Finite Group Databasebeta

← #13160889

Subgroups — #13160889

Order 1728.

2156 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 2156 conjugacy classes of subgroups — too many to draw. Only its 188 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×D4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723503C211yesmaximal, characteristic, complemented
3C2×#88785C211yesmaximal, characteristic, complemented
4C3×#63659C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic, complemented
8C3⋊D3×#102—31nomaximal
9#10591416—34nomaximal
10C23×C3⋊(C3⋊D3)—21no—
11C2×C33×D4C2211yescharacteristic
12C2×#6312—21no—
13D4×C3⋊(C3⋊D3)—22no—
14D4×C3⋊(C3⋊D3)C414yescomplemented
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×#6289—21no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
20C32×C4⋊Dic3C2211yescharacteristic, complemented
21C2×C32×C3⋊C8C2211yescharacteristic, complemented
22#88806—21no—
23#88785—22no—
24C2×D4×C3⋊D3—31no—
25C2×D4×C3⋊D3—34no—
26C2×D4×C3⋊D3—64no—
27C2×C8×C3⋊D3—31no—
28C4⋊C4×C3⋊D3—31no—
29#63900—34no—
30C2×#3710—34no—
31C32×#102—31no—
32#63659D311yescharacteristic, complemented
33#63648—31no—
34#63636—31no—
35C3×#832—34no—
36C3×#832D314yescomplemented
37#63564—34no—
38#63564—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⋊D3)D411yescharacteristic
45C33×D4—21no—
46C33×D4C2×C412yescomplemented
47C2×C33×C4C2311yescharacteristic
48#6312—24no—
49C2×C3⋊(C3⋊Dic3)D411yescharacteristic
50C4⋊(C3⋊(C3⋊D3))—21no—
51C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
52C4×C3⋊(C3⋊D3)C2×C412yes—
53C4×C3⋊(C3⋊D3)D412yes—
54C2×C3×C3⋊Dic3—21no—
55C2×C32×Dic3—21no—
56#6289—42no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59D3×#102—94no—
60D3×#102—91no—
61C23×C3⋊D3—61no—
62C23×C3⋊D3—64no—
63C23×C3⋊D3—68no—
64C2×C32×D4—24no—
65C2×C32×D4C2×D314yes—
66C2×C32×D4C2×D311yescharacteristic
67C2×#638—61no—
68C2×#638—64no—
69C2×#638—68no—
70D4×C3⋊D3—34no—
71D4×C3⋊D3—316no—
72D4×C3⋊D3—62no—
73D4×C3⋊D3—68no—
74D4×C3⋊D3—616no—
75D4×C3⋊D3—616no—
76C2×C4⋊(C3⋊D3)—31no—
77C2×C4⋊(C3⋊D3)—34no—
78C2×C4⋊(C3⋊D3)—64no—
79C2×C4×C3⋊D3—31no—
80C2×C4×C3⋊D3—34no—
81C2×C4×C3⋊D3—64no—
82C2×C4×C3⋊D3—61no—
83C2×#624—64no—
84C2×C32×C8—31no—
85C32×C4⋊C4—31no—
86C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
87#3750—61no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
89C8×C3⋊D3—62no—
90C3×C4⋊Dic3—34no—
91C3×C4⋊Dic3C2×D314yescomplemented
92C2×C3×C3⋊C8—34no—
93C2×C3×C3⋊C8C2×D314yescomplemented
94#3724—64no—
95#3710—68no—
96C22×C33—22no—
97C22×C33—41no—
98C22×C33—41no—
99C22×C33C2×D411yescharacteristic
100C2×C3⋊(C3⋊D3)—22no—
101C2×C3⋊(C3⋊D3)—41no—
102C2×C3⋊(C3⋊D3)—41no—
103C2×C3⋊(C3⋊D3)—48no—
104C2×C3⋊(C3⋊D3)C22⋊C412yes—
105C2×C3⋊(C3⋊D3)D812yes—
106C2×C3⋊(C3⋊D3)SD1612yes—
107C33×C4C22×C411yescharacteristic
108C33×C4C2×D411yescharacteristic
109C3⋊(C3⋊Dic3)C22⋊C412yes—
110C3×C3⋊Dic3—41no—
111C32×Dic3—41no—
112C2×D3×D4—94no—
113C2×D3×D4—91no—
114C2×D3×D4—184no—
115C2×D3×C8—94no—
116C2×D3×C8—91no—
117D3×C4⋊C4—94no—
118D3×C4⋊C4—91no—
119C3×#102—34no—
120C3×#102—91no—
121#832—34no—
122#832C3⋊D311yescharacteristic, complemented
123#821—94no—
124#821—91no—
125#809—94no—
126#809—91no—
127C23×C32—28no—
128C23×C32—24no—
129C23×C32—21no—
130C22×C3⋊D3—31no—
131C22×C3⋊D3—34no—
132C22×C3⋊D3—64no—
133C22×C3⋊D3—64no—
134C22×C3⋊D3—616no—
135C22×C3⋊D3—68no—
136C22×C3⋊D3—632no—
137C22×C3⋊D3—64no—
138C22×C3⋊D3—61no—
139C22×C3⋊D3—122no—
140C22×C3⋊D3—122no—
141C22×C3⋊D3—128no—
142C22×C3⋊D3—128no—
143C22×C3⋊D3—1216no—
144C22×C3⋊D3—1216no—
145C32×D4—28no—
146C32×D4—24no—
147C32×D4—21no—
148C32×D4—28no—
149C32×D4C4×D318yescomplemented
150C32×D4C4×D312yescomplemented
151C2×C32×C4—24no—
152C2×C32×C4—61no—
153C2×C32×C4C22×D314yes—
154C2×C32×C4C22×D311yescharacteristic
155#638—64no—
156#638—616no—
157#638—632no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—31no—
160C2×C3⋊Dic3—34no—
161C2×C3⋊Dic3—64no—
162C4⋊(C3⋊D3)—32no—
163C4⋊(C3⋊D3)—38no—
164C4⋊(C3⋊D3)—68no—
165C4⋊(C3⋊D3)—68no—
166C4⋊(C3⋊D3)—64no—
167C4⋊(C3⋊D3)—61no—
168C4×C3⋊D3—32no—
169C4×C3⋊D3—32no—
170C4×C3⋊D3—38no—
171C4×C3⋊D3—38no—
172C4×C3⋊D3—68no—
173C4×C3⋊D3—68no—
174C4×C3⋊D3—122no—
175C2×C3×Dic3—24no—
176C2×C3×Dic3—64no—
177#624—128no—
178C32×C8—61no—
179C3⋊(C3⋊C8)—21no—
180C3×C3⋊C8—24no—
181C3×C3⋊C8—64no—
182C2×#102—271noSylow
183C2×C33—42no—
184C2×C33C2×C22⋊C411yescharacteristic
185C2×C33C2×D811yescharacteristic
186C2×C33C2×SD1611yescharacteristic
187C3⋊(C3⋊D3)—42no—
188C3⋊(C3⋊D3)#10214yescomplemented
189C23×D3—184no—
190C23×D3—181no—
191C23×D3—188no—
192C2×C3×D4—24no—
193C2×C3×D4D3214yes—
194C2×C3×D4C2×C3⋊D311yescharacteristic
195C2×#67—184no—
196C2×#67—181no—
197C2×#67—188no—
198D3×D4—916no—
199D3×D4—94no—
200D3×D4—1816no—
201D3×D4—188no—
202D3×D4—182no—
203D3×D4—1816no—
204C2×D12—94no—
205C2×D12—91no—
206C2×D12—184no—
207C2×C4×D3—94no—
208C2×C4×D3—91no—
209C2×C4×D3—184no—
210C2×C4×D3—181no—
211C2×C4×D3—184no—
212C2×C3×C8—34no—
213C2×C3×C8—91no—
214C3×C4⋊C4—34no—
215C3×C4⋊C4—91no—
216C4⋊Dic3—34no—
217C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
218Dic3⋊C4—181no—
219Dic3⋊C4—184no—
220C2×C3⋊C8—34no—
221C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
222D3×C8—182no—
223D3×C8—188no—
224C22×C32—216no—
225C22×C32—28no—
226C22×C32—22no—
227C22×C32—24no—
228C22×C32—48no—
229C22×C32—48no—
230C22×C32—44no—
231C22×C32—41no—
232C22×C32—44no—
233C22×C32—41no—
234C22×C32D3×D414yes—
235C22×C32D3×D411yescharacteristic
236C2×C3⋊D3—32no—
237C2×C3⋊D3—32no—
238C2×C3⋊D3—32no—
239C2×C3⋊D3—38no—
240C2×C3⋊D3—38no—
241C2×C3⋊D3—38no—
242C2×C3⋊D3—68no—
243C2×C3⋊D3—68no—
244C2×C3⋊D3—68no—
245C2×C3⋊D3—616no—
246C2×C3⋊D3—68no—
247C2×C3⋊D3—62no—
248C2×C3⋊D3—128no—
249C2×C3⋊D3—1232no—
250C2×C3⋊D3—1264no—
251C2×C3⋊D3—128no—
252C2×C3⋊D3—128no—
253C2×C3⋊D3—124no—
254C2×C3⋊D3—121no—
255C2×C3⋊D3—124no—
256C2×C3⋊D3—121no—
257C32×C4—24no—
258C32×C4—24no—
259C32×C4—121no—
260C32×C4C2×C4×D314yes—
261C32×C4C2×C4×D311yescharacteristic
262C32×C4D3×D414yes—
263C32×C4D3×D411yescharacteristic
264C3⋊Dic3—32no—
265C3⋊Dic3—38no—
266C3⋊Dic3—41no—
267C3⋊Dic3—68no—
268C3×Dic3—44no—
269C3×Dic3—124no—
270C22×D4—271no—
271C22×C8—271no—
272C2×C4⋊C4—271no—
273#102—31no—
274#102—91no—
275#102—271no—
276#102—271no—
277C33C2×#10211yescharacteristic, complemented, Sylow
278C23×C3—28no—
279C23×C3—24no—
280C23×C3—21no—
281C22×D3—94no—
282C22×D3—91no—
283C22×D3—184no—
284C22×D3—184no—
285C22×D3—1816no—
286C22×D3—188no—
287C22×D3—1832no—
288C22×D3—184no—
289C22×D3—181no—
290C22×D3—362no—
291C22×D3—362no—
292C22×D3—368no—
293C22×D3—368no—
294C22×D3—3616no—
295C22×D3—3616no—
296C3×D4—28no—
297C3×D4—24no—
298C3×D4—21no—
299C3×D4—28no—
300C3×D4#62418yescomplemented
301C3×D4C4×C3⋊D312yescomplemented
302C2×C3×C4—24no—
303C2×C3×C4—64no—
304C2×C3×C4—181no—
305C2×C3×C4C2×D3214yes—
306C2×C3×C4C22×C3⋊D311yescharacteristic
307#67—184no—
308#67—1816no—
309#67—1832no—
310C2×Dic3—21no—
311C2×Dic3—64no—
312C2×Dic3—94no—
313C2×Dic3—91no—
314C2×Dic3—184no—
315D12—98no—
316D12—92no—
317D12—188no—
318D12—188no—
319D12—184no—
320D12—181no—
321C4×D3—98no—
322C4×D3—98no—
323C4×D3—92no—
324C4×D3—92no—
325C4×D3—188no—
326C4×D3—188no—
327C4×D3—362no—
328C4×D3—368no—
329C3×C8—64no—
330C3×C8—181no—
331C3⋊C8—21no—
332C3⋊C8—64no—
333C2×C32—24no—
334C2×C32—24no—
335C2×C32—24no—
336C2×C32—416no—
337C2×C32—48no—
338C2×C32—42no—
339C2×C32D3×C22⋊C414yes—
340C2×C32D3×C22⋊C411yescharacteristic
341C2×C32D3×D814yes—
342C2×C32D3×D811yescharacteristic
343C2×C32D3×SD1614yes—
344C2×C32D3×SD1611yescharacteristic
345C3⋊D3—34no—
346C3⋊D3—316no—
347C3⋊D3—616no—
348C3⋊D3—1216no—
349C3⋊D3—128no—
350C3⋊D3—122no—
351C24—541no—
352C2×D4—274no—
353C2×D4—271no—
354C2×D4—542no—
355C2×D4—541no—
356C2×D4D3×C3⋊D311yescharacteristic
357C22×C4—271no—
358C22×C4—541no—
359C2×C8—31no—
360C2×C8—91no—
361C2×C8—542no—
362C4⋊C4—31no—
363C4⋊C4—91no—
364C4⋊C4—541no—
365C22×C3—216no—
366C22×C3—22no—
367C22×C3—28no—
368C22×C3—24no—
369C22×C3—48no—
370C22×C3—48no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—44no—
374C22×C3—44no—
375C22×C3#381114yes—
376C22×C3D4×C3⋊D311yescharacteristic
377C2×D3—92no—
378C2×D3—92no—
379C2×D3—92no—
380C2×D3—98no—
381C2×D3—98no—
382C2×D3—98no—
383C2×D3—188no—
384C2×D3—188no—
385C2×D3—188no—
386C2×D3—1816no—
387C2×D3—182no—
388C2×D3—188no—
389C2×D3—3632no—
390C2×D3—368no—
391C2×D3—3664no—
392C2×D3—368no—
393C2×D3—368no—
394C2×D3—361no—
395C2×D3—361no—
396C2×D3—364no—
397C2×D3—364no—
398C3×C4—24no—
399C3×C4—24no—
400C3×C4—124no—
401C3×C4—361no—
402C3×C4C2×#62414yes—
403C3×C4#381114yes—
404C3×C4C2×C4×C3⋊D311yescharacteristic
405C3×C4D4×C3⋊D311yescharacteristic
406Dic3—41no—
407Dic3—92no—
408Dic3—98no—
409Dic3—124no—
410Dic3—188no—
411C32—24no—
412C32D3×#10214yescomplemented
413C32D3×#10211yescharacteristic, complemented
414C23—21no—
415C23—271no—
416C23—544no—
417C23—541no—
418C23—1082no—
419C23—1082no—
420D4—21no—
421D4—272no—
422D4—544no—
423D4—541no—
424D4#628912yescomplemented
425C2×C4—61no—
426C2×C4—181no—
427C2×C4—271no—
428C2×C4—272no—
429C2×C4—272no—
430C2×C4—1082no—
431C2×C4C2×D3×C3⋊D311yescharacteristic
432C8—61no—
433C8—181no—
434C2×C3—24no—
435C2×C3—24no—
436C2×C3—24no—
437C2×C3—416no—
438C2×C3—48no—
439C2×C3—42no—
440C2×C3#6392614yes—
441C2×C3#6393014yes—
442C2×C3#6397414yes—
443C2×C3C22⋊C4×C3⋊D311yescharacteristic
444C2×C3D8×C3⋊D311yescharacteristic
445C2×C3SD16×C3⋊D311yescharacteristic
446D3—916no—
447D3—94no—
448D3—1816no—
449D3—3616no—
450D3—368no—
451D3—362no—
452C22—22no—
453C22—41no—
454C22—41no—
455C22—272no—
456C22—272no—
457C22—272no—
458C22—542no—
459C22—1088no—
460C22—1081no—
461C22—1081no—
462C22#8903911yescharacteristic
463C4—121no—
464C4—272no—
465C4—361no—
466C4C2×#628911yescharacteristic
467C4#8903911yescharacteristic
468C3—24no—
469C3#1059141614yescomplemented
470C3C3⋊D3×#10211yescharacteristic, complemented
471C2—42no—
472C2—274no—
473C2—1082no—
474C2#1172354711yescharacteristic
475C2#1172355311yescharacteristic
476C2#1172362711yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×D4×C3⋊(C3⋊D3)C2
cosocleC2×C33×C4C23
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C2×#102
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. #11723503C2semidirect
3. C2×#88785C2semidirect
4. C3×#63659C2semidirect
5. C32×#832C2semidirect
6. #11722036C2semidirect
7. #11722035C2semidirect
14. D4×C3⋊(C3⋊D3)C4semidirect
18. C3×C4⋊(C3⋊Dic3)C22semidirect
19. C2×C3×C3⋊(C3⋊C8)C22semidirect
20. C32×C4⋊Dic3C22semidirect
21. C2×C32×C3⋊C8C22semidirect
32. #63659D3semidirect
36. C3×#832D3semidirect
46. C33×D4C2×C4semidirect
51. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
86. C4⋊(C3⋊Dic3)C2×D3semidirect
88. C2×C3⋊(C3⋊C8)C2×D3semidirect
91. C3×C4⋊Dic3C2×D3semidirect
93. C2×C3×C3⋊C8C2×D3semidirect
122. #832C3⋊D3semidirect
149. C32×D4C4×D3semidirect
150. C32×D4C4×D3semidirect
188. C3⋊(C3⋊D3)#102semidirect
217. C4⋊Dic3C2×C3⋊D3semidirect
221. C2×C3⋊C8C2×C3⋊D3semidirect
277. C33C2×#102semidirect
300. C3×D4#624semidirect
301. C3×D4C4×C3⋊D3semidirect
412. C32D3×#102semidirect
413. C32D3×#102semidirect
424. D4#6289semidirect
469. C3#10591416semidirect
470. C3C3⋊D3×#102semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊(C3⋊D3)2
#117235032
C2×#887852
C3×#636592
C32×#8322
#117220362
#117220352
C3⋊D3×#1023
#105914163