Finite Group Databasebeta

← #13156320

Subgroups — #13156320

Order 1728.

576 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 104 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
1#11723406C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic
3#11722078C211yesmaximal, characteristic
4#11722041C212yesmaximal
5C3×#63576C212yesmaximal
6#10590488—32nomaximal
7#10590432—31nomaximal
8C2×#6308C2211yescharacteristic
9C2×C3×#627—21no—
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2212yes—
12C32×C4⋊Dic3C2211yescharacteristic
13C32×C4⋊Dic3C2212yes—
14#88806—21no—
15Dic3×C3⋊Dic3—21no—
16C3×#3724—21no—
17#64096—31no—
18C3×#888—32no—
19#63919D311yescharacteristic
20#63844—32no—
21#63844—61no—
22#63637—32no—
23C3×#807—32no—
24C3×#807—32no—
25#63576D312yes—
26#63572—31no—
27Dic3⋊(C3⋊C8)—32no—
28#63567—32no—
29#63567—32no—
30#63567—62no—
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33#6308—21no—
34#6308Q812yescomplemented
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C32×Dic3—21no—
38C2×C32×Dic3—22no—
39C3×#627—42no—
40C3×C3⋊(C3⋊C8)—22no—
41#5091—91no—
42#4875—92no—
43C2×#634—31no—
44C2×#634—32no—
45C2×#634—61no—
46C2×#634—62no—
47C2×#634—121no—
48C2×C3×Dic6—62no—
49C2×#627—21no—
50C32×C4⋊C4—32no—
51C32×C4⋊C4—31no—
52#3750—61no—
53C4×C3⋊Dic3—61no—
54C4×C3⋊Dic3C2×D311yescharacteristic
55C2×C3⋊(C3⋊C8)C2×D312yes—
56C3×C4⋊Dic3—21no—
57C3×C4⋊Dic3—22no—
58C3×C4⋊Dic3C2×D312yes—
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—
68#3724—62no—
69#3724—62no—
70Dic32—62no—
71Dic32—62no—
72C22×C33C2×D411yescharacteristic
73C33×C4C2×Q811yescharacteristic
74C33×C4#4111yescharacteristic
75C3⋊(C3⋊Dic3)—22no—
76C3×C3⋊Dic3—21no—
77C3×C3⋊Dic3Q1612yes—
78C32×Dic3—41no—
79C32×Dic3—42no—
80C3×C4⋊Q8—91no—
81#888—32no—
82#887—92no—
83#887—181no—
84C4×Dic6—91no—
85C3×#107—92no—
86#816—92no—
87Dic3⋊C8—92no—
88#810—92no—
89#810—92no—
90#810—182no—
91#807—32no—
92#807—32no—
93C4⋊(C3⋊C8)—91no—
94C2×C32×C4—22no—
95C2×C32×C4—21no—
96C2×C32×C4—41no—
97C2×C32×C4—61no—
98C2×C32×C4—62no—
99C2×C32×C4C22×D312yes—
100C2×C32×C4C22×D311yescharacteristic
101C2×C3⋊Dic3—61no—
102C2×C3⋊Dic3—62no—
103C2×C3⋊Dic3—62no—
104C2×C3⋊Dic3—122no—
105C2×C3⋊Dic3—122no—
106C2×C3⋊Dic3#6711yescharacteristic
107C2×C3⋊Dic3#6711yescharacteristic
108#634—32no—
109#634—34no—
110#634—62no—
111#634—64no—
112#634—61no—
113#634—62no—
114#634—62no—
115#634—122no—
116#634—122no—
117#634—122no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—22no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—32no—
124C2×C3×Dic3—42no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C3×Dic6—124no—
128#627—42no—
129C3⋊(C3⋊C8)—22no—
130C3×C3⋊C8—62no—
131C3×C3⋊C8—62no—
132C3×C3⋊C8—122no—
133#477—271noSylow
134C2×C33C22⋊Q811yescharacteristic
135C2×C33C2×Q1611yescharacteristic
136C2×C33AGL(1,ℤ/8)11yescharacteristic
137C2×C3×Q8—181no—
138C2×Dic6—62no—
139C2×Dic6—91no—
140C2×Dic6—92no—
141C2×Dic6—181no—
142C2×Dic6—182no—
143C2×Dic6—361no—
144C2×C3×C8—92no—
145C3×C4⋊C4—32no—
146C3×C4⋊C4—32no—
147C3×C4⋊C4—32no—
148C3×C4⋊C4—61no—
149C3×C4⋊C4—62no—
150C3×C4⋊C4—181no—
151C3×C42—91no—
152C4⋊Dic3D3211yescharacteristic
153C4⋊Dic3D3212yes—
154Dic3⋊C4—62no—
155Dic3⋊C4—181no—
156Dic3⋊C4—182no—
157Dic3⋊C4—182no—
158C4×Dic3—32no—
159C4×Dic3—61no—
160C4×Dic3—181no—
161C4×Dic3—182no—
162C4×Dic3—182no—
163C2×C3⋊C8—32no—
164C2×C3⋊C8—32no—
165C2×C3⋊C8—62no—
166C22×C32—22no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32D3×D412yes—
170C22×C32C2×#6711yescharacteristic
171C32×C4—21no—
172C32×C4—22no—
173C32×C4—21no—
174C32×C4—22no—
175C32×C4—41no—
176C32×C4—41no—
177C32×C4—121no—
178C32×C4—122no—
179C32×C4#23512yes—
180C32×C4D3×Q812yes—
181C32×C4D3×Q811yescharacteristic
182C32×C4#23911yescharacteristic
183C3⋊Dic3—21no—
184C3⋊Dic3—62no—
185C3⋊Dic3—64no—
186C3⋊Dic3—64no—
187C3⋊Dic3—124no—
188C3⋊Dic3—124no—
189C3⋊Dic3#21612yes—
190C3×Dic3—34no—
191C3×Dic3—42no—
192C3×Dic3—42no—
193C3×Dic3—42no—
194C3×Dic3—42no—
195C3×Dic3—62no—
196C3×Dic3—62no—
197C3×Dic3—82no—
198C3×Dic3—121no—
199C4⋊Q8—91no—
200C4×Q8—271no—
201#107—92no—
202C4⋊C8—271no—
203#103—272no—
204C33#47711yescharacteristic, complemented, Sylow
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×D3211yescharacteristic
217C2×C3×C4C2×D3212yes—
218C2×Dic3—22no—
219C2×Dic3—21no—
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—92no—
230Dic6—94no—
231Dic6—124no—
232Dic6—182no—
233Dic6—184no—
234Dic6—182no—
235Dic6—181no—
236Dic6—182no—
237Dic6—362no—
238Dic6—362no—
239Dic6—362no—
240C3×C8—182no—
241C3⋊C8—62no—
242C3⋊C8—62no—
243C3⋊C8—122no—
244C2×C32—21no—
245C2×C32—22no—
246C2×C32—21no—
247C2×C32—22no—
248C2×C32—22no—
249C2×C32—21no—
250C2×C32—41no—
251C2×C32—41no—
252C2×C32—41no—
253C2×C32#89612yes—
254C2×C32#91112yes—
255C2×C32D3×Q1612yes—
256C2×C32#93211yescharacteristic
257C2×C32C2×#21611yescharacteristic
258C2×C32#94611yescharacteristic
259C2×Q8—181no—
260C2×Q8—271no—
261C2×C8—92no—
262C4⋊C4—32no—
263C4⋊C4—31no—
264C4⋊C4—181no—
265C4⋊C4—541no—
266C42—91no—
267C42—541no—
268C22×C3—21no—
269C22×C3—22no—
270C22×C3—41no—
271C22×C3D3×#6712yes—
272C22×C3#381111yescharacteristic
273C3×C4—21no—
274C3×C4—22no—
275C3×C4—22no—
276C3×C4—21no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—92no—
280C3×C4—122no—
281C3×C4—122no—
282C3×C4—122no—
283C3×C4—122no—
284C3×C4—181no—
285C3×C4—242no—
286C3×C4#379711yescharacteristic
287C3×C4#379712yes—
288C3×C4#379811yescharacteristic
289C3×C4#379912yes—
290Dic3—34no—
291Dic3—41no—
292Dic3—42no—
293Dic3—62no—
294Dic3—62no—
295Dic3—121no—
296Dic3—184no—
297Dic3—182no—
298Dic3—184no—
299Dic3—364no—
300Dic3—364no—
301C32—22no—
302C32—21no—
303C32—41no—
304C32#487512yescomplemented
305C32#509111yescharacteristic, complemented
306Q8—272no—
307Q8—362no—
308Q8—541no—
309C2×C4—61no—
310C2×C4—62no—
311C2×C4—91no—
312C2×C4—91no—
313C2×C4—541no—
314C2×C4D3311yescharacteristic
315C8—182no—
316C2×C3—21no—
317C2×C3—22no—
318C2×C3—21no—
319C2×C3—22no—
320C2×C3—21no—
321C2×C3—22no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—41no—
325C2×C3#6386511yescharacteristic
326C2×C3#6388812yes—
327C2×C3#6392711yescharacteristic
328C2×C3#6393212yes—
329C2×C3D3×#21612yes—
330C2×C3#6394411yescharacteristic
331C22#8895211yescharacteristic
332C4—92no—
333C4—122no—
334C4—121no—
335C4—181no—
336C4—542no—
337C4#8895811yescharacteristic
338C4#8896511yescharacteristic
339C3—22no—
340C3—21no—
341C3—41no—
342C3#1059043211yescharacteristic, complemented
343C3#1059048812yescomplemented
344C2#1172284611yescharacteristic
345C2#1172287211yescharacteristic
346C2#1172297711yescharacteristic
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
34. #6308Q8semidirect
204. C33#477semidirect
304. C32#4875semidirect
305. C32#5091semidirect
342. C3#10590432semidirect
343. C3#10590488semidirect

Maximal subgroups

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

subgroupindex
#117234062
C3×#639192
#117220782
#117220412
C3×#635762
#105904883
#105904323