Finite Group Databasebeta

← #13156667

Subgroups — #13156667

Order 1728.

530 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C4⋊#6296C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic
3#11722056C212yesmaximal
4C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic
5C3×#63575C212yesmaximal
6#10590902—31nomaximal
7#10590411—32nomaximal
8C2×#6296—21no—
9C2×C3×#627—21no—
10C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×C4⋊Dic3C2212yes—
14C2×C32×C3⋊C8C2211yescharacteristic
15#88815—21no—
16C3×#3724—21no—
17C3×#888—32no—
18#63919—31no—
19#63919D311yescharacteristic
20#63918—32no—
21C8×C3⋊Dic3—31no—
22#63634—32no—
23C3×#818—32no—
24C3×#807—32no—
25#63576—32no—
26#63576—32no—
27#63576—62no—
28#63575D312yes—
29Dic3×C3⋊C8—32no—
30Dic3×C3⋊C8—61no—
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C32×Dic3—22no—
36#6296—42no—
37C3×#627—42no—
38C3⋊(C3⋊(C3⋊C8))Q812yescomplemented
39C32×C3⋊C8Q812yescomplemented
40#5078—91no—
41#4874—92no—
42C2×C3×Dic6—62no—
43C2×#627—21no—
44C2×#627—62no—
45C2×#627—61no—
46C2×C32×C8—31no—
47C32×C4⋊C4—32no—
48C4⋊(C3⋊Dic3)C2×D312yes—
49C4×C3⋊Dic3C2×D311yescharacteristic
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—
55C3×C4⋊Dic3—22no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—32no—
58C3×C4⋊Dic3—62no—
59C3×C4⋊Dic3C2×D312yes—
60C3×C4⋊Dic3C2×D312yes—
61C3×Dic3⋊C4—62no—
62C3×C4×Dic3—32no—
63C3×C4×Dic3—61no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8C2×D312yes—
66#3724—21no—
67#3724—61no—
68Dic3⋊Dic3—62no—
69C22×C33C2×D411yescharacteristic
70C33×C4C2×Q811yescharacteristic
71C33×C4C2×Q811yescharacteristic
72C3×C3⋊Dic3—42no—
73C3×C3⋊Dic3D812yes—
74C3×C3⋊Dic3Q1612yes—
75C32×Dic3—42no—
76C3×C4⋊Q8—91no—
77#888—32no—
78#888—92no—
79#869—91no—
80C3×#107—92no—
81#818—32no—
82C8×Dic3—92no—
83C8×Dic3—181no—
84#807—32no—
85#807—92no—
86#807—92no—
87#807—92no—
88#807—182no—
89C4×C3⋊C8—91no—
90C2×C32×C4—21no—
91C2×C32×C4—22no—
92C2×C32×C4—41no—
93C2×C32×C4—62no—
94C2×C32×C4C22×D312yes—
95C2×C32×C4C22×D311yescharacteristic
96C2×C3⋊Dic3—22no—
97C2×C3⋊Dic3#6711yescharacteristic
98C2×C3⋊Dic3#6711yescharacteristic
99C2×C3×Dic3—22no—
100C2×C3×Dic3—22no—
101C2×C3×Dic3—32no—
102C2×C3×Dic3—32no—
103C2×C3×Dic3—42no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—122no—
109C3×Dic6—124no—
110#627—42no—
111#627—124no—
112#627—122no—
113C32×C8—32no—
114C3⋊(C3⋊C8)—34no—
115C3⋊(C3⋊C8)—32no—
116C3⋊(C3⋊C8)—62no—
117C3⋊(C3⋊C8)—64no—
118C3⋊(C3⋊C8)—122no—
119C3×C3⋊C8—22no—
120C3×C3⋊C8Dic614yescomplemented
121#500—271noSylow
122C2×C33C4⋊Q811yescharacteristic
123C2×C33C2×D811yescharacteristic
124C2×C33C2×Q1611yescharacteristic
125C2×C3×Q8—181no—
126C2×Dic6—62no—
127C2×Dic6—182no—
128C2×Dic6—181no—
129C2×C3×C8—32no—
130C2×C3×C8—61no—
131C3×C4⋊C4—32no—
132C3×C4⋊C4—32no—
133C3×C4⋊C4—62no—
134C3×C4⋊C4—92no—
135C3×C4⋊C4—181no—
136C3×C42—91no—
137C4⋊Dic3—32no—
138C4⋊Dic3—32no—
139C4⋊Dic3—62no—
140C4⋊Dic3—181no—
141C4⋊Dic3D3212yes—
142Dic3⋊C4—62no—
143Dic3⋊C4—182no—
144C4×Dic3—32no—
145C4×Dic3—61no—
146C2×C3⋊C8—91no—
147C2×C3⋊C8—92no—
148C2×C3⋊C8—182no—
149C2×C3⋊C8—181no—
150C2×C3⋊C8—361no—
151C2×C3⋊C8D3211yescharacteristic
152C22×C32—21no—
153C22×C32—22no—
154C22×C32—41no—
155C22×C32D3×D412yes—
156C22×C32C2×#6711yescharacteristic
157C32×C4—22no—
158C32×C4—22no—
159C32×C4—21no—
160C32×C4—21no—
161C32×C4—41no—
162C32×C4—41no—
163C32×C4—122no—
164C32×C4C2×Dic612yes—
165C32×C4D3×Q811yescharacteristic
166C32×C4D3×Q812yes—
167C32×C4D3×Q811yescharacteristic
168C3⋊Dic3—42no—
169C3⋊Dic3#21312yes—
170C3⋊Dic3#21612yes—
171C3×Dic3—34no—
172C3×Dic3—34no—
173C3×Dic3—42no—
174C3×Dic3—42no—
175C3×Dic3—62no—
176C3×Dic3—62no—
177C3×Dic3—82no—
178C3×Dic3—122no—
179C3×Dic3—122no—
180C3×Dic3—242no—
181C4⋊Q8—91no—
182C4⋊Q8—271no—
183#107—92no—
184#107—272no—
185C4×C8—271no—
186C33#50011yescharacteristic, complemented, Sylow
187C3×Q8—362no—
188C2×C3×C4—21no—
189C2×C3×C4—22no—
190C2×C3×C4—41no—
191C2×C3×C4—62no—
192C2×C3×C4—62no—
193C2×C3×C4—91no—
194C2×C3×C4—91no—
195C2×C3×C4—122no—
196C2×C3×C4—182no—
197C2×C3×C4C2×D3212yes—
198C2×C3×C4C2×D3211yescharacteristic
199C2×Dic3—22no—
200C2×Dic3—32no—
201C2×Dic3—32no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3—62no—
205C2×Dic3—62no—
206C2×Dic3—122no—
207Dic6—124no—
208Dic6—364no—
209Dic6—362no—
210C3×C8—34no—
211C3×C8—62no—
212C3⋊C8—94no—
213C3⋊C8—92no—
214C3⋊C8—184no—
215C3⋊C8—182no—
216C3⋊C8—362no—
217C3⋊C8#62712yescomplemented
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32—22no—
223C2×C32—22no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32#88812yes—
228C2×C32D3×D812yes—
229C2×C32D3×Q1612yes—
230C2×C32C2×#21311yescharacteristic
231C2×C32C2×#21611yescharacteristic
232C2×C32#94411yescharacteristic
233C2×Q8—181no—
234C2×Q8—541no—
235C2×C8—31no—
236C2×C8—271no—
237C4⋊C4—32no—
238C4⋊C4—92no—
239C4⋊C4—181no—
240C4⋊C4—541no—
241C42—91no—
242C22×C3—21no—
243C22×C3—22no—
244C22×C3—41no—
245C22×C3#380211yescharacteristic
246C22×C3D3×#6712yes—
247C3×C4—21no—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—21no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—92no—
254C3×C4—92no—
255C3×C4—122no—
256C3×C4—122no—
257C3×C4—242no—
258C3×C4—362no—
259C3×C4D3×Dic612yes—
260C3×C4#379712yes—
261C3×C4#379711yescharacteristic
262C3×C4C2×#62711yescharacteristic
263Dic3—34no—
264Dic3—34no—
265Dic3—42no—
266Dic3—62no—
267Dic3—62no—
268Dic3—122no—
269Dic3—122no—
270Dic3—242no—
271C32—21no—
272C32—22no—
273C32—41no—
274C32#487412yescomplemented
275C32#507811yescharacteristic, complemented
276Q8—362no—
277Q8—1082no—
278C2×C4—62no—
279C2×C4—91no—
280C2×C4—91no—
281C2×C4—182no—
282C2×C4D3311yescharacteristic
283C8—32no—
284C8—272no—
285C2×C3—22no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—22no—
289C2×C3—21no—
290C2×C3—22no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—41no—
294C2×C3#6379711yescharacteristic
295C2×C3#6380211yescharacteristic
296C2×C3#6383912yes—
297C2×C3#6391911yescharacteristic
298C2×C3D3×#21312yes—
299C2×C3D3×#21612yes—
300C22#8895311yescharacteristic
301C4—92no—
302C4—92no—
303C4—122no—
304C4—362no—
305C4D3×#62711yescharacteristic
306C4#8895811yescharacteristic
307C3—22no—
308C3—21no—
309C3—41no—
310C3#1059041112yescomplemented
311C3#1059090211yescharacteristic, complemented
312C2#1172283111yescharacteristic
313C2#1172284711yescharacteristic
314C2#1172303511yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#500
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
38. C3⋊(C3⋊(C3⋊C8))Q8semidirect
39. C32×C3⋊C8Q8semidirect
120. C3×C3⋊C8Dic6semidirect
186. C33#500semidirect
217. C3⋊C8#627semidirect
274. C32#4874semidirect
275. C32#5078semidirect
310. C3#10590411semidirect
311. C3#10590902semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
C3×#639192
#117220562
C3⋊C8×C3⋊Dic32
C3×#635752
#105909023
#105904113