Finite Group Databasebeta

← #13161269

Subgroups — #13161269

Order 1728.

1068 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 196 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×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2#11723404C211yesmaximal, characteristic
3C2×#88787C211yesmaximal, characteristic
4#11722205C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722039C211yesmaximal, characteristic, complemented
8#10592501—31nomaximal
9#10591557—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×Q8×C3⋊D3—22no—
12C3×Q8×C3⋊D3C414yescomplemented
13C2×C3×C4×C3⋊D3—21no—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×#634—21no—
16C2×C3×#634C2211yescharacteristic
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88803—21no—
22Dic3×C3⋊Dic3—21no—
23#88787—22no—
24C2×Q8×C3⋊D3D311yescharacteristic
25C2×C3×D3×Q8—34no—
26C2×#3743—31no—
27#64101—31no—
28#63841—34no—
29C2×#3711—34no—
30C32×#103—31no—
31#63662—31no—
32#63662—34no—
33#63662—64no—
34#63643—31no—
35#63637—31no—
36C3×#835D314yescomplemented
37#63566—34no—
38#63565—34no—
39C22×C3×C3⋊D3D411yescharacteristic
40C33×Q8—21no—
41C33×Q8C2×C412yes—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×C4×C3⋊D3—24no—
48C3×C4×C3⋊D3C2×C412yes—
49C3×C4×C3⋊D3D412yes—
50C3×#634—21no—
51C3×#634—24no—
52C3×#634C2×C412yes—
53C2×C32×Dic3—21no—
54C3⋊(C3⋊(C3⋊C8))—21no—
55C32×C3⋊C8—21no—
56#5225—91no—
57#4802—94no—
58C2×C32×Q8—24no—
59C2×C32×Q8C2×D314yes—
60C2×C32×Q8C2×D311yescharacteristic
61Q8×C3⋊D3—22no—
62Q8×C3⋊D3Dic314yescomplemented
63C2×C4×C3⋊D3—21no—
64C2×C4×C3⋊D3C2×D311yescharacteristic
65C2×#634—21no—
66C2×#634C2×D311yescharacteristic
67C3×D3×Q8—316no—
68C3×D3×Q8—68no—
69C2×C3×C4×D3—34no—
70C2×C3×C4×D3—64no—
71C2×C3×Dic6—34no—
72C2×C3×Dic6—64no—
73C2×C32×C8—31no—
74C32×C4⋊C4—31no—
75#3752—61no—
76C4⋊(C3⋊Dic3)—31no—
77C4⋊(C3⋊Dic3)—34no—
78C4⋊(C3⋊Dic3)—64no—
79C4×C3⋊Dic3—61no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—34no—
82C2×C3⋊(C3⋊C8)—64no—
83#3743—62no—
84C3×C4⋊Dic3C2×D314yes—
85C2×C3×C3⋊C8C2×D314yes—
86(C2×D3)⋊Dic3—64no—
87Dic32—64no—
88#3711—68no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3C22⋊C412yes—
93C33×C4—22no—
94C33×C4C22×C411yescharacteristic
95C33×C4C2×D411yescharacteristic
96C3⋊(C3⋊Dic3)—41no—
97C3×C3⋊Dic3—22no—
98C3×C3⋊Dic3C22⋊C412yes—
99C32×Dic3—41no—
100C22×C3×Q8—91no—
101C2×D3×Q8—34no—
102#926—91no—
103C2×#208—91no—
104C2×ΓL(1,𝔽25)—94no—
105#892—94no—
106C3×#103—34no—
107#835—94no—
108#835—91no—
109#835—91no—
110#835—91no—
111#835—184no—
112#835C3⋊D311yescharacteristic, complemented
113#816—94no—
114#810—94no—
115C22×C3⋊D3#6711yescharacteristic
116C22×C3×D3—34no—
117C32×Q8—21no—
118C32×Q8—28no—
119C32×Q8—24no—
120C32×Q8—44no—
121C32×Q8C4×D318yes—
122C32×Q8C2×Dic312yes—
123C2×C32×C4—21no—
124C2×C32×C4—24no—
125C2×C32×C4—24no—
126C2×C32×C4—44no—
127C2×C32×C4—61no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—61no—
132C2×C3⋊Dic3—64no—
133C2×C3⋊Dic3—124no—
134C2×C3⋊Dic3#6711yescharacteristic
135C4×C3⋊D3—24no—
136C4×C3⋊D3C2×Dic312yes—
137C4×C3⋊D3#6712yes—
138#634—21no—
139#634—24no—
140#634C2×Dic312yes—
141C2×C3×Dic3—24no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—64no—
144C3×C4×D3—38no—
145C3×C4×D3—38no—
146C3×C4×D3—616no—
147C3×Dic6—38no—
148C3×Dic6—616no—
149C3×Dic6—64no—
150C32×C8—61no—
151C3⋊(C3⋊C8)—61no—
152C3⋊(C3⋊C8)—64no—
153C3⋊(C3⋊C8)—124no—
154C3×C3⋊C8—24no—
155#419—271noSylow
156C2×C33C2×C22⋊C411yescharacteristic
157C2×C33#13711yescharacteristic
158C2×C33#13711yescharacteristic
159C3×C3⋊D3—22no—
160C2×C3×Q8—24no—
161C2×C3×Q8—94no—
162C2×C3×Q8—91no—
163C2×C3×Q8—181no—
164C2×C3×Q8—182no—
165C2×C3×Q8D3214yes—
166C2×C3×Q8C2×C3⋊D311yescharacteristic
167C22×C3×C4—91no—
168C22×C3×C4—181no—
169D3×Q8—316no—
170D3×Q8—68no—
171C2×C4×D3—34no—
172C2×C4×D3—64no—
173C2×Dic6—34no—
174C2×Dic6—64no—
175C2×C3×C8—34no—
176C3×C4⋊C4—34no—
177#217—181no—
178(C2×D3)⋊C4—184no—
179C4⋊Dic3—94no—
180C4⋊Dic3—91no—
181C4⋊Dic3—184no—
182C4⋊Dic3C2×C3⋊D311yescharacteristic
183C4×Dic3—181no—
184C4×Dic3—184no—
185#208—182no—
186C2×C3⋊C8—94no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—184no—
189C2×C3⋊C8C2×C3⋊D311yescharacteristic
190ΓL(1,𝔽25)—188no—
191C22×C32—24no—
192C22×C32D3×D414yes—
193C22×C32C2×#6711yescharacteristic
194C2×C3⋊D3—21no—
195C2×C3⋊D3—21no—
196C2×C3⋊D3#21712yes—
197C2×C3×D3—38no—
198C2×C3×D3—64no—
199C2×C3×D3—64no—
200C32×C4—22no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—28no—
204C32×C4—48no—
205C32×C4—121no—
206C32×C4C2×C4×D314yes—
207C32×C4D3×D414yes—
208C32×C4C22×Dic311yescharacteristic
209C32×C4C2×#6711yescharacteristic
210C3⋊Dic3—22no—
211C3⋊Dic3—121no—
212C3⋊Dic3—124no—
213C3⋊Dic3—244no—
214C3⋊Dic3#21712yes—
215C3×Dic3—38no—
216C3×Dic3—44no—
217C3×Dic3—68no—
218C22×Q8—91no—
219C2×M16—271no—
220#117—271no—
221#103—31no—
222#103—271no—
223#103—271no—
224#103—271no—
225C33#41911yescharacteristic, complemented, Sylow
226C23×C3—91no—
227C22×D3—34no—
228C3×Q8—24no—
229C3×Q8—21no—
230C3×Q8—28no—
231C3×Q8—44no—
232C3×Q8—92no—
233C3×Q8—184no—
234C3×Q8—181no—
235C3×Q8D3×Dic318yes—
236C3×Q8C4×C3⋊D312yes—
237C2×C3×C4—24no—
238C2×C3×C4—21no—
239C2×C3×C4—24no—
240C2×C3×C4—44no—
241C2×C3×C4—64no—
242C2×C3×C4—91no—
243C2×C3×C4—92no—
244C2×C3×C4—92no—
245C2×C3×C4—184no—
246C2×C3×C4—181no—
247C2×C3×C4C2×D3214yes—
248C2×C3×C4C22×C3⋊D311yescharacteristic
249C2×Dic3—21no—
250C2×Dic3—34no—
251C2×Dic3—64no—
252C2×Dic3—184no—
253C2×Dic3—181no—
254C2×Dic3—364no—
255C4×D3—38no—
256C4×D3—38no—
257C4×D3—616no—
258Dic6—38no—
259Dic6—616no—
260Dic6—64no—
261C3×C8—64no—
262C3⋊C8—21no—
263C3⋊C8—184no—
264C3⋊C8—181no—
265C3⋊C8—364no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—24no—
269C2×C32D3×C22⋊C414yes—
270C2×C32#91514yes—
271C2×C32#91814yes—
272C2×C32#94211yescharacteristic
273C2×C32#94211yescharacteristic
274C2×C32C2×#21711yescharacteristic
275C3⋊D3—22no—
276C3×D3—68no—
277C2×Q8—94no—
278C2×Q8—91no—
279C2×Q8—181no—
280C2×Q8—182no—
281C2×Q8D3×C3⋊D311yescharacteristic
282C22×C4—91no—
283C22×C4—181no—
284M16—542no—
285C2×C8—31no—
286C2×C8—271no—
287C4⋊C4—31no—
288C4⋊C4—271no—
289C22⋊C4—541no—
290C42—541no—
291C22×C3—24no—
292C22×C3—92no—
293C22×C3—181no—
294C22×C3—181no—
295C22×C3D3×#6714yes—
296C22×C3D4×C3⋊D311yescharacteristic
297C2×D3—38no—
298C2×D3—64no—
299C2×D3—64no—
300C3×C4—22no—
301C3×C4—28no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—48no—
305C3×C4—92no—
306C3×C4—124no—
307C3×C4—182no—
308C3×C4C2×D3×Dic314yes—
309C3×C4D3×#6714yes—
310C3×C4C2×C4×C3⋊D311yescharacteristic
311C3×C4D4×C3⋊D311yescharacteristic
312Dic3—38no—
313Dic3—41no—
314Dic3—68no—
315Dic3—361no—
316Dic3—364no—
317Dic3—724no—
318C32—24no—
319C32#480214yescomplemented
320C32#522511yescharacteristic, complemented
321C23—91no—
322Q8—21no—
323Q8—92no—
324Q8—184no—
325Q8—181no—
326Q8Dic3×C3⋊D312yes—
327C2×C4—21no—
328C2×C4—61no—
329C2×C4—91no—
330C2×C4—92no—
331C2×C4—92no—
332C2×C4—184no—
333C2×C4—181no—
334C2×C4—541no—
335C2×C4C2×D3×C3⋊D311yescharacteristic
336C8—61no—
337C8—541no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—182no—
342C2×C3#6394614yes—
343C2×C3#6394814yes—
344C2×C3D3×#21714yes—
345C2×C3C22⋊C4×C3⋊D311yescharacteristic
346C2×C3#6412411yescharacteristic
347C2×C3#6412711yescharacteristic
348D3—68no—
349C22—92no—
350C22—181no—
351C22—181no—
352C22C3⋊D3×#6711yescharacteristic
353C4—22no—
354C4—92no—
355C4—121no—
356C4—182no—
357C4—1081no—
358C4C2×Dic3×C3⋊D311yescharacteristic
359C4C3⋊D3×#6711yescharacteristic
360C3—24no—
361C3#1059155714yescomplemented
362C3#1059250111yescharacteristic, complemented
363C2—182no—
364C2#1172358411yescharacteristic
365C2#1172358611yescharacteristic
366C2C3⋊D3×#21711yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#419
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
4. #11722205C2semidirect
5. C32×#835C2semidirect
6. #11722040C2semidirect
7. #11722039C2semidirect
12. C3×Q8×C3⋊D3C4semidirect
36. C3×#835D3semidirect
62. Q8×C3⋊D3Dic3semidirect
112. #835C3⋊D3semidirect
225. C33#419semidirect
319. C32#4802semidirect
320. C32#5225semidirect
361. C3#10591557semidirect
362. C3#10592501semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
#117234042
C2×#887872
#117222052
C32×#8352
#117220402
#117220392
#105925013
#105915573