Finite Group Databasebeta

← #13156325

Subgroups — #13156325

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
1C4⋊#6296C211yesmaximal, characteristic
2Dic6×C3⋊Dic3C211yesmaximal, characteristic
3#11722055C212yesmaximal
4C3×#63572C211yesmaximal, characteristic
5C3×#63566C212yesmaximal
6#10591631—31nomaximal
7#10590804—32nomaximal
8C2×C32×Dic6C2211yescharacteristic
9C2×#6296—21no—
10C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14#88815—21no—
15Dic3⋊(C3⋊Dic3)—21no—
16Dic3×C3⋊Dic3—21no—
17Q8×C3⋊Dic3—31no—
18#63919—31no—
19#63918—32no—
20Dic3×Dic6—32no—
21Dic3×Dic6—61no—
22#63645—32no—
23C3×#835—32no—
24C3×#816—32no—
25C3×Dic3⋊C8—32no—
26#63575—32no—
27#63575—32no—
28#63575—62no—
29#63572D311yescharacteristic
30#63566D312yes—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C32×Dic3—21no—
37C32×Dic6—21no—
38C32×Dic6Q812yescomplemented
39#6296—42no—
40C32×C3⋊C8—22no—
41#4850—91no—
42#4793—92no—
43C2×C32×Q8—31no—
44C2×C3×Dic6—21no—
45C2×C3×Dic6C2×D312yes—
46C2×#627—62no—
47C2×#627—61no—
48C2×C32×C8—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)C2×D312yes—
56C4×C3⋊Dic3—61no—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—62no—
61C3×C4×Dic3—32no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8—22no—
64C2×C3×C3⋊C8C2×D312yes—
65C2×C3×C3⋊C8C2×D312yes—
66#3724—61no—
67Dic3⋊Dic3—62no—
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⋊Dic3—42no—
78C3×C3⋊Dic3Q1612yes—
79C32×Dic3—22no—
80Q8×Dic3—92no—
81Q8×Dic3—181no—
82#888—92no—
83#869—91no—
84C4×Dic6—91no—
85C3×C4⋊C8—91no—
86C3×#103—92no—
87#835—32no—
88#818—92no—
89#818—92no—
90#818—182no—
91#816—32no—
92Dic3⋊C8—32no—
93#807—92no—
94C32×Q8—32no—
95C32×Q8—61no—
96C2×C32×C4—22no—
97C2×C32×C4—21no—
98C2×C32×C4—41no—
99C2×C32×C4—61no—
100C2×C32×C4C22×D312yes—
101C2×C32×C4C22×D311yescharacteristic
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3—61no—
104C2×C3⋊Dic3—62no—
105C2×C3⋊Dic3—62no—
106C2×C3⋊Dic3—122no—
107C2×C3⋊Dic3—122no—
108C2×C3⋊Dic3D1211yescharacteristic
109C2×C3⋊Dic3D1211yescharacteristic
110C2×C3×Dic3—22no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—122no—
119C3×Dic6—22no—
120C3×Dic6—22no—
121C3×Dic6—22no—
122C3×Dic6Dic614yescomplemented
123#627—124no—
124#627—122no—
125C32×C8—62no—
126C3×C3⋊C8—24no—
127C3×C3⋊C8—22no—
128C3×C3⋊C8—22no—
129#477—271noSylow
130C2×C33C22⋊Q811yescharacteristic
131C2×C33C2×Q1611yescharacteristic
132C2×C33AGL(1,ℤ/8)11yescharacteristic
133C2×C3×Q8—32no—
134C2×C3×Q8—61no—
135C2×Dic6—182no—
136C2×Dic6—181no—
137C2×Dic6D3211yescharacteristic
138C2×C3×C8—32no—
139C2×C3×C8—32no—
140C2×C3×C8—62no—
141C3×C4⋊C4—92no—
142C3×C42—91no—
143C4⋊Dic3—32no—
144C4⋊Dic3—32no—
145C4⋊Dic3—62no—
146C4⋊Dic3—92no—
147C4⋊Dic3—91no—
148C4⋊Dic3—181no—
149C4⋊Dic3—182no—
150C4⋊Dic3—182no—
151C4⋊Dic3—181no—
152C4⋊Dic3—182no—
153C4⋊Dic3—361no—
154Dic3⋊C4—181no—
155Dic3⋊C4—182no—
156C4×Dic3—32no—
157C4×Dic3—61no—
158C4×Dic3—181no—
159C4×Dic3—182no—
160C4×Dic3—182no—
161C2×C3⋊C8D3212yes—
162C22×C32—22no—
163C22×C32—21no—
164C22×C32—41no—
165C22×C32C2×D1211yescharacteristic
166C22×C32D3×D412yes—
167C32×C4—22no—
168C32×C4—22no—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4—41no—
172C32×C4—41no—
173C32×C4—62no—
174C32×C4C2×Dic612yes—
175C32×C4#23712yes—
176C32×C4#23711yescharacteristic
177C32×C4D3×Q811yescharacteristic
178C3⋊Dic3—21no—
179C3⋊Dic3—42no—
180C3⋊Dic3—121no—
181C3⋊Dic3—122no—
182C3⋊Dic3—122no—
183C3⋊Dic3—242no—
184C3⋊Dic3—242no—
185C3⋊Dic3Dic1212yes—
186C3×Dic3—24no—
187C3×Dic3—24no—
188C3×Dic3—34no—
189C3×Dic3—62no—
190C3×Dic3—62no—
191C3×Dic3—121no—
192C3×Dic3—122no—
193C3×Dic3—122no—
194C3×Dic3—242no—
195C4⋊Q8—271no—
196C4×Q8—271no—
197#107—272no—
198C4⋊C8—91no—
199#103—92no—
200C33#47711yescharacteristic, complemented, Sylow
201C3×Q8—34no—
202C3×Q8—62no—
203C3×Q8—62no—
204C3×Q8—62no—
205C2×C3×C4—21no—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—91no—
211C2×C3×C4—91no—
212C2×C3×C4—182no—
213C2×C3×C4C2×D3212yes—
214C2×C3×C4C2×D3211yescharacteristic
215C2×Dic3—21no—
216C2×Dic3—32no—
217C2×Dic3—32no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220C2×Dic3—62no—
221C2×Dic3—62no—
222C2×Dic3—122no—
223C2×Dic3—182no—
224C2×Dic3—182no—
225C2×Dic3—181no—
226C2×Dic3—362no—
227C2×Dic3—362no—
228Dic6—21no—
229Dic6—364no—
230Dic6—362no—
231Dic6#62712yescomplemented
232C3×C8—64no—
233C3×C8—62no—
234C3×C8—62no—
235C3⋊C8—22no—
236C2×C32—22no—
237C2×C32—22no—
238C2×C32—22no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—41no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32#87812yes—
246C2×C32#89711yescharacteristic
247C2×C32C2×Dic1211yescharacteristic
248C2×C32#90811yescharacteristic
249C2×C32#91412yes—
250C2×C32D3×Q1612yes—
251C2×Q8—31no—
252C2×Q8—541no—
253C2×C8—32no—
254C4⋊C4—92no—
255C4⋊C4—271no—
256C4⋊C4—541no—
257C4⋊C4—541no—
258C42—91no—
259C42—541no—
260C22×C3—21no—
261C22×C3—22no—
262C22×C3—41no—
263C22×C3D3×D1212yes—
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—181no—
275C3×C4—362no—
276C3×C4D3×Dic612yes—
277C3×C4#380512yes—
278C3×C4#380511yescharacteristic
279C3×C4C2×#62711yescharacteristic
280Dic3—22no—
281Dic3—34no—
282Dic3—62no—
283Dic3—62no—
284Dic3—121no—
285Dic3—122no—
286Dic3—122no—
287Dic3—242no—
288Dic3—362no—
289Dic3—362no—
290Dic3—361no—
291Dic3—722no—
292Dic3—722no—
293C32—22no—
294C32—21no—
295C32—41no—
296C32#479312yescomplemented
297C32#485011yescharacteristic, complemented
298Q8—32no—
299Q8—61no—
300Q8—1082no—
301C2×C4—61no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—182no—
305C2×C4—541no—
306C2×C4D3311yescharacteristic
307C8—62no—
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#6379412yes—
318C2×C3D3×Dic1212yes—
319C2×C3#6389312yes—
320C2×C3#6394111yescharacteristic
321C2×C3#6394411yescharacteristic
322C2×C3#6398211yescharacteristic
323C22#8895511yescharacteristic
324C4—62no—
325C4—92no—
326C4—181no—
327C4—362no—
328C4—1081no—
329C4D3×#62711yescharacteristic
330C4#8896311yescharacteristic
331C3—22no—
332C3—21no—
333C3—41no—
334C3#1059080412yescomplemented
335C3#1059163111yescharacteristic, complemented
336C2#1172284911yescharacteristic
337C2#1172288111yescharacteristic
338C2#1172298111yescharacteristic
subgroupG/H
centerC22#88955
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
122. C3×Dic6Dic6semidirect
200. C33#477semidirect
231. Dic6#627semidirect
296. C32#4793semidirect
297. C32#4850semidirect
334. C3#10590804semidirect
335. C3#10591631semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
Dic6×C3⋊Dic32
#117220552
C3×#635722
C3×#635662
#105916313
#105908043