Finite Group Databasebeta

← #13156321

Subgroups — #13156321

Order 1728.

552 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 110 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
1Dic6×C3⋊Dic3C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic
3#11722078C211yesmaximal, characteristic
4#11722057C212yesmaximal
5C3×#63565C212yesmaximal
6#10591631—31nomaximal
7#10590489—32nomaximal
8C2×C32×Dic6C2211yescharacteristic
9C2×C3×#627—21no—
10C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4⋊Dic3C2212yes—
14Dic3⋊(C3⋊Dic3)—21no—
15Dic3×C3⋊Dic3—21no—
16C3×#3724—21no—
17Q8×C3⋊Dic3—31no—
18C3×#888—32no—
19#63919D311yescharacteristic
20Dic3×Dic6—32no—
21Dic3×Dic6—61no—
22#63634—32no—
23C3×#835—32no—
24C3×#810—32no—
25#63575—32no—
26#63575—32no—
27#63575—62no—
28#63572—31no—
29Dic3⋊(C3⋊C8)—32no—
30#63565D312yes—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C32×Dic3—21no—
36C2×C32×Dic3—22no—
37C32×Dic6—21no—
38C32×Dic6Q812yescomplemented
39C3×#627—42no—
40C3×C3⋊(C3⋊C8)—22no—
41#5091—91no—
42#4793—92no—
43C2×C32×Q8—31no—
44C2×C3×Dic6—21no—
45C2×C3×Dic6—62no—
46C2×C3×Dic6C2×D312yes—
47C2×#627—21no—
48C32×C4⋊C4—32no—
49C4⋊(C3⋊Dic3)—31no—
50C4⋊(C3⋊Dic3)—32no—
51C4⋊(C3⋊Dic3)—61no—
52C4⋊(C3⋊Dic3)—61no—
53C4⋊(C3⋊Dic3)—62no—
54C4⋊(C3⋊Dic3)—121no—
55C4×C3⋊Dic3—61no—
56C4×C3⋊Dic3C2×D311yescharacteristic
57C2×C3⋊(C3⋊C8)C2×D312yes—
58C3×C4⋊Dic3—22no—
59C3×C4⋊Dic3C2×D312yes—
60C3×C4⋊Dic3C2×D312yes—
61C3×Dic3⋊C4—62no—
62C3×C4×Dic3—32no—
63C3×C4×Dic3—61no—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—62no—
67#3724—21no—
68Dic3⋊Dic3—62no—
69Dic3⋊Dic3—62no—
70Dic32—62no—
71Dic32—62no—
72C22×C33C2×D411yescharacteristic
73C33×C4C2×Q811yescharacteristic
74C33×C4#4111yescharacteristic
75C3⋊(C3⋊Dic3)—41no—
76C3×C3⋊Dic3—21no—
77C3×C3⋊Dic3Q1612yes—
78C32×Dic3—22no—
79C32×Dic3—42no—
80C3×C4⋊Q8—91no—
81Q8×Dic3—92no—
82Q8×Dic3—181no—
83#888—32no—
84C4×Dic6—91no—
85C3×#103—92no—
86#835—32no—
87#818—92no—
88Dic3⋊C8—92no—
89#810—32no—
90#807—92no—
91#807—92no—
92#807—182no—
93C4⋊(C3⋊C8)—91no—
94C32×Q8—32no—
95C32×Q8—61no—
96C2×C32×C4—22no—
97C2×C32×C4—21no—
98C2×C32×C4—41no—
99C2×C32×C4—61no—
100C2×C32×C4—62no—
101C2×C32×C4C22×D312yes—
102C2×C32×C4C22×D311yescharacteristic
103C2×C3⋊Dic3—61no—
104C2×C3⋊Dic3—62no—
105C2×C3⋊Dic3—62no—
106C2×C3⋊Dic3—122no—
107C2×C3⋊Dic3—122no—
108C2×C3⋊Dic3#6711yescharacteristic
109C2×C3⋊Dic3#6711yescharacteristic
110C2×C3×Dic3—22no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—22no—
114C2×C3×Dic3—32no—
115C2×C3×Dic3—32no—
116C2×C3×Dic3—42no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C3×Dic6—22no—
120C3×Dic6—22no—
121C3×Dic6—22no—
122C3×Dic6—124no—
123C3×Dic6Dic614yescomplemented
124#627—42no—
125C3⋊(C3⋊C8)—22no—
126C3×C3⋊C8—64no—
127C3×C3⋊C8—62no—
128C3×C3⋊C8—62no—
129#477—271noSylow
130C2×C33C22⋊Q811yescharacteristic
131C2×C33C2×Q1611yescharacteristic
132C2×C33AGL(1,ℤ/8)11yescharacteristic
133C2×C3×Q8—32no—
134C2×C3×Q8—61no—
135C2×C3×Q8—181no—
136C2×Dic6—62no—
137C2×Dic6D3211yescharacteristic
138C2×C3×C8—92no—
139C3×C4⋊C4—32no—
140C3×C4⋊C4—32no—
141C3×C4⋊C4—62no—
142C3×C4⋊C4—181no—
143C3×C42—91no—
144C4⋊Dic3—91no—
145C4⋊Dic3—92no—
146C4⋊Dic3—181no—
147C4⋊Dic3—182no—
148C4⋊Dic3—182no—
149C4⋊Dic3—182no—
150C4⋊Dic3—361no—
151C4⋊Dic3D3212yes—
152Dic3⋊C4—62no—
153Dic3⋊C4—181no—
154C4×Dic3—32no—
155C4×Dic3—61no—
156C4×Dic3—181no—
157C4×Dic3—182no—
158C4×Dic3—182no—
159C2×C3⋊C8—32no—
160C2×C3⋊C8—32no—
161C2×C3⋊C8—62no—
162C22×C32—22no—
163C22×C32—21no—
164C22×C32—41no—
165C22×C32D3×D412yes—
166C22×C32C2×#6711yescharacteristic
167C32×C4—22no—
168C32×C4—22no—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4—41no—
172C32×C4—41no—
173C32×C4—62no—
174C32×C4—122no—
175C32×C4C2×Dic612yes—
176C32×C4#23712yes—
177C32×C4D3×Q811yescharacteristic
178C32×C4#23911yescharacteristic
179C3⋊Dic3—21no—
180C3⋊Dic3—121no—
181C3⋊Dic3—122no—
182C3⋊Dic3—122no—
183C3⋊Dic3—242no—
184C3⋊Dic3—242no—
185C3⋊Dic3#21612yes—
186C3×Dic3—24no—
187C3×Dic3—24no—
188C3×Dic3—34no—
189C3×Dic3—42no—
190C3×Dic3—42no—
191C3×Dic3—62no—
192C3×Dic3—62no—
193C3×Dic3—82no—
194C3×Dic3—121no—
195C4⋊Q8—91no—
196C4×Q8—271no—
197#107—272no—
198C4⋊C8—271no—
199#103—92no—
200C33#47711yescharacteristic, complemented, Sylow
201C3×Q8—34no—
202C3×Q8—62no—
203C3×Q8—62no—
204C3×Q8—62no—
205C3×Q8—362no—
206C2×C3×C4—22no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4—62no—
213C2×C3×C4—91no—
214C2×C3×C4—91no—
215C2×C3×C4—122no—
216C2×C3×C4C2×D3212yes—
217C2×C3×C4C2×D3211yescharacteristic
218C2×Dic3—21no—
219C2×Dic3—22no—
220C2×Dic3—32no—
221C2×Dic3—32no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—182no—
225C2×Dic3—182no—
226C2×Dic3—181no—
227C2×Dic3—362no—
228C2×Dic3—362no—
229Dic6—21no—
230Dic6—124no—
231Dic6#62712yescomplemented
232C3×C8—182no—
233C3⋊C8—64no—
234C3⋊C8—62no—
235C3⋊C8—62no—
236C2×C32—22no—
237C2×C32—22no—
238C2×C32—21no—
239C2×C32—22no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—41no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32#87812yes—
246C2×C32#91412yes—
247C2×C32D3×Q1612yes—
248C2×C32#93211yescharacteristic
249C2×C32C2×#21611yescharacteristic
250C2×C32#94611yescharacteristic
251C2×Q8—31no—
252C2×Q8—181no—
253C2×C8—92no—
254C4⋊C4—32no—
255C4⋊C4—181no—
256C4⋊C4—271no—
257C4⋊C4—541no—
258C42—91no—
259C42—541no—
260C22×C3—21no—
261C22×C3—22no—
262C22×C3—41no—
263C22×C3D3×#6712yes—
264C22×C3#381111yescharacteristic
265C3×C4—21no—
266C3×C4—22no—
267C3×C4—22no—
268C3×C4—21no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4—64no—
272C3×C4—64no—
273C3×C4—92no—
274C3×C4—122no—
275C3×C4—122no—
276C3×C4—181no—
277C3×C4—242no—
278C3×C4D3×Dic612yes—
279C3×C4#379612yes—
280C3×C4#380511yescharacteristic
281C3×C4C2×#62711yescharacteristic
282Dic3—22no—
283Dic3—34no—
284Dic3—42no—
285Dic3—62no—
286Dic3—62no—
287Dic3—121no—
288Dic3—362no—
289Dic3—362no—
290Dic3—361no—
291Dic3—722no—
292Dic3—722no—
293C32—22no—
294C32—21no—
295C32—41no—
296C32#479312yescomplemented
297C32#509111yescharacteristic, complemented
298Q8—32no—
299Q8—61no—
300Q8—362no—
301C2×C4—61no—
302C2×C4—62no—
303C2×C4—91no—
304C2×C4—91no—
305C2×C4—541no—
306C2×C4D3311yescharacteristic
307C8—182no—
308C2×C3—21no—
309C2×C3—22no—
310C2×C3—21no—
311C2×C3—22no—
312C2×C3—21no—
313C2×C3—22no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3#6388712yes—
318C2×C3#6393412yes—
319C2×C3#6394111yescharacteristic
320C2×C3D3×#21612yes—
321C2×C3#6394411yescharacteristic
322C2×C3#6398211yescharacteristic
323C22#8895211yescharacteristic
324C4—62no—
325C4—92no—
326C4—122no—
327C4—181no—
328C4—1081no—
329C4D3×#62711yescharacteristic
330C4#8896411yescharacteristic
331C3—22no—
332C3—21no—
333C3—41no—
334C3#1059048912yescomplemented
335C3#1059163111yescharacteristic, complemented
336C2#1172284611yescharacteristic
337C2#1172287311yescharacteristic
338C2#1172297611yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#477
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. C32×Dic6Q8semidirect
123. C3×Dic6Dic6semidirect
200. C33#477semidirect
231. Dic6#627semidirect
296. C32#4793semidirect
297. C32#5091semidirect
334. C3#10590489semidirect
335. C3#10591631semidirect

Maximal subgroups

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

subgroupindex
Dic6×C3⋊Dic32
C3×#639192
#117220782
#117220572
C3×#635652
#105916313
#105904893