Finite Group Databasebeta

← #10593997

Subgroups — #10593997

Order 576.

707 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D32C211yesmaximal, characteristic, complemented
2C2×#3798C211yesmaximal, characteristic, complemented
3C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C3×C4×D12C211yesmaximal, characteristic, complemented
5C3×#872C211yesmaximal, characteristic, complemented
6#63923C211yesmaximal, characteristic, complemented
7#63922C211yesmaximal, characteristic, complemented
8#63910C211yesmaximal, characteristic, complemented
9C4×#626C211yesmaximal, characteristic, complemented
10C4×#626C211yesmaximal, characteristic, complemented
11#63895C211yesmaximal, characteristic, complemented
12#63893C211yesmaximal, characteristic, complemented
13#63891C211yesmaximal, characteristic, complemented
14#63890C211yesmaximal, characteristic, complemented
15#63886C211yesmaximal, characteristic, complemented
16#5547—31nomaximal
17#5524—31nomaximal
18C22×D32C2211yescharacteristic
19C2×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×C4×C3⋊D3C2211yescharacteristic
21C2×C4×C3⋊D3C2211yescharacteristic, complemented
22C2×C3×D12C2211yescharacteristic
23C2×C3×C4×D3C2211yescharacteristic
24C2×C3×C4×D3C2211yescharacteristic
25C2×C3×C4×D3C2211yescharacteristic, complemented
26C2×#627C2211yescharacteristic, complemented
27C2×#626C2211yescharacteristic
28C2×#626C2211yescharacteristic
29C2×#626C2211yescharacteristic
30C2×#626C2211yescharacteristic, complemented
31C2×#625C2211yescharacteristic
32C2×#624C2211yescharacteristic
33C2×D3×Dic3C2211yescharacteristic
34C2×D3×Dic3C2211yescharacteristic
35C4×D32—24no—
36#3798—24no—
37C32×C42C2211yescharacteristic, complemented
38#3752C2211yescharacteristic
39#3752C2211yescharacteristic, complemented
40C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
41C3×(C2×D3)⋊C4C2211yescharacteristic
42C3×(C2×D3)⋊C4C2211yescharacteristic
43C3×(C2×D3)⋊C4C2211yescharacteristic
44C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C3×C4⋊Dic3C2211yescharacteristic, complemented
46C3×Dic3⋊C4C2211yescharacteristic, complemented
47C3×Dic3⋊C4C2211yescharacteristic, complemented
48C3×C4×Dic3C2211yescharacteristic, complemented
49Dic3⋊Dic3C2211yescharacteristic, complemented
50Dic3⋊Dic3C2211yescharacteristic, complemented
51#3722C2211yescharacteristic
52#3722C2211yescharacteristic, complemented
53(C2×D3)⋊Dic3C2211yescharacteristic
54(C2×D3)⋊Dic3C2211yescharacteristic, complemented
55C2×#235—31no—
56C2×#235—31no—
57C22×C4×D3—31no—
58C22×C4×D3—31no—
59C3×C4×D4—31no—
60C3×#117—31no—
61#939—31no—
62#937—31no—
63C4×#67—31no—
64C4×#67—31no—
65#897—31no—
66#897—31no—
67#896—31no—
68#895—31no—
69#895—31no—
70#894—31no—
71#886—31no—
72#886—31no—
73#884—31no—
74#883—31no—
75#882—31no—
76#882—31no—
77#882—31no—
78#878—31no—
79C4×D12—31no—
80C4×D12—31no—
81C4×D12—31no—
82C4×D12—31no—
83C4×D12—61no—
84C4×D12D311yescharacteristic, complemented
85#872D311yescharacteristic, complemented
86C22×C3⋊D3C2311yescharacteristic
87C22×C3⋊D3C2311yescharacteristic
88C22×C3×D3C2311yescharacteristic
89C22×C3×D3C2311yescharacteristic
90C22×C3×D3C2311yescharacteristic
91C2×D32—22no—
92C2×D32—22no—
93C2×D32—22no—
94C2×C32×C4C2311yescharacteristic
95C2×C32×C4C2311yescharacteristic
96C2×C32×C4C2311yescharacteristic
97C2×C3⋊Dic3C2311yescharacteristic
98C2×C3⋊Dic3C2311yescharacteristic
99C4⋊(C3⋊D3)—22no—
100C4×C3⋊D3—22no—
101C4×C3⋊D3—22no—
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C3×D12—22no—
108C3×C4×D3—22no—
109C3×C4×D3—22no—
110C3×C4×D3D414yescomplemented
111#627—22no—
112#626—22no—
113#626—22no—
114#626—22no—
115#626—22no—
116#625—22no—
117#624—22no—
118D3×Dic3—22no—
119D3×Dic3—22no—
120#525—91noSylow
121C23×D3—31no—
122C23×D3—31no—
123C2×C3×D4—31no—
124C22×C3×C4—31no—
125C22×C3×C4—31no—
126C22×C3×C4—31no—
127C2×#67—31no—
128C2×#67—31no—
129C2×#67—31no—
130C2×#67—31no—
131C2×#67—31no—
132C2×#67—31no—
133C22×Dic3—31no—
134C22×Dic3—31no—
135#235—64no—
136#235—64no—
137C2×D12—31no—
138C2×D12—31no—
139C2×D12—31no—
140C2×D12—31no—
141C2×D12—31no—
142C2×D12—31no—
143C2×D12—61no—
144C2×D12C2×D311yescharacteristic
145C2×C4×D3—31no—
146C2×C4×D3—31no—
147C2×C4×D3—31no—
148C2×C4×D3—31no—
149C2×C4×D3—31no—
150C2×C4×D3—31no—
151C2×C4×D3—31no—
152C2×C4×D3—31no—
153C2×C4×D3—61no—
154C2×C4×D3—61no—
155C2×C4×D3—64no—
156C2×C4×D3—64no—
157C2×C4×D3C2×D311yescharacteristic
158C2×C4×D3C2×D311yescharacteristic
159C2×C4×D3C2×D311yescharacteristic, complemented
160C2×Dic6—31no—
161C2×Dic6—31no—
162C3×C4⋊C4—31no—
163C3×C4⋊C4—31no—
164C3×C4⋊C4—31no—
165C3×C22⋊C4—31no—
166C3×C22⋊C4—31no—
167C3×C22⋊C4—31no—
168C3×C22⋊C4—31no—
169C3×C42—21no—
170C3×C42—31no—
171C3×C42C2×D311yescharacteristic, complemented
172C3×C42C2×D311yescharacteristic, complemented
173#217—31no—
174#217—31no—
175(C2×D3)⋊C4—31no—
176(C2×D3)⋊C4—31no—
177(C2×D3)⋊C4—31no—
178(C2×D3)⋊C4—31no—
179(C2×D3)⋊C4—31no—
180(C2×D3)⋊C4—31no—
181(C2×D3)⋊C4—31no—
182(C2×D3)⋊C4—31no—
183(C2×D3)⋊C4—31no—
184(C2×D3)⋊C4—31no—
185(C2×D3)⋊C4—61no—
186(C2×D3)⋊C4—61no—
187(C2×D3)⋊C4C2×D311yescharacteristic
188(C2×D3)⋊C4C2×D311yescharacteristic
189(C2×D3)⋊C4C2×D311yescharacteristic
190(C2×D3)⋊C4C2×D311yescharacteristic, complemented
191C4⋊Dic3—31no—
192C4⋊Dic3—31no—
193C4⋊Dic3—31no—
194C4⋊Dic3—31no—
195C4⋊Dic3—61no—
196C4⋊Dic3C2×D311yescharacteristic, complemented
197Dic3⋊C4—31no—
198Dic3⋊C4—31no—
199Dic3⋊C4C2×D311yescharacteristic, complemented
200Dic3⋊C4C2×D311yescharacteristic, complemented
201C4×Dic3C2×D311yescharacteristic, complemented
202C22×C32C2411yescharacteristic
203C2×C3⋊D3—21no—
204C2×C3⋊D3—21no—
205C2×C3⋊D3—21no—
206C2×C3⋊D3—21no—
207C2×C3⋊D3—21no—
208C2×C3⋊D3#4112yescomplemented
209C2×C3×D3—21no—
210C2×C3×D3—21no—
211C2×C3×D3—21no—
212C2×C3×D3—21no—
213C2×C3×D3—21no—
214C2×C3×D3—21no—
215C2×C3×D3—21no—
216C2×C3×D3C2×D412yes—
217C2×C3×D3#4112yescomplemented
218D32—28no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4C2×D412yes—
222C3⋊Dic3—21no—
223C3⋊Dic3—21no—
224C3×Dic3—21no—
225C3×Dic3—21no—
226C3×Dic3—21no—
227C3×Dic3—21no—
228C3×Dic3C2×D412yes—
229C2×#41—91no—
230C23×C4—91no—
231#123—91no—
232#123—91no—
233C22⋊Q8—91no—
234C22⋊Q8—91no—
235#121—91no—
236#121—91no—
237#120—91no—
238#120—91no—
239C4×D4—31no—
240C4×D4—91no—
241C4×D4—91no—
242C4×D4—91no—
243#117—31no—
244C23×C3—31no—
245C23×C3—31no—
246C23×C3—31no—
247C22×D3—31no—
248C22×D3—31no—
249C22×D3—31no—
250C22×D3—31no—
251C22×D3—61no—
252C22×D3—61no—
253C22×D3—62no—
254C22×D3—62no—
255C22×D3—62no—
256C22×D3—62no—
257C22×D3—62no—
258C22×D3—62no—
259C22×D3C22×D311yescharacteristic
260C22×D3C22×D311yescharacteristic
261C22×D3C22×D311yescharacteristic
262C3×D4—62no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—31no—
267C2×C3×C4—34no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—62no—
273C2×C3×C4—62no—
274C2×C3×C4C22×D311yescharacteristic
275C2×C3×C4C22×D311yescharacteristic
276C2×C3×C4C22×D311yescharacteristic
277C2×C3×C4C22×D311yescharacteristic
278C2×C3×C4C22×D311yescharacteristic
279C2×C3×C4C22×D311yescharacteristic
280#67—62no—
281#67—62no—
282#67—62no—
283#67—62no—
284#67—62no—
285#67—62no—
286C2×Dic3—31no—
287C2×Dic3—31no—
288C2×Dic3—31no—
289C2×Dic3—31no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—62no—
293C2×Dic3—62no—
294C2×Dic3C22×D311yescharacteristic
295C2×Dic3C22×D311yescharacteristic
296C2×Dic3C22×D311yescharacteristic
297C2×Dic3C22×D311yescharacteristic
298C2×Dic3C22×D311yescharacteristic
299D12—22no—
300D12—62no—
301D12—62no—
302D12—64no—
303D12—62no—
304D12—62no—
305D12—62no—
306D12—62no—
307C4×D3—22no—
308C4×D3—22no—
309C4×D3—62no—
310C4×D3—62no—
311C4×D3—62no—
312C4×D3—62no—
313C4×D3—62no—
314C4×D3—62no—
315C4×D3—62no—
316C4×D3—62no—
317C4×D3—122no—
318C4×D3—122no—
319C4×D3D1214yescomplemented
320Dic6—62no—
321Dic6—62no—
322C2×C32C22×D411yescharacteristic
323C2×C32C2×#4111yescharacteristic
324C2×C32C2×#4111yescharacteristic
325C3⋊D3—22no—
326C3⋊D3—41no—
327C3×D3—22no—
328C3×D3—22no—
329C3×D3—41no—
330C24—91no—
331#41—184no—
332C2×Q8—91no—
333C2×D4—31no—
334C2×D4—91no—
335C2×D4—91no—
336C2×D4—91no—
337C2×D4—91no—
338C2×D4—91no—
339C2×D4—91no—
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—91no—
344C22×C4—91no—
345C22×C4—91no—
346C22×C4—91no—
347C22×C4—91no—
348C22×C4—184no—
349C4⋊C4—31no—
350C4⋊C4—31no—
351C4⋊C4—31no—
352C4⋊C4—91no—
353C4⋊C4—91no—
354C4⋊C4—91no—
355C22⋊C4—31no—
356C22⋊C4—31no—
357C22⋊C4—31no—
358C22⋊C4—31no—
359C22⋊C4—91no—
360C22⋊C4—91no—
361C22⋊C4—91no—
362C22⋊C4—91no—
363C22⋊C4—91no—
364C22⋊C4—91no—
365C42—31no—
366C42D3211yescharacteristic, complemented
367C22×C3—21no—
368C22×C3—32no—
369C22×C3—32no—
370C22×C3—61no—
371C22×C3—61no—
372C22×C3—61no—
373C22×C3—61no—
374C22×C3—61no—
375C22×C3—61no—
376C22×C3—61no—
377C22×C3C23×D311yescharacteristic
378C22×C3C23×D311yescharacteristic
379C2×D3—21no—
380C2×D3—21no—
381C2×D3—21no—
382C2×D3—21no—
383C2×D3—21no—
384C2×D3—21no—
385C2×D3—21no—
386C2×D3—32no—
387C2×D3—32no—
388C2×D3—62no—
389C2×D3—68no—
390C2×D3—68no—
391C2×D3—61no—
392C2×D3—61no—
393C2×D3—61no—
394C2×D3—61no—
395C2×D3—62no—
396C2×D3—61no—
397C2×D3—61no—
398C2×D3—61no—
399C2×D3—61no—
400C2×D3—61no—
401C2×D3—61no—
402C2×D3—121no—
403C2×D3—121no—
404C2×D3—121no—
405C2×D3—121no—
406C2×D3C2×D1212yes—
407C2×D3#23512yescomplemented
408C3×C4—22no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—22no—
414C3×C4—32no—
415C3×C4—41no—
416C3×C4—61no—
417C3×C4—61no—
418C3×C4—61no—
419C3×C4—61no—
420C3×C4C2×D1212yes—
421C3×C4D3×D412yes—
422Dic3—21no—
423Dic3—21no—
424Dic3—21no—
425Dic3—21no—
426Dic3—61no—
427Dic3—61no—
428Dic3—61no—
429Dic3—61no—
430Dic3—121no—
431Dic3—121no—
432Dic3C2×D1212yes—
433C32#52511yescharacteristic, complemented, Sylow
434C23—31no—
435C23—31no—
436C23—31no—
437C23—91no—
438C23—91no—
439C23—182no—
440C23—182no—
441C23—182no—
442Q8—182no—
443D4—62no—
444D4—182no—
445D4—182no—
446D4—182no—
447D4—182no—
448D4—182no—
449D4—182no—
450C2×C4—31no—
451C2×C4—34no—
452C2×C4—31no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—31no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—91no—
459C2×C4—91no—
460C2×C4—182no—
461C2×C4—182no—
462C2×C4—182no—
463C2×C4—182no—
464C2×C4—182no—
465C2×C4C2×D3211yescharacteristic
466C2×C4C2×D3211yescharacteristic
467C2×C4C2×D3211yescharacteristic
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—62no—
472C2×C3—62no—
473C2×C3—121no—
474C2×C3C22×D1211yescharacteristic
475C2×C3C2×#23511yescharacteristic
476C2×C3C2×D3×D411yescharacteristic
477C2×C3D3×#4111yescharacteristic
478C2×C3D3×#4111yescharacteristic
479C2×C3D3×#4111yescharacteristic
480D3—22no—
481D3—22no—
482D3—41no—
483D3—62no—
484D3—62no—
485D3—122no—
486D3—122no—
487D3—121no—
488D3—121no—
489C22—32no—
490C22—32no—
491C22—61no—
492C22—61no—
493C22—61no—
494C22—61no—
495C22—61no—
496C22—61no—
497C22—61no—
498C22—92no—
499C22—188no—
500C22—181no—
501C22—181no—
502C22—181no—
503C22—181no—
504C22—181no—
505C22C22×D3211yescharacteristic
506C4—21no—
507C4—21no—
508C4—32no—
509C4—61no—
510C4—61no—
511C4—61no—
512C4—61no—
513C4—181no—
514C4—181no—
515C4D3×D1212yes—
516C3—21no—
517C3#552411yescharacteristic, complemented
518C3#554711yescharacteristic, complemented
519C2—62no—
520C2—62no—
521C2—121no—
522C2—182no—
523C2—361no—
524C2C2×D3×D1211yescharacteristic
525C2D3×#23511yescharacteristic
526C2#6430611yescharacteristic
subgroupG/H
centerC2×C4C2×D32
commutator, socle, cosocleC22×C32C24
FrattiniC22C22×D32
FittingC32×C42C22
hypercenterC42D32
hypocenterC32#525
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×C4×D32C2semidirect
2. C2×#3798C2semidirect
3. C4×C4⋊(C3⋊D3)C2semidirect
4. C3×C4×D12C2semidirect
5. C3×#872C2semidirect
6. #63923C2semidirect
7. #63922C2semidirect
8. #63910C2semidirect
9. C4×#626C2semidirect
10. C4×#626C2semidirect
11. #63895C2semidirect
12. #63893C2semidirect
13. #63891C2semidirect
14. #63890C2semidirect
15. #63886C2semidirect
21. C2×C4×C3⋊D3C22semidirect
25. C2×C3×C4×D3C22semidirect
26. C2×#627C22semidirect
30. C2×#626C22semidirect
37. C32×C42C22semidirect
39. #3752C22semidirect
40. C4⋊(C3⋊Dic3)C22semidirect
44. C3×(C2×D3)⋊C4C22semidirect
45. C3×C4⋊Dic3C22semidirect
46. C3×Dic3⋊C4C22semidirect
47. C3×Dic3⋊C4C22semidirect
48. C3×C4×Dic3C22semidirect
49. Dic3⋊Dic3C22semidirect
50. Dic3⋊Dic3C22semidirect
52. #3722C22semidirect
54. (C2×D3)⋊Dic3C22semidirect
84. C4×D12D3semidirect
85. #872D3semidirect
110. C3×C4×D3D4semidirect
159. C2×C4×D3C2×D3semidirect
171. C3×C42C2×D3semidirect
172. C3×C42C2×D3semidirect
190. (C2×D3)⋊C4C2×D3semidirect
196. C4⋊Dic3C2×D3semidirect
199. Dic3⋊C4C2×D3semidirect
200. Dic3⋊C4C2×D3semidirect
201. C4×Dic3C2×D3semidirect
208. C2×C3⋊D3#41semidirect
217. C2×C3×D3#41semidirect
319. C4×D3D12semidirect
366. C42D32semidirect
407. C2×D3#235semidirect
433. C32#525semidirect
517. C3#5524semidirect
518. C3#5547semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D322
C2×#37982
C4×C4⋊(C3⋊D3)2
C3×C4×D122
C3×#8722
#639232
#639222
#639102
C4×#6262
#638952
#638932
#638912
#638902
#638862
#55473
#55243