Finite Group Databasebeta

← #13155710

Subgroups — #13155710

Order 1728.

642 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 112 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×#88812C211yesmaximal, characteristic
2#11723410C212yesmaximal
3#11722021C211yesmaximal, characteristic
4C3×#63574C211yesmaximal, characteristic
5C3×#63567C212yesmaximal
6#10590791—31nomaximal
7#10590485—32nomaximal
8C2×C3×#634C2212yes—
9C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
11C32×C4⋊Dic3C2212yes—
12C2×C32×C3⋊C8C2211yescharacteristic
13#88812C414yescomplemented
14#88807—22no—
15Dic3×C3⋊Dic3—22no—
16#64096—32no—
17#63843—32no—
18#63843—32no—
19#63843—62no—
20C2×#3719—32no—
21C2×#3718—31no—
22#63642—31no—
23C3×#816—32no—
24C3×#810—32no—
25C3×#808—32no—
26#63574D311yescharacteristic
27#63567D312yes—
28#63555—32no—
29#63555—61no—
30C2×C33×C4C2311yescharacteristic
31C2×C3⋊(C3⋊Dic3)D412yes—
32C2×C3×C3⋊Dic3—22no—
33C3×#634—22no—
34C3×#634C2×C414yes—
35C2×C32×Dic3—22no—
36C3×C3⋊(C3⋊C8)C2×C412yes—
37C32×C3⋊C8—21no—
38#4853—91no—
39#4789—92no—
40C2×#634C2×D312yes—
41C2×C3×Dic6—32no—
42C2×C3×Dic6—32no—
43C2×C3×Dic6—62no—
44C2×C32×C8—31no—
45C32×C4⋊C4—32no—
46#3750—62no—
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×C4⋊Dic3—22no—
55C3×C4⋊Dic3C2×D312yes—
56C3×C4⋊Dic3C2×D312yes—
57C2×C3×C3⋊C8—21no—
58C2×C3×C3⋊C8—32no—
59C2×C3×C3⋊C8—61no—
60C2×C3×C3⋊C8C2×D312yes—
61Dic3⋊Dic3—62no—
62Dic3⋊Dic3—62no—
63Dic3⋊Dic3—64no—
64Dic32—62no—
65Dic32—62no—
66Dic32—64no—
67#3719—38no—
68#3718—34no—
69C22×C33C2×D411yescharacteristic
70C33×C4C22×C411yescharacteristic
71C33×C4#4111yescharacteristic
72C3⋊(C3⋊Dic3)—22no—
73C3×C3⋊Dic3—24no—
74C32×Dic3—42no—
75Q8×Dic3—92no—
76C2×#216—92no—
77C2×Dic12—91no—
78#887—92no—
79#887—92no—
80#887—182no—
81C3×#106—91no—
82C3×#103—92no—
83#816—32no—
84#815—92no—
85#815—91no—
86#815—181no—
87#810—32no—
88#808—32no—
89C2×C32×C4—21no—
90C2×C32×C4—22no—
91C2×C32×C4—41no—
92C2×C32×C4—62no—
93C2×C32×C4C22×D312yes—
94C2×C32×C4C22×D311yescharacteristic
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—32no—
97C2×C3⋊Dic3—32no—
98C2×C3⋊Dic3—32no—
99C2×C3⋊Dic3—62no—
100C2×C3⋊Dic3—62no—
101C2×C3⋊Dic3—62no—
102C2×C3⋊Dic3—122no—
103#634—22no—
104#634C4×D314yes—
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—34no—
112C3×Dic6—34no—
113C3×Dic6—64no—
114C3×Dic6—64no—
115C3×Dic6—62no—
116C3×Dic6—62no—
117C32×C8—61no—
118C3⋊(C3⋊C8)C4×D312yes—
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—92no—
127C2×Dic6—32no—
128C2×Dic6—32no—
129C2×Dic6—62no—
130C2×C3×C8—32no—
131C2×C3×C8—61no—
132C2×C3×C8—91no—
133C3×C4⋊C4—32no—
134C3×C4⋊C4—32no—
135C3×C4⋊C4—62no—
136#216—98no—
137C4⋊Dic3—182no—
138C4⋊Dic3D3212yes—
139Dic3⋊C4—182no—
140Dic3⋊C4—182no—
141Dic3⋊C4—184no—
142C4×Dic3—91no—
143C4×Dic3—92no—
144C4×Dic3—181no—
145C4×Dic3—182no—
146C4×Dic3—182no—
147C4×Dic3—182no—
148C4×Dic3—182no—
149C4×Dic3—184no—
150C4×Dic3—361no—
151C2×C3⋊C8—32no—
152C2×C3⋊C8—61no—
153C2×C3⋊C8D3211yescharacteristic
154Dic12—94no—
155C22×C32—21no—
156C22×C32—22no—
157C22×C32—41no—
158C22×C32D3×D412yes—
159C22×C32D3×D411yescharacteristic
160C32×C4—22no—
161C32×C4—21no—
162C32×C4—22no—
163C32×C4—21no—
164C32×C4—41no—
165C32×C4—41no—
166C32×C4—122no—
167C32×C4C2×C4×D312yes—
168C32×C4C2×C4×D311yescharacteristic
169C32×C4#23712yes—
170C32×C4#23911yescharacteristic
171C3⋊Dic3—24no—
172C3⋊Dic3—62no—
173C3⋊Dic3—64no—
174C3⋊Dic3—62no—
175C3⋊Dic3—62no—
176C3⋊Dic3—122no—
177C3⋊Dic3—122no—
178C3⋊Dic3—124no—
179C3×Dic3—44no—
180C3×Dic3—42no—
181C3×Dic3—42no—
182C3×Dic3—68no—
183C3×Dic3—64no—
184C3×Dic3—64no—
185C2×Q16—271no—
186C4×Q8—272no—
187#106—91no—
188#103—92no—
189#97—271no—
190C33#44111yescharacteristic, complemented, Sylow
191C3×Q8—94no—
192C3×Q8—182no—
193C2×C3×C4—21no—
194C2×C3×C4—22no—
195C2×C3×C4—41no—
196C2×C3×C4—64no—
197C2×C3×C4—62no—
198C2×C3×C4—62no—
199C2×C3×C4—182no—
200C2×C3×C4C2×D3212yes—
201C2×C3×C4C2×D3211yescharacteristic
202C2×Dic3—22no—
203C2×Dic3—64no—
204C2×Dic3—62no—
205C2×Dic3—62no—
206C2×Dic3—92no—
207C2×Dic3—92no—
208C2×Dic3—92no—
209C2×Dic3—182no—
210C2×Dic3—182no—
211C2×Dic3—182no—
212C2×Dic3—362no—
213Dic6—34no—
214Dic6—34no—
215Dic6—64no—
216Dic6—64no—
217Dic6—62no—
218Dic6—62no—
219C3×C8—62no—
220C3×C8—62no—
221C3×C8—92no—
222C3⋊C8—21no—
223C3⋊C8—34no—
224C3⋊C8—62no—
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—21no—
228C2×C32—22no—
229C2×C32—22no—
230C2×C32—41no—
231C2×C32—42no—
232C2×C32#67⋊C412yes—
233C2×C32#89311yescharacteristic
234C2×C32#91512yes—
235C2×C32#91512yes—
236C2×C32#91812yes—
237C2×Q8—92no—
238Q16—274no—
239C2×C8—31no—
240C2×C8—91no—
241C4⋊C4—32no—
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—124no—
257C3×C4—122no—
258C3×C4—122no—
259C3×C4—184no—
260C3×C4#379612yes—
261C3×C4C4×D3212yes—
262C3×C4C4×D3211yescharacteristic
263C3×C4#380511yescharacteristic
264Dic3—42no—
265Dic3—68no—
266Dic3—64no—
267Dic3—64no—
268Dic3—184no—
269Dic3—182no—
270Dic3—182no—
271Dic3—182no—
272Dic3—362no—
273Dic3—362no—
274Dic3—364no—
275C32—22no—
276C32—21no—
277C32—41no—
278C32#478912yescomplemented
279C32#485311yescharacteristic, complemented
280Q8—94no—
281Q8—182no—
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#6388112yes—
297C2×C3#6390211yescharacteristic
298C2×C3#6393112yes—
299C2×C3#6395112yes—
300C22#8895411yescharacteristic
301C4—122no—
302C4—184no—
303C4—542no—
304C4#8894911yescharacteristic
305C4#8896411yescharacteristic
306C3—22no—
307C3—21no—
308C3—41no—
309C3#1059048512yescomplemented
310C3#1059079111yescharacteristic, complemented
311C2#1172289312yes—
312C2#1172293811yescharacteristic
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
13. #88812C4semidirect
190. C33#441semidirect
278. C32#4789semidirect
279. C32#4853semidirect
309. C3#10590485semidirect
310. C3#10590791semidirect

Maximal subgroups

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

subgroupindex
C2×#888122
#117234102
#117220212
C3×#635742
C3×#635672
#105907913
#105904853