181no—
299Q8—181no—
300Q8—541no—
301Q8—541no—
302C2×C4—21no—
303C2×C4—61no—
304C2×C4—92no—
305C2×C4—181no—
306C2×C4—541no—
307C2×C4C2×D3×C3⋊D311yescharacteristic
308C8—32no—
309C8—272no—
310C2×C3—28no—
311C2×C3—24no—
312C2×C3D3×#21618yes—
313C2×C3#6397914yes—
314C2×C3Q16×C3⋊D312yes—
315C2×C3#6414911yescharacteristic
316C22C3⋊D3×#6711yescharacteristic
317C4—41no—
318C4—94no—
319C4—121no—
320C4—361no—
321C4—1081no—
322C4D3×#63811yescharacteristic
323C4#8903911yescharacteristic
324C3—24no—
325C3#1059162614yescomplemented
326C3#1059288211yescharacteristic, complemented
327C2C3⋊D3×#21612yes—
328C2#1172363311yescharacteristic
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#494
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
210. C33#494semidirect
292. C32#5092semidirect
293. C32#5183semidirect
325. C3#10591626semidirect
326. C3#10592882semidirect

Maximal subgroups

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

subgroupindex
C2×#888632
C3×#641532
C2×C32×#2162
#117234552
C2×#888002
C2×#887982
C3⋊C8×C3⋊Dic32
#105928823
#105916263