Finite Group Databasebeta

← #13155666

Subgroups — #13155666

Order 1728.

654 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 114 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#11722018C211yesmaximal, characteristic
4C3×#63576C211yesmaximal, characteristic
5C3×#63567C212yesmaximal
6#10590790—31nomaximal
7#10590482—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—
22C8×C3⋊Dic3—31no—
23C3×#816—32no—
24C3×#810—32no—
25C3×#807—32no—
26#63576D311yescharacteristic
27#63567D312yes—
28#63553—32no—
29#63553—61no—
30C2×C33×C4C2311yescharacteristic
31C2×C3⋊(C3⋊Dic3)D411yescharacteristic
32C2×C3⋊(C3⋊Dic3)D411yescharacteristic
33C2×C3×C3⋊Dic3—22no—
34C3×#634—22no—
35C3×#634C2×C414yes—
36C2×C32×Dic3—22no—
37C3×C3⋊(C3⋊C8)C2×C412yes—
38C32×C3⋊C8—21no—
39#4873—91no—
40#4790—92no—
41C2×#634C2×D312yes—
42C2×C3×Dic6—32no—
43C2×C3×Dic6—32no—
44C2×C3×Dic6—62no—
45C2×C32×C8—31no—
46C32×C4⋊C4—32no—
47#3750—62no—
48C4×C3⋊Dic3—32no—
49C4×C3⋊Dic3—31no—
50C4×C3⋊Dic3—62no—
51C4×C3⋊Dic3—61no—
52C4×C3⋊Dic3—62no—
53C4×C3⋊Dic3—121no—
54C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
55C3×C4⋊Dic3—22no—
56C3×C4⋊Dic3C2×D312yes—
57C3×C4⋊Dic3C2×D312yes—
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#3719—38no—
69#3718—34no—
70C22×C33C2×D411yescharacteristic
71C33×C4C22×C411yescharacteristic
72C33×C4#4111yescharacteristic
73C3⋊(C3⋊Dic3)—21no—
74C3⋊(C3⋊Dic3)Q1612yescomplemented
75C3×C3⋊Dic3—24no—
76C32×Dic3—42no—
77Q8×Dic3—92no—
78C2×#216—92no—
79C2×Dic12—91no—
80#887—92no—
81#887—92no—
82#887—182no—
83C3×#107—91no—
84C3×#103—92no—
85#816—32no—
86C8×Dic3—92no—
87C8×Dic3—91no—
88C8×Dic3—181no—
89#810—32no—
90#807—32no—
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—31no—
99C2×C3⋊Dic3—32no—
100C2×C3⋊Dic3—31no—
101C2×C3⋊Dic3—32no—
102C2×C3⋊Dic3—62no—
103C2×C3⋊Dic3—61no—
104C2×C3⋊Dic3—61no—
105C2×C3⋊Dic3—62no—
106C2×C3⋊Dic3—121no—
107C2×C3⋊Dic3—121no—
108#634—22no—
109#634C4×D314yes—
110C2×C3×Dic3—24no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—64no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C3×Dic6—34no—
117C3×Dic6—34no—
118C3×Dic6—64no—
119C3×Dic6—64no—
120C3×Dic6—62no—
121C3×Dic6—62no—
122C32×C8—61no—
123C3⋊(C3⋊C8)C4×D312yes—
124C3×C3⋊C8—22no—
125C3×C3⋊C8—22no—
126C3×C3⋊C8—34no—
127C3×C3⋊C8—62no—
128C4×Q16—271noSylow
129C2×C33C4×D411yescharacteristic
130C2×C33C2×Q1611yescharacteristic
131C2×C33#13511yescharacteristic
132C2×C3×Q8—92no—
133C2×Dic6—32no—
134C2×Dic6—32no—
135C2×Dic6—62no—
136C2×C3×C8—32no—
137C2×C3×C8—61no—
138C2×C3×C8—91no—
139C3×C4⋊C4—32no—
140C3×C4⋊C4—32no—
141C3×C4⋊C4—62no—
142#216—98no—
143C4⋊Dic3—182no—
144C4⋊Dic3D3212yes—
145Dic3⋊C4—182no—
146Dic3⋊C4—182no—
147Dic3⋊C4—184no—
148C4×Dic3—91no—
149C4×Dic3—92no—
150C4×Dic3—181no—
151C4×Dic3—182no—
152C4×Dic3—182no—
153C4×Dic3—182no—
154C4×Dic3—182no—
155C4×Dic3—184no—
156C4×Dic3—361no—
157C2×C3⋊C8—32no—
158C2×C3⋊C8—61no—
159C2×C3⋊C8D3211yescharacteristic
160Dic12—94no—
161C22×C32—21no—
162C22×C32—22no—
163C22×C32—41no—
164C22×C32D3×D412yes—
165C22×C32D3×D411yescharacteristic
166C32×C4—22no—
167C32×C4—21no—
168C32×C4—22no—
169C32×C4—21no—
170C32×C4—41no—
171C32×C4—41no—
172C32×C4—122no—
173C32×C4C2×C4×D312yes—
174C32×C4C2×C4×D311yescharacteristic
175C32×C4#23712yes—
176C32×C4#23911yescharacteristic
177C3⋊Dic3—24no—
178C3⋊Dic3—34no—
179C3⋊Dic3—32no—
180C3⋊Dic3—62no—
181C3⋊Dic3—64no—
182C3⋊Dic3—61no—
183C3⋊Dic3—62no—
184C3⋊Dic3—62no—
185C3⋊Dic3—122no—
186C3⋊Dic3—122no—
187C3⋊Dic3—122no—
188C3×Dic3—44no—
189C3×Dic3—42no—
190C3×Dic3—42no—
191C3×Dic3—68no—
192C3×Dic3—64no—
193C3×Dic3—64no—
194C2×Q16—271no—
195C4×Q8—272no—
196#107—91no—
197#103—92no—
198C4×C8—271no—
199C33C4×Q1611yescharacteristic, complemented, Sylow
200C3×Q8—94no—
201C3×Q8—182no—
202C2×C3×C4—21no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4—64no—
206C2×C3×C4—62no—
207C2×C3×C4—62no—
208C2×C3×C4—182no—
209C2×C3×C4C2×D3212yes—
210C2×C3×C4C2×D3211yescharacteristic
211C2×Dic3—22no—
212C2×Dic3—64no—
213C2×Dic3—62no—
214C2×Dic3—62no—
215C2×Dic3—91no—
216C2×Dic3—92no—
217C2×Dic3—91no—
218C2×Dic3—92no—
219C2×Dic3—181no—
220C2×Dic3—181no—
221C2×Dic3—182no—
222C2×Dic3—182no—
223C2×Dic3—361no—
224C2×Dic3—361no—
225Dic6—34no—
226Dic6—34no—
227Dic6—64no—
228Dic6—64no—
229Dic6—62no—
230Dic6—62no—
231C3×C8—62no—
232C3×C8—62no—
233C3×C8—92no—
234C3⋊C8—21no—
235C3⋊C8—34no—
236C3⋊C8—62no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32—22no—
240C2×C32—21no—
241C2×C32—22no—
242C2×C32—22no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32#67⋊C412yes—
247C2×C32#89311yescharacteristic
248C2×C32#91211yescharacteristic
249C2×C32#91612yes—
250C2×C32D3×Q1612yes—
251C2×C32D3×Q1611yescharacteristic
252C2×Q8—92no—
253Q16—274no—
254C2×C8—31no—
255C2×C8—91no—
256C4⋊C4—32no—
257C4⋊C4—542no—
258C42—271no—
259C42—542no—
260C22×C3—22no—
261C22×C3—21no—
262C22×C3—41no—
263C22×C3#380211yescharacteristic
264C22×C3#381112yes—
265C3×C4—21no—
266C3×C4—22no—
267C3×C4—21no—
268C3×C4—22no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4—124no—
272C3×C4—122no—
273C3×C4—122no—
274C3×C4—184no—
275C3×C4#379612yes—
276C3×C4C4×D3212yes—
277C3×C4C4×D3211yescharacteristic
278C3×C4#380511yescharacteristic
279Dic3—42no—
280Dic3—68no—
281Dic3—64no—
282Dic3—64no—
283Dic3—92no—
284Dic3—94no—
285Dic3—182no—
286Dic3—184no—
287Dic3—182no—
288Dic3—181no—
289Dic3—182no—
290Dic3—362no—
291Dic3—362no—
292Dic3—362no—
293C32—22no—
294C32—21no—
295C32—41no—
296C32#479012yescomplemented
297C32#487311yescharacteristic, complemented
298Q8—94no—
299Q8—182no—
300C2×C4—62no—
301C2×C4—182no—
302C2×C4—271no—
303C2×C4—271no—
304C2×C4D3311yescharacteristic
305C8—61no—
306C8—92no—
307C2×C3—21no—
308C2×C3—22no—
309C2×C3—21no—
310C2×C3—22no—
311C2×C3—22no—
312C2×C3—21no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3#6380211yescharacteristic
317C2×C3#6380911yescharacteristic
318C2×C3#6388112yes—
319C2×C3#6390211yescharacteristic
320C2×C3#6393612yes—
321C2×C3#6394412yes—
322C22#8895411yescharacteristic
323C4—122no—
324C4—184no—
325C4—272no—
326C4—541no—
327C4#8894911yescharacteristic
328C4#8896411yescharacteristic
329C3—21no—
330C3—22no—
331C3—41no—
332C3#1059048212yescomplemented
333C3#1059079011yescharacteristic, complemented
334C2#1172284811yescharacteristic
335C2#1172286111yescharacteristic
336C2#1172293811yescharacteristic
subgroupG/H
centerC22#88954
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socle, prime radicalC22×C33C2×D4
hypocenterC33C4×Q16
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
74. C3⋊(C3⋊Dic3)Q16semidirect
199. C33C4×Q16semidirect
296. C32#4790semidirect
297. C32#4873semidirect
332. C3#10590482semidirect
333. C3#10590790semidirect

Maximal subgroups

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

subgroupindex
C2×#888122
#117234102
#117220182
C3×#635762
C3×#635672
#105907903
#105904823