Finite Group Databasebeta

← #13161042

Subgroups — #13161042

Order 1728.

1340 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1340 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×D4×C3⋊D3C211yesmaximal, characteristic
2#11723404C211yesmaximal, characteristic, complemented
3C2×#88787C211yesmaximal, characteristic, complemented
4#11722202C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic, complemented
8#10592469—31nomaximal
9#10591421—34nomaximal
10C23×C3×C3⋊D3—21no—
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—21no—
13C3×D4×C3⋊D3—22no—
14C3×D4×C3⋊D3C414yescomplemented
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
18C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
19C32×C4⋊Dic3C2211yescharacteristic, complemented
20C2×C32×C3⋊C8C2211yescharacteristic, complemented
21#88803—21no—
22Dic3×C3⋊Dic3—21no—
23#88787—22no—
24C2×D4×C3⋊D3D311yescharacteristic
25C2×C3×D3×D4—34no—
26C2×#3743—31no—
27#64101—31no—
28#63841—34no—
29C2×#3711—34no—
30C32×#102—31no—
31#63659—31no—
32#63659—34no—
33#63659—64no—
34#63648—31no—
35#63636—31no—
36C3×#832D314yescomplemented
37#63563—34no—
38#63562—34no—
39C23×C33—21no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3—24no—
42C22×C3×C3⋊D3—42no—
43C22×C3×C3⋊D3—42no—
44C22×C3×C3⋊D3D411yescharacteristic
45C33×D4—21no—
46C33×D4C2×C412yes—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49C3×#638—24no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)C2×C412yes—
53C3×C4×C3⋊D3C2×C412yes—
54C3×C4×C3⋊D3D412yes—
55C2×C32×Dic3—21no—
56C3⋊(C3⋊(C3⋊C8))—21no—
57C32×C3⋊C8—21no—
58#5215—91no—
59#4770—94no—
60C23×C3⋊D3—21no—
61C23×C3×D3—64no—
62C2×C32×D4—24no—
63C2×C32×D4C2×D314yes—
64C2×C32×D4C2×D311yescharacteristic
65C2×#638—21no—
66D4×C3⋊D3—22no—
67D4×C3⋊D3Dic314yescomplemented
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×C3×#67—64no—
71C3×D3×D4—316no—
72C3×D3×D4—68no—
73C2×C3×D12—34no—
74C2×C3×C4×D3—34no—
75C2×C32×C8—31no—
76C32×C4⋊C4—31no—
77#3752—61no—
78C4⋊(C3⋊Dic3)—31no—
79C4⋊(C3⋊Dic3)—34no—
80C4⋊(C3⋊Dic3)—64no—
81C4×C3⋊Dic3—61no—
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—34no—
84C2×C3⋊(C3⋊C8)—64no—
85#3743—62no—
86C3×C4⋊Dic3C2×D314yescomplemented
87C2×C3×C3⋊C8C2×D314yescomplemented
88(C2×D3)⋊Dic3—64no—
89Dic32—64no—
90#3711—68no—
91C22×C33—22no—
92C22×C33—41no—
93C22×C33—41no—
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3—21no—
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3—22no—
98C2×C3×C3⋊D3—48no—
99C2×C3×C3⋊D3—41no—
100C2×C3×C3⋊D3—41no—
101C2×C3×C3⋊D3C22⋊C412yes—
102C33×C4C22×C411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3⋊(C3⋊Dic3)—41no—
105C3×C3⋊Dic3C22⋊C412yes—
106C32×Dic3—41no—
107C22×C3×D4—91no—
108C2×D3×D4—34no—
109#926—91no—
110C2×#208—91no—
111C2×ΓL(1,𝔽25)—94no—
112#892—94no—
113C3×#102—34no—
114#832—94no—
115#832—91no—
116#832—91no—
117#832—91no—
118#832—184no—
119#832C3⋊D311yescharacteristic, complemented
120#821—94no—
121#809—94no—
122C23×C32—21no—
123C23×C32—24no—
124C23×C32—44no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3—24no—
127C22×C3⋊D3—42no—
128C22×C3⋊D3—42no—
129C22×C3⋊D3#6711yescharacteristic
130C22×C3×D3—34no—
131C22×C3×D3—616no—
132C22×C3×D3—64no—
133C22×C3×D3—128no—
134C22×C3×D3—128no—
135C32×D4—21no—
136C32×D4—28no—
137C32×D4—24no—
138C32×D4—44no—
139C32×D4C4×D318yes—
140C32×D4C2×Dic312yes—
141C2×C32×C4—24no—
142C2×C32×C4—61no—
143C2×C32×C4C22×D314yes—
144C2×C32×C4C22×D311yescharacteristic
145#638—24no—
146C2×C3⋊Dic3—61no—
147C2×C3⋊Dic3—64no—
148C2×C3⋊Dic3—124no—
149C2×C3⋊Dic3#6711yescharacteristic
150C4⋊(C3⋊D3)—21no—
151C4⋊(C3⋊D3)C2×Dic312yes—
152C4×C3⋊D3C2×Dic312yes—
153C4×C3⋊D3#6712yes—
154C3×#67—616no—
155C2×C3×Dic3—24no—
156C2×C3×Dic3—34no—
157C3×D12—38no—
158C3×D12—64no—
159C3×C4×D3—38no—
160C3×C4×D3—38no—
161C32×C8—61no—
162C3⋊(C3⋊C8)—61no—
163C3⋊(C3⋊C8)—64no—
164C3⋊(C3⋊C8)—124no—
165C3×C3⋊C8—24no—
166#418—271noSylow
167C2×C33—42no—
168C2×C33C2×C22⋊C411yescharacteristic
169C2×C33AGL(1,ℤ/8)11yescharacteristic
170C2×C33AGL(1,ℤ/8)11yescharacteristic
171C3×C3⋊D3—22no—
172C3×C3⋊D3—42no—
173C24×C3—181no—
174C23×D3—64no—
175C2×C3×D4—24no—
176C2×C3×D4—94no—
177C2×C3×D4—91no—
178C2×C3×D4—182no—
179C2×C3×D4—181no—
180C2×C3×D4D3214yes—
181C2×C3×D4C2×C3⋊D311yescharacteristic
182C22×C3×C4—91no—
183C2×#67—64no—
184D3×D4—316no—
185D3×D4—68no—
186C2×D12—34no—
187C2×C4×D3—34no—
188C2×C3×C8—34no—
189C3×C4⋊C4—34no—
190#217—181no—
191(C2×D3)⋊C4—184no—
192C4⋊Dic3—94no—
193C4⋊Dic3—91no—
194C4⋊Dic3—184no—
195C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
196C4×Dic3—181no—
197C4×Dic3—184no—
198#208—182no—
199C2×C3⋊C8—94no—
200C2×C3⋊C8—91no—
201C2×C3⋊C8—184no—
202C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
203ΓL(1,𝔽25)—188no—
204C22×C32—22no—
205C22×C32—24no—
206C22×C32—28no—
207C22×C32—48no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—44no—
211C22×C32—44no—
212C22×C32—84no—
213C22×C32—84no—
214C22×C32D3×D414yes—
215C22×C32C2×#6711yescharacteristic
216C2×C3⋊D3—21no—
217C2×C3⋊D3—21no—
218C2×C3⋊D3—22no—
219C2×C3⋊D3—48no—
220C2×C3⋊D3—41no—
221C2×C3⋊D3—41no—
222C2×C3⋊D3#21712yes—
223C2×C3×D3—38no—
224C2×C3×D3—64no—
225C2×C3×D3—64no—
226C2×C3×D3—68no—
227C2×C3×D3—1232no—
228C2×C3×D3—124no—
229C2×C3×D3—124no—
230C32×C4—24no—
231C32×C4—24no—
232C32×C4—121no—
233C32×C4C2×C4×D314yes—
234C32×C4D3×D414yes—
235C32×C4C22×Dic311yescharacteristic
236C32×C4C2×#6711yescharacteristic
237C3⋊Dic3—121no—
238C3⋊Dic3—124no—
239C3⋊Dic3—244no—
240C3⋊Dic3#21712yes—
241C3×Dic3—38no—
242C3×Dic3—44no—
243C22×D4—91no—
244C2×M16—271no—
245#117—271no—
246#102—31no—
247#102—271no—
248#102—271no—
249#102—271no—
250C33#41811yescharacteristic, complemented, Sylow
251C23×C3—24no—
252C23×C3—21no—
253C23×C3—44no—
254C23×C3—91no—
255C23×C3—184no—
256C23×C3—181no—
257C23×C3—362no—
258C23×C3—362no—
259C22×D3—34no—
260C22×D3—616no—
261C22×D3—64no—
262C22×D3—128no—
263C22×D3—128no—
264C3×D4—24no—
265C3×D4—21no—
266C3×D4—28no—
267C3×D4—44no—
268C3×D4—92no—
269C3×D4—184no—
270C3×D4—181no—
271C3×D4D3×Dic318yes—
272C3×D4C4×C3⋊D312yes—
273C2×C3×C4—24no—
274C2×C3×C4—64no—
275C2×C3×C4—91no—
276C2×C3×C4—92no—
277C2×C3×C4—92no—
278C2×C3×C4C2×D3214yes—
279C2×C3×C4C22×C3⋊D311yescharacteristic
280#67—616no—
281C2×Dic3—21no—
282C2×Dic3—34no—
283C2×Dic3—184no—
284C2×Dic3—181no—
285C2×Dic3—364no—
286D12—38no—
287D12—64no—
288C4×D3—38no—
289C4×D3—38no—
290C3×C8—64no—
291C3⋊C8—21no—
292C3⋊C8—184no—
293C3⋊C8—181no—
294C3⋊C8—364no—
295C2×C32—24no—
296C2×C32—24no—
297C2×C32—24no—
298C2×C32—42no—
299C2×C32—48no—
300C2×C32—88no—
301C2×C32D3×C22⋊C414yes—
302C2×C32#91114yes—
303C2×C32#91414yes—
304C2×C32#93211yescharacteristic
305C2×C32#93211yescharacteristic
306C2×C32C2×#21711yescharacteristic
307C3⋊D3—22no—
308C3⋊D3—42no—
309C3×D3—68no—
310C3×D3—128no—
311C24—181no—
312C2×D4—94no—
313C2×D4—91no—
314C2×D4—182no—
315C2×D4—181no—
316C2×D4D3×C3⋊D311yescharacteristic
317C22×C4—91no—
318M16—542no—
319C2×C8—31no—
320C2×C8—271no—
321C4⋊C4—31no—
322C4⋊C4—271no—
323C22⋊C4—541no—
324C42—541no—
325C22×C3—22no—
326C22×C3—28no—
327C22×C3—24no—
328C22×C3—48no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—44no—
332C22×C3—44no—
333C22×C3—84no—
334C22×C3—84no—
335C22×C3—92no—
336C22×C3—181no—
337C22×C3—181no—
338C22×C3—182no—
339C22×C3—368no—
340C22×C3—361no—
341C22×C3—361no—
342C22×C3D3×#6714yes—
343C22×C3D4×C3⋊D311yescharacteristic
344C2×D3—38no—
345C2×D3—64no—
346C2×D3—64no—
347C2×D3—68no—
348C2×D3—1232no—
349C2×D3—124no—
350C2×D3—124no—
351C3×C4—24no—
352C3×C4—24no—
353C3×C4—92no—
354C3×C4—124no—
355C3×C4C2×D3×Dic314yes—
356C3×C4D3×#6714yes—
357C3×C4C2×C4×C3⋊D311yescharacteristic
358C3×C4D4×C3⋊D311yescharacteristic
359Dic3—38no—
360Dic3—41no—
361Dic3—361no—
362Dic3—364no—
363Dic3—724no—
364C32—24no—
365C32#477014yescomplemented
366C32#521511yescharacteristic, complemented
367C23—21no—
368C23—91no—
369C23—184no—
370C23—181no—
371C23—362no—
372C23—362no—
373D4—21no—
374D4—92no—
375D4—184no—
376D4—181no—
377D4Dic3×C3⋊D312yes—
378C2×C4—61no—
379C2×C4—91no—
380C2×C4—92no—
381C2×C4—92no—
382C2×C4—541no—
383C2×C4C2×D3×C3⋊D311yescharacteristic
384C8—61no—
385C8—541no—
386C2×C3—24no—
387C2×C3—24no—
388C2×C3—24no—
389C2×C3—48no—
390C2×C3—42no—
391C2×C3—88no—
392C2×C3—182no—
393C2×C3—362no—
394C2×C3#6393214yes—
395C2×C3#6393414yes—
396C2×C3D3×#21714yes—
397C2×C3C22⋊C4×C3⋊D311yescharacteristic
398C2×C3#6412011yescharacteristic
399C2×C3#6412311yescharacteristic
400D3—68no—
401D3—128no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—92no—
406C22—181no—
407C22—181no—
408C22—182no—
409C22—368no—
410C22—361no—
411C22—361no—
412C22C3⋊D3×#6711yescharacteristic
413C4—92no—
414C4—121no—
415C4—1081no—
416C4C2×Dic3×C3⋊D311yescharacteristic
417C4C3⋊D3×#6711yescharacteristic
418C3—24no—
419C3#1059142114yescomplemented
420C3#1059246911yescharacteristic, complemented
421C2—42no—
422C2—182no—
423C2—362no—
424C2#1172356011yescharacteristic
425C2#1172356211yescharacteristic
426C2C3⋊D3×#21711yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC2×C33×C4C23
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#418
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723404C2semidirect
3. C2×#88787C2semidirect
4. #11722202C2semidirect
5. C32×#832C2semidirect
6. #11722034C2semidirect
7. #11722033C2semidirect
14. C3×D4×C3⋊D3C4semidirect
17. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
18. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
19. C32×C4⋊Dic3C22semidirect
20. C2×C32×C3⋊C8C22semidirect
36. C3×#832D3semidirect
67. D4×C3⋊D3Dic3semidirect
86. C3×C4⋊Dic3C2×D3semidirect
87. C2×C3×C3⋊C8C2×D3semidirect
119. #832C3⋊D3semidirect
195. C4⋊Dic3C2×C3⋊D3semidirect
202. C2×C3⋊C8C2×C3⋊D3semidirect
250. C33#418semidirect
365. C32#4770semidirect
366. C32#5215semidirect
419. C3#10591421semidirect
420. C3#10592469semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
#117234042
C2×#887872
#117222022
C32×#8322
#117220342
#117220332
#105924693
#105914213