Finite Group Databasebeta

← #13155719

Subgroups — #13155719

Order 1728.

690 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 118 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×#88810C211yesmaximal, characteristic
2C3×#63894C212yesmaximal, complemented
3#11722033C212yesmaximal, complemented
4#11722019C211yesmaximal, characteristic, complemented
5C3×#63574C211yesmaximal, characteristic, complemented
6#10590589—32nomaximal
7#10590344—31nomaximal
8C2×C3×C4⋊(C3⋊D3)C2212yes—
9C2×C3×#624—22no—
10C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×C4⋊Dic3C2212yes—
13C32×C4×Dic3C2211yescharacteristic
14#88810C414yescomplemented
15C3×#3722—22no—
16C3×#893—32no—
17C3×C4×D12—32no—
18#63894D312yescomplemented
19C2×#3714—32no—
20C2×#3713—31no—
21#63636—32no—
22#63630—31no—
23C3×#808—32no—
24#63574D311yescharacteristic, complemented
25#63562—32no—
26#63562—32no—
27#63562—62no—
28#63554—32no—
29#63554—61no—
30C22×C3×C3⋊D3—22no—
31C2×C33×C4C2311yescharacteristic
32C3×C4⋊(C3⋊D3)—22no—
33C3×C4⋊(C3⋊D3)C2×C414yescomplemented
34C2×C32×Dic3—22no—
35C2×C32×Dic3D412yes—
36C3×#624—44no—
37C3⋊(C3⋊(C3⋊C8))—21no—
38C3×C3⋊(C3⋊C8)C2×C412yescomplemented
39#5014—92no—
40#4869—91no—
41C2×C4⋊(C3⋊D3)C2×D312yes—
42C2×C3×D12—32no—
43C2×C3×D12—32no—
44C2×C3×D12—62no—
45C2×C3×C4×D3—62no—
46C2×C3×C4×D3—62no—
47C2×#624—22no—
48C32×C4⋊C4—32no—
49C32×C42—31no—
50C2×C3⋊(C3⋊C8)—32no—
51C2×C3⋊(C3⋊C8)—31no—
52C2×C3⋊(C3⋊C8)—61no—
53C2×C3⋊(C3⋊C8)—62no—
54C2×C3⋊(C3⋊C8)—121no—
55C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
56C3×(C2×D3)⋊C4—62no—
57C3×(C2×D3)⋊C4—62no—
58C3×C4⋊Dic3—22no—
59C3×C4⋊Dic3C2×D312yes—
60C3×C4⋊Dic3C2×D312yes—
61C3×C4×Dic3—21no—
62C3×C4×Dic3C2×D312yes—
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—61no—
65#3722—22no—
66#3714—38no—
67#3713—34no—
68C22×C33C2×D411yescharacteristic
69C2×C3×C3⋊D3—24no—
70C2×C3×C3⋊D3—42no—
71C2×C3×C3⋊D3—42no—
72C33×C4C22×C411yescharacteristic
73C33×C4#4111yescharacteristic
74C32×Dic3—22no—
75C32×Dic3—42no—
76C3×C4×D4—92no—
77C2×#213—92no—
78C2×D24—91no—
79#893—32no—
80C4×D12—32no—
81C3×#106—91no—
82#832—92no—
83#815—91no—
84#809—92no—
85#809—92no—
86#809—182no—
87#808—32no—
88#803—92no—
89#803—181no—
90C22×C3⋊D3—22no—
91C22×C3×D3—62no—
92C22×C3×D3—62no—
93C22×C3×D3—122no—
94C2×C32×C4—21no—
95C2×C32×C4—22no—
96C2×C32×C4—32no—
97C2×C32×C4—41no—
98C2×C32×C4—62no—
99C2×C32×C4C22×D312yes—
100C2×C32×C4C22×D311yescharacteristic
101C4⋊(C3⋊D3)—22no—
102C4⋊(C3⋊D3)C4×D314yescomplemented
103C2×C3×Dic3—22no—
104C2×C3×Dic3—22no—
105C2×C3×Dic3—22no—
106C2×C3×Dic3—42no—
107C2×C3×Dic3#6712yes—
108C2×C3×Dic3#6712yes—
109C3×D12—34no—
110C3×D12—34no—
111C3×D12—64no—
112C3×D12—62no—
113C3×D12—62no—
114C3×D12—122no—
115C3×C4×D3—124no—
116C3×C4×D3—124no—
117#624—44no—
118C3⋊(C3⋊C8)—61no—
119C3⋊(C3⋊C8)—62no—
120C3⋊(C3⋊C8)—62no—
121C3⋊(C3⋊C8)—122no—
122C3⋊(C3⋊C8)—122no—
123C3⋊(C3⋊C8)C4×D312yescomplemented
124C3×C3⋊C8—34no—
125C3×C3⋊C8—62no—
126#442—271noSylow
127C2×C33C4×D411yescharacteristic
128C2×C33AGL(1,ℤ/8)12yes—
129C3×C3⋊D3—44no—
130C2×C3×D4—92no—
131C22×C3×C4—182no—
132C2×D12—32no—
133C2×D12—32no—
134C2×D12—62no—
135C2×C4×D3—62no—
136C2×C4×D3—62no—
137C2×C3×C8—91no—
138C3×C4⋊C4—32no—
139C3×C4⋊C4—32no—
140C3×C4⋊C4—62no—
141C3×C22⋊C4—182no—
142C3×C42—32no—
143C3×C42—61no—
144#213—98no—
145(C2×D3)⋊C4—62no—
146(C2×D3)⋊C4—62no—
147C4⋊Dic3D3212yes—
148C4×Dic3D3211yescharacteristic
149C2×C3⋊C8—32no—
150C2×C3⋊C8—61no—
151C2×C3⋊C8—92no—
152C2×C3⋊C8—91no—
153C2×C3⋊C8—181no—
154C2×C3⋊C8—182no—
155C2×C3⋊C8—361no—
156D24—94no—
157C22×C32—21no—
158C22×C32—22no—
159C22×C32—41no—
160C22×C32D3×D411yescharacteristic
161C22×C32C2×#6712yes—
162C2×C3⋊D3—24no—
163C2×C3⋊D3—42no—
164C2×C3⋊D3—42no—
165C2×C3×D3—64no—
166C2×C3×D3—64no—
167C2×C3×D3—124no—
168C2×C3×D3—122no—
169C2×C3×D3—122no—
170C2×C3×D3—122no—
171C2×C3×D3—122no—
172C2×C3×D3—242no—
173C2×C3×D3—242no—
174C32×C4—21no—
175C32×C4—22no—
176C32×C4—21no—
177C32×C4—22no—
178C32×C4—41no—
179C32×C4—41no—
180C32×C4—62no—
181C32×C4—122no—
182C32×C4C2×C4×D311yescharacteristic
183C32×C4C2×C4×D312yes—
184C32×C4#23512yes—
185C32×C4#23911yescharacteristic
186C3×Dic3—22no—
187C3×Dic3—22no—
188C3×Dic3—42no—
189C3×Dic3—42no—
190C3×Dic3—42no—
191C3×Dic3—82no—
192C2×D8—271no—
193C4×D4—92no—
194#106—91no—
195#102—272no—
196#97—271no—
197C33#44211yescharacteristic, complemented, Sylow
198C23×C3—182no—
199C22×D3—62no—
200C22×D3—62no—
201C22×D3—122no—
202C3×D4—94no—
203C3×D4—182no—
204C2×C3×C4—21no—
205C2×C3×C4—22no—
206C2×C3×C4—32no—
207C2×C3×C4—32no—
208C2×C3×C4—41no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4—122no—
213C2×C3×C4—364no—
214C2×C3×C4C2×D3211yescharacteristic
215C2×C3×C4C2×D3212yes—
216C2×Dic3—22no—
217C2×Dic3#62512yes—
218D12—34no—
219D12—34no—
220D12—64no—
221D12—62no—
222D12—62no—
223D12—122no—
224C4×D3—124no—
225C4×D3—124no—
226C3×C8—92no—
227C3⋊C8—34no—
228C3⋊C8—62no—
229C3⋊C8—182no—
230C3⋊C8—181no—
231C3⋊C8—182no—
232C3⋊C8—362no—
233C3⋊C8—362no—
234C2×C32—22no—
235C2×C32—21no—
236C2×C32—22no—
237C2×C32—22no—
238C2×C32—22no—
239C2×C32—41no—
240C2×C32—42no—
241C2×C32#89311yescharacteristic
242C2×C32#91412yes—
243C2×C32C4×#6712yes—
244C2×C32#93212yes—
245C2×C32#94912yes—
246C3⋊D3—44no—
247C3×D3—124no—
248C3×D3—124no—
249C3×D3—244no—
250C2×D4—92no—
251C22×C4—182no—
252D8—274no—
253C2×C8—91no—
254C2×C8—271no—
255C4⋊C4—32no—
256C22⋊C4—182no—
257C42—31no—
258C22×C3—21no—
259C22×C3—22no—
260C22×C3—41no—
261C22×C3—184no—
262C22×C3—362no—
263C22×C3—362no—
264C22×C3C2×#62511yescharacteristic
265C22×C3D3×#6712yes—
266C2×D3—64no—
267C2×D3—64no—
268C2×D3—124no—
269C2×D3—122no—
270C2×D3—122no—
271C2×D3—122no—
272C2×D3—122no—
273C2×D3—242no—
274C2×D3—242no—
275C3×C4—21no—
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—22no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—62no—
282C3×C4—62no—
283C3×C4—122no—
284C3×C4—122no—
285C3×C4—122no—
286C3×C4—242no—
287C3×C4#379811yescharacteristic
288C3×C4#379912yes—
289C3×C4C4×D3211yescharacteristic
290C3×C4C4×D3212yes—
291Dic3—22no—
292Dic3—42no—
293C32—22no—
294C32—21no—
295C32—41no—
296C32#486911yescharacteristic, complemented
297C32#501412yescomplemented
298C23—182no—
299D4—94no—
300D4—182no—
301C2×C4—32no—
302C2×C4—62no—
303C2×C4—364no—
304C2×C4D3311yescharacteristic
305C8—92no—
306C8—541no—
307C2×C3—22no—
308C2×C3—22no—
309C2×C3—22no—
310C2×C3—22no—
311C2×C3—21no—
312C2×C3—42no—
313C2×C3—41no—
314C2×C3—364no—
315C2×C3#6382312yes—
316C2×C3C4×#62511yescharacteristic
317C2×C3#6390412yes—
318C2×C3#6393412yes—
319C2×C3#6393912yes—
320D3—124no—
321D3—124no—
322D3—244no—
323C22—184no—
324C22—362no—
325C22—362no—
326C22D3×#62511yescharacteristic
327C4—62no—
328C4—122no—
329C4#8894911yescharacteristic
330C4#8896511yescharacteristic
331C3—22no—
332C3—21no—
333C3—41no—
334C3#1059034411yescharacteristic, complemented
335C3#1059058912yescomplemented
336C2—364no—
337C2#1172286712yes—
338C2#1172293111yescharacteristic
subgroupG/H
centerC22D3×#625
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×#88810C2
hypocenterC33#442
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63894C2semidirect
3. #11722033C2semidirect
4. #11722019C2semidirect
5. C3×#63574C2semidirect
14. #88810C4semidirect
18. #63894D3semidirect
24. #63574D3semidirect
33. C3×C4⋊(C3⋊D3)C2×C4semidirect
38. C3×C3⋊(C3⋊C8)C2×C4semidirect
102. C4⋊(C3⋊D3)C4×D3semidirect
123. C3⋊(C3⋊C8)C4×D3semidirect
197. C33#442semidirect
296. C32#4869semidirect
297. C32#5014semidirect
334. C3#10590344semidirect
335. C3#10590589semidirect

Maximal subgroups

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

subgroupindex
C2×#888102
C3×#638942
#117220332
#117220192
C3×#635742
#105905893
#105903443