Finite Group Databasebeta

← #13160292

Subgroups — #13160292

Order 1728.

1092 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1092 conjugacy classes of subgroups — too many to draw. Only its 198 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×C4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C3×#63657C211yesmaximal, characteristic
3C32×#830C211yesmaximal, characteristic
4#11722050C212yesmaximal, complemented
5#11722021C212yesmaximal, complemented
6#10592418—31nomaximal
7#10591066—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C3⋊(C3⋊Dic3)C411yescharacteristic
10C22×C3⋊(C3⋊Dic3)C411yescharacteristic
11C4×C3⋊(C3⋊Dic3)C411yescharacteristic
12C4×C3⋊(C3⋊Dic3)C411yescharacteristic
13C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16C2×C32×C3⋊C8C2212yes—
17C2×C4×C3⋊Dic3—31no—
18C2×C4×C3⋊Dic3—34no—
19C2×C4×C3⋊Dic3—64no—
20C32×C22⋊C8—31no—
21#63657D311yescharacteristic
22#63642—32no—
23#63641—32no—
24C3×#830—34no—
25C3×#830D314yes—
26#63572—38no—
27#63555—38no—
28C23×C33C2×C411yescharacteristic
29C2×C33×C4—22no—
30C2×C33×C4C2×C411yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—22no—
33C2×C3⋊(C3⋊Dic3)—22no—
34C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
35C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
36C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
37C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
38C3×C3⋊(C3⋊C8)—22no—
39C32×C3⋊C8—22no—
40#4719—94no—
41#4719—91no—
42C22×C32×C4—24no—
43C22×C32×C4C2×D314yes—
44C22×C32×C4C2×D311yescharacteristic
45C22×C3⋊Dic3—31no—
46C22×C3⋊Dic3—31no—
47C22×C3⋊Dic3—34no—
48C22×C3⋊Dic3—34no—
49C22×C3⋊Dic3—64no—
50C22×C3⋊Dic3—64no—
51C2×C32×C8—32no—
52C4×C3⋊Dic3—31no—
53C4×C3⋊Dic3—31no—
54C4×C3⋊Dic3—34no—
55C4×C3⋊Dic3—34no—
56C4×C3⋊Dic3—31no—
57C4×C3⋊Dic3—31no—
58C4×C3⋊Dic3—34no—
59C4×C3⋊Dic3—34no—
60C4×C3⋊Dic3—64no—
61C4×C3⋊Dic3—64no—
62C4×C3⋊Dic3—64no—
63C4×C3⋊Dic3—64no—
64C2×C3⋊(C3⋊C8)C2×D312yes—
65C2×C3×C3⋊C8—38no—
66C2×C3×C3⋊C8C2×D318yes—
67C22×C33—21no—
68C22×C33—21no—
69C22×C33M1612yes—
70C22×C33C22×C411yescharacteristic
71C33×C4—21no—
72C33×C4#4112yes—
73C3⋊(C3⋊Dic3)—21no—
74C3⋊(C3⋊Dic3)—21no—
75C3⋊(C3⋊Dic3)—21no—
76C3⋊(C3⋊Dic3)M1612yes—
77C2×C4×Dic3—94no—
78C2×C4×Dic3—91no—
79C2×C4×Dic3—184no—
80C3×C22⋊C8—34no—
81C3×C22⋊C8—91no—
82#830—34no—
83#830C3⋊D311yescharacteristic
84#815—98no—
85#815—92no—
86Dic3⋊C8—98no—
87Dic3⋊C8—92no—
88C23×C32—24no—
89C23×C32C4×D314yes—
90C23×C32C4×D311yescharacteristic
91C2×C32×C4—216no—
92C2×C32×C4—28no—
93C2×C32×C4—22no—
94C2×C32×C4—24no—
95C2×C32×C4—24no—
96C2×C32×C4C4×D314yes—
97C2×C32×C4C4×D311yescharacteristic
98C2×C32×C4C22×D314yes—
99C2×C32×C4C22×D311yescharacteristic
100C2×C3⋊Dic3—31no—
101C2×C3⋊Dic3—31no—
102C2×C3⋊Dic3—31no—
103C2×C3⋊Dic3—31no—
104C2×C3⋊Dic3—34no—
105C2×C3⋊Dic3—34no—
106C2×C3⋊Dic3—34no—
107C2×C3⋊Dic3—34no—
108C2×C3⋊Dic3—64no—
109C2×C3⋊Dic3—64no—
110C2×C3⋊Dic3—64no—
111C2×C3⋊Dic3—64no—
112C2×C3⋊Dic3—62no—
113C2×C3⋊Dic3—68no—
114C2×C3⋊Dic3—616no—
115C2×C3⋊Dic3—616no—
116C2×C3⋊Dic3—68no—
117C2×C3⋊Dic3—62no—
118C32×C8—62no—
119C3⋊(C3⋊C8)—22no—
120C3×C3⋊C8—28no—
121C3×C3⋊C8—68no—
122#432—271noSylow
123C2×C33—22no—
124C2×C33#11711yescharacteristic
125C2×C33C2×M1611yescharacteristic
126C2×C33C2×M1611yescharacteristic
127C22×C3×C4—24no—
128C22×C3×C4D3214yes—
129C22×C3×C4C2×C3⋊D311yescharacteristic
130C22×Dic3—94no—
131C22×Dic3—94no—
132C22×Dic3—91no—
133C22×Dic3—91no—
134C22×Dic3—184no—
135C22×Dic3—184no—
136C2×C3×C8—38no—
137C2×C3×C8—92no—
138C4×Dic3—94no—
139C4×Dic3—94no—
140C4×Dic3—91no—
141C4×Dic3—91no—
142C4×Dic3—94no—
143C4×Dic3—94no—
144C4×Dic3—91no—
145C4×Dic3—91no—
146C4×Dic3—184no—
147C4×Dic3—184no—
148C4×Dic3—184no—
149C4×Dic3—184no—
150C2×C3⋊C8—38no—
151C2×C3⋊C8C2×C3⋊D312yes—
152C22×C32—28no—
153C22×C32—28no—
154C22×C32—24no—
155C22×C32—21no—
156C22×C32—24no—
157C22×C32—21no—
158C22×C32—28no—
159C22×C32—24no—
160C22×C32ΓL(1,𝔽25)18yes—
161C22×C32ΓL(1,𝔽25)12yes—
162C22×C32C2×C4×D314yes—
163C22×C32C2×C4×D311yescharacteristic
164C32×C4—28no—
165C32×C4—24no—
166C32×C4—21no—
167C32×C4—28no—
168C32×C4#23718yes—
169C32×C4#23712yes—
170C3⋊Dic3—32no—
171C3⋊Dic3—38no—
172C3⋊Dic3—68no—
173C3⋊Dic3—68no—
174C3⋊Dic3—68no—
175C3⋊Dic3—68no—
176C3⋊Dic3—64no—
177C3⋊Dic3—64no—
178C3⋊Dic3—61no—
179C3⋊Dic3—61no—
180C3⋊Dic3—64no—
181C3⋊Dic3—61no—
182C2×C42—271no—
183C4⋊C8—272no—
184C22⋊C8—31no—
185C22⋊C8—91no—
186#97—272no—
187C33#43211yescharacteristic, complemented, Sylow
188C23×C3—24no—
189C23×C3#62414yes—
190C23×C3C4×C3⋊D311yescharacteristic
191C2×C3×C4—216no—
192C2×C3×C4—28no—
193C2×C3×C4—22no—
194C2×C3×C4—24no—
195C2×C3×C4—24no—
196C2×C3×C4#62414yes—
197C2×C3×C4C4×C3⋊D311yescharacteristic
198C2×C3×C4C2×D3214yes—
199C2×C3×C4C22×C3⋊D311yescharacteristic
200C2×Dic3—94no—
201C2×Dic3—91no—
202C2×Dic3—94no—
203C2×Dic3—94no—
204C2×Dic3—94no—
205C2×Dic3—91no—
206C2×Dic3—91no—
207C2×Dic3—91no—
208C2×Dic3—184no—
209C2×Dic3—184no—
210C2×Dic3—184no—
211C2×Dic3—184no—
212C2×Dic3—188no—
213C2×Dic3—182no—
214C2×Dic3—1816no—
215C2×Dic3—1816no—
216C2×Dic3—188no—
217C2×Dic3—182no—
218C3×C8—68no—
219C3×C8—182no—
220C3⋊C8—22no—
221C3⋊C8—68no—
222C2×C32—216no—
223C2×C32—28no—
224C2×C32—22no—
225C2×C32—24no—
226C2×C32—24no—
227C2×C32—24no—
228C2×C32#87714yes—
229C2×C32#87711yescharacteristic
230C2×C32C2×ΓL(1,𝔽25)14yes—
231C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
232C2×C32D3×M1614yes—
233C2×C32D3×M1611yescharacteristic
234C22×C4—271no—
235C22×C4—271no—
236C22×C4D3×C3⋊D311yescharacteristic
237C2×C8—32no—
238C2×C8—92no—
239C42—271no—
240C42—271no—
241C42—271no—
242C42—271no—
243C22×C3—28no—
244C22×C3—28no—
245C22×C3—24no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—24no—
249C22×C3—24no—
250C22×C3—28no—
251C22×C3#371218yes—
252C22×C3#374312yes—
253C22×C3C2×#62414yes—
254C22×C3C2×C4×C3⋊D311yescharacteristic
255C3×C4—28no—
256C3×C4—21no—
257C3×C4—24no—
258C3×C4—28no—
259C3×C4#380518yes—
260C3×C4#383012yes—
261Dic3—98no—
262Dic3—92no—
263Dic3—188no—
264Dic3—188no—
265Dic3—188no—
266Dic3—188no—
267Dic3—184no—
268Dic3—184no—
269Dic3—184no—
270Dic3—181no—
271Dic3—181no—
272Dic3—181no—
273C32—24no—
274C32#471914yescomplemented
275C32#471911yescharacteristic, complemented
276C23#628911yescharacteristic
277C2×C4—22no—
278C2×C4—271no—
279C2×C4—271no—
280C2×C4—271no—
281C2×C4—271no—
282C2×C4—542no—
283C2×C4—542no—
284C2×C4#628911yescharacteristic
285C2×C4C2×D3×C3⋊D311yescharacteristic
286C8—62no—
287C8—182no—
288C2×C3—216no—
289C2×C3—28no—
290C2×C3—22no—
291C2×C3—24no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3#6381814yes—
295C2×C3C2×#371214yes—
296C2×C3#6395714yes—
297C2×C3#6408611yescharacteristic
298C2×C3C2×#374311yescharacteristic
299C2×C3M16×C3⋊D311yescharacteristic
300C22—21no—
301C22—21no—
302C22#8878812yes—
303C22C2×#628911yescharacteristic
304C4—21no—
305C4—272no—
306C4—541no—
307C4—541no—
308C4—541no—
309C4#8902912yes—
310C3—24no—
311C3#1059106614yescomplemented
312C3#1059241811yescharacteristic, complemented
313C2—22no—
314C2#1172335811yescharacteristic
315C2C2×#8878811yescharacteristic
316C2#1172359811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC22×C33×C4C22
prime radicalC23×C33C2×C4
cosocleC2×C33×C4C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#432
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11722050C2semidirect
5. #11722021C2semidirect
187. C33#432semidirect
274. C32#4719semidirect
275. C32#4719semidirect
311. C3#10591066semidirect
312. C3#10592418semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊(C3⋊Dic3)2
C3×#636572
C32×#8302
#117220502
#117220212
#105924183
#105910663