Finite Group Databasebeta

← #13155697

Subgroups — #13155697

Order 1728.

642 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 124 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
1#11723397C212yesmaximal
2C2×C3×#3718C211yesmaximal, characteristic
3#11722079C211yesmaximal, characteristic
4#11722021C211yesmaximal, characteristic
5C3×#63566C212yesmaximal
6#10591582—32nomaximal
7#10590791—31nomaximal
8C2×C32×Dic6C2212yes—
9C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C3×C4⋊(C3⋊Dic3)C2212yes—
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C2×C32×C3⋊C8C2211yescharacteristic
13Dic3⋊(C3⋊Dic3)—22no—
14Dic3×C3⋊Dic3—22no—
15C3×#3718C414yescomplemented
16Q8×C3⋊Dic3—32no—
17C2×C3×#216—32no—
18#63843—32no—
19#63843—32no—
20#63843—62no—
21C2×#3718D311yescharacteristic
22#63642—31no—
23C3×#835—32no—
24C3×#816—32no—
25#63574—31no—
26#63573—32no—
27#63566D312yes—
28#63555—32no—
29#63555—61no—
30C2×C33×C4C2311yescharacteristic
31C2×C3⋊(C3⋊Dic3)D412yes—
32C2×C3×C3⋊Dic3—22no—
33C2×C32×Dic3—22no—
34C32×Dic6—22no—
35C32×Dic6C2×C414yes—
36C3×C3⋊(C3⋊C8)C2×C412yes—
37C32×C3⋊C8—21no—
38#5184—91no—
39#4789—92no—
40C2×C32×Q8—32no—
41C2×C3×Dic6—22no—
42C2×C3×Dic6C2×D312yes—
43C2×C3×Dic6C2×D312yes—
44C2×C32×C8—31no—
45C4⋊(C3⋊Dic3)—62no—
46C4⋊(C3⋊Dic3)C2×D312yes—
47C4×C3⋊Dic3—32no—
48C4×C3⋊Dic3—31no—
49C4×C3⋊Dic3—62no—
50C4×C3⋊Dic3—61no—
51C4×C3⋊Dic3—62no—
52C4×C3⋊Dic3—121no—
53C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
54C3×#216—38no—
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—62no—
58C2×C3×C3⋊C8—21no—
59C2×C3×C3⋊C8—32no—
60C2×C3×C3⋊C8—61no—
61C2×C3×C3⋊C8C2×D312yes—
62Dic3⋊Dic3—62no—
63Dic3⋊Dic3—62no—
64Dic3⋊Dic3—64no—
65Dic32—62no—
66Dic32—62no—
67Dic32—64no—
68#3718Dic314yescomplemented
69C22×C33C2×D411yescharacteristic
70C33×C4C22×C411yescharacteristic
71C33×C4#4111yescharacteristic
72C3⋊(C3⋊Dic3)—22no—
73C3×C3⋊Dic3—42no—
74C32×Dic3—24no—
75C2×C3×Q16—91no—
76Q8×Dic3—92no—
77Q8×Dic3—92no—
78Q8×Dic3—182no—
79C2×#216—32no—
80#887—92no—
81C3×#103—92no—
82#835—32no—
83#817—91no—
84#816—32no—
85#815—92no—
86#815—91no—
87#815—181no—
88#808—92no—
89C32×Q8—34no—
90C32×Q8—62no—
91C2×C32×C4—21no—
92C2×C32×C4—22no—
93C2×C32×C4—41no—
94C2×C32×C4—62no—
95C2×C32×C4C22×D312yes—
96C2×C32×C4C22×D311yescharacteristic
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3—32no—
99C2×C3⋊Dic3—32no—
100C2×C3⋊Dic3—32no—
101C2×C3⋊Dic3—62no—
102C2×C3⋊Dic3—62no—
103C2×C3⋊Dic3—62no—
104C2×C3⋊Dic3—122no—
105C2×C3×Dic3—24no—
106C2×C3×Dic3—22no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—64no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C3×Dic6—24no—
112C3×Dic6—22no—
113C3×Dic6—22no—
114C3×Dic6—24no—
115C3×Dic6C4×D314yes—
116C3×Dic6C2×Dic314yes—
117C32×C8—61no—
118C3⋊(C3⋊C8)C2×Dic312yes—
119C3×C3⋊C8—22no—
120C3×C3⋊C8—22no—
121C3×C3⋊C8—34no—
122C3×C3⋊C8—62no—
123#441—271noSylow
124C2×C33C4×D411yescharacteristic
125C2×C33#13712yes—
126C2×C3×Q8—32no—
127C2×C3×Q8—32no—
128C2×C3×Q8—62no—
129C2×Dic6D3212yes—
130C3×Q16—94no—
131C2×C3×C8—32no—
132C2×C3×C8—61no—
133C2×C3×C8—91no—
134C3×C4⋊C4—92no—
135#216—38no—
136C4⋊Dic3—32no—
137C4⋊Dic3—32no—
138C4⋊Dic3—62no—
139C4⋊Dic3—182no—
140C4⋊Dic3—182no—
141C4⋊Dic3—184no—
142Dic3⋊C4—182no—
143C4×Dic3—92no—
144C4×Dic3—91no—
145C4×Dic3—181no—
146C4×Dic3—182no—
147C4×Dic3—182no—
148C4×Dic3—182no—
149C4×Dic3—182no—
150C4×Dic3—184no—
151C4×Dic3—361no—
152C2×C3⋊C8—32no—
153C2×C3⋊C8—61no—
154C2×C3⋊C8D3211yescharacteristic
155C22×C32—21no—
156C22×C32—22no—
157C22×C32—41no—
158C22×C32D3×D412yes—
159C22×C32D3×D411yescharacteristic
160C32×C4—21no—
161C32×C4—22no—
162C32×C4—21no—
163C32×C4—22no—
164C32×C4—41no—
165C32×C4—41no—
166C32×C4—64no—
167C32×C4C2×C4×D312yes—
168C32×C4#23712yes—
169C32×C4#23711yescharacteristic
170C32×C4C22×Dic311yescharacteristic
171C3⋊Dic3—42no—
172C3⋊Dic3—62no—
173C3⋊Dic3—64no—
174C3⋊Dic3—62no—
175C3⋊Dic3—62no—
176C3⋊Dic3—122no—
177C3⋊Dic3—122no—
178C3⋊Dic3—124no—
179C3×Dic3—28no—
180C3×Dic3—24no—
181C3×Dic3—24no—
182C3×Dic3—124no—
183C3×Dic3—122no—
184C3×Dic3—122no—
185C2×Q16—91no—
186C4×Q8—272no—
187#106—271no—
188#103—92no—
189#97—271no—
190C33#44111yescharacteristic, complemented, Sylow
191C3×Q8—34no—
192C3×Q8—34no—
193C3×Q8—64no—
194C3×Q8—64no—
195C3×Q8—62no—
196C3×Q8—62no—
197C2×C3×C4—21no—
198C2×C3×C4—22no—
199C2×C3×C4—41no—
200C2×C3×C4—64no—
201C2×C3×C4—62no—
202C2×C3×C4—62no—
203C2×C3×C4—182no—
204C2×C3×C4C2×D3211yescharacteristic
205C2×C3×C4C2×D3212yes—
206C2×Dic3—22no—
207C2×Dic3—64no—
208C2×Dic3—62no—
209C2×Dic3—62no—
210C2×Dic3—92no—
211C2×Dic3—92no—
212C2×Dic3—92no—
213C2×Dic3—182no—
214C2×Dic3—182no—
215C2×Dic3—182no—
216C2×Dic3—362no—
217Dic6—22no—
218Dic6D3×Dic314yes—
219C3×C8—62no—
220C3×C8—62no—
221C3×C8—92no—
222C3⋊C8—21no—
223C3⋊C8—34no—
224C3⋊C8—62no—
225C2×C32—22no—
226C2×C32—21no—
227C2×C32—22no—
228C2×C32—22no—
229C2×C32—22no—
230C2×C32—41no—
231C2×C32—42no—
232C2×C32#67⋊C412yes—
233C2×C32#91512yes—
234C2×C32#91812yes—
235C2×C32#91812yes—
236C2×C32D4×Dic311yescharacteristic
237C2×Q8—32no—
238Q16—94no—
239C2×C8—31no—
240C2×C8—91no—
241C4⋊C4—92no—
242C4⋊C4—542no—
243C42—271no—
244C42—542no—
245C22×C3—22no—
246C22×C3—21no—
247C22×C3—41no—
248C22×C3#380211yescharacteristic
249C22×C3#381112yes—
250C3×C4—21no—
251C3×C4—22no—
252C3×C4—21no—
253C3×C4—22no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—68no—
257C3×C4—64no—
258C3×C4—64no—
259C3×C4—362no—
260C3×C4C4×D3211yescharacteristic
261C3×C4C2×D3×Dic312yes—
262C3×C4#380511yescharacteristic
263C3×C4#380512yes—
264Dic3—24no—
265Dic3—124no—
266Dic3—122no—
267Dic3—122no—
268Dic3—184no—
269Dic3—182no—
270Dic3—182no—
271Dic3—182no—
272Dic3—362no—
273Dic3—362no—
274Dic3—364no—
275C32—21no—
276C32—22no—
277C32—41no—
278C32#478912yescomplemented
279C32#518411yescharacteristic, complemented
280Q8—34no—
281Q8—62no—
282C2×C4—62no—
283C2×C4—182no—
284C2×C4—272no—
285C2×C4D3311yescharacteristic
286C8—61no—
287C8—92no—
288C2×C3—22no—
289C2×C3—22no—
290C2×C3—22no—
291C2×C3—22no—
292C2×C3—21no—
293C2×C3—42no—
294C2×C3—41no—
295C2×C3#6381112yes—
296C2×C3#6390211yescharacteristic
297C2×C3#6394512yes—
298C2×C3#6395112yes—
299C2×C3#67⋊Dic312yes—
300C22#8895411yescharacteristic
301C4—64no—
302C4—362no—
303C4—542no—
304C4D32×Dic311yescharacteristic
305C4#8896311yescharacteristic
306C3—21no—
307C3—22no—
308C3—41no—
309C3#1059079111yescharacteristic, complemented
310C3#1059158212yescomplemented
311C2#1172289412yes—
312C2#1172292511yescharacteristic
subgroupG/H
centerC22#88954
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socle, prime radicalC22×C33C2×D4
hypocenterC33#441
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
15. C3×#3718C4semidirect
68. #3718Dic3semidirect
190. C33#441semidirect
278. C32#4789semidirect
279. C32#5184semidirect
309. C3#10590791semidirect
310. C3#10591582semidirect

Maximal subgroups

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

subgroupindex
#117233972
C2×C3×#37182
#117220792
#117220212
C3×#635662
#105915823
#105907913