Finite Group Databasebeta

← #10594473

Subgroups — #10594473

Order 576.

522 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
2C3×#898C211yesmaximal, characteristic, complemented
3C3×#894C211yesmaximal, characteristic, complemented
4#63916C211yesmaximal, characteristic, complemented
5#63912C211yesmaximal, characteristic, complemented
6#63912C211yesmaximal, characteristic, complemented
7C4×#626C211yesmaximal, characteristic, complemented
8#63891C211yesmaximal, characteristic, complemented
9#63885C211yesmaximal, characteristic, complemented
10#63881C211yesmaximal, characteristic, complemented
11#63881C211yesmaximal, characteristic, complemented
12#63879C211yesmaximal, characteristic, complemented
13#63877C211yesmaximal, characteristic, complemented
14#63861C211yesmaximal, characteristic, complemented
15#63847C211yesmaximal, characteristic, complemented
16#5709—31nomaximal
17#5661—31nomaximal
18C2×C4×C3⋊D3C2211yescharacteristic
19C2×C4×C3⋊D3C2211yescharacteristic
20C2×C4×C3⋊D3C2211yescharacteristic
21C2×C3×D12C2211yescharacteristic
22C2×C3×C4×D3C2211yescharacteristic
23C2×#626C2211yescharacteristic
24C2×#626C2211yescharacteristic
25C2×#626C2211yescharacteristic
26C2×#625C2211yescharacteristic
27C2×#625C2211yescharacteristic
28C2×D3×Dic3C2211yescharacteristic
29C2×D3×Dic3C2211yescharacteristic
30C32×C4⋊C4C2211yescharacteristic
31C4⋊(C3⋊Dic3)C2211yescharacteristic
32#3750C2211yescharacteristic
33#3750C2211yescharacteristic
34C3×(C2×D3)⋊C4C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×(C2×D3)⋊C4C2211yescharacteristic
39C3×(C2×D3)⋊C4C2211yescharacteristic
40C3×C4⋊Dic3C2211yescharacteristic
41C3×Dic3⋊C4C2211yescharacteristic
42C3×Dic3⋊C4C2211yescharacteristic
43C3×C4×Dic3C2211yescharacteristic
44#3724C2211yescharacteristic
45#3724C2211yescharacteristic
46Dic3⋊Dic3C2211yescharacteristic
47#3722C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51Dic32C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×#126—31no—
54C3×#123—31no—
55#939—31no—
56#938—31no—
57C4×#67—31no—
58C4×#67—31no—
59#898—31no—
60#898D311yescharacteristic, complemented
61#895—31no—
62#894D311yescharacteristic, complemented
63#893—31no—
64#893—31no—
65D3×C4⋊C4—31no—
66D3×C4⋊C4—31no—
67D3×C4⋊C4—61no—
68#889—31no—
69#889—31no—
70#886—31no—
71#884—31no—
72#884—31no—
73#884—31no—
74#884—31no—
75#884—31no—
76#883—31no—
77#67⋊C4—31no—
78#67⋊C4—31no—
79#67⋊C4—31no—
80#879—31no—
81#877—31no—
82C4×D12—31no—
83#872—31no—
84C22×C3⋊D3C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C2×C32×C4C2311yescharacteristic
89C2×C32×C4C2311yescharacteristic
90C2×C32×C4C2311yescharacteristic
91C2×C3⋊Dic3C2311yescharacteristic
92C2×C3⋊Dic3C2311yescharacteristic
93C2×C3⋊Dic3C2311yescharacteristic
94C4×C3⋊D3—22no—
95C4×C3⋊D3—41no—
96C4×C3⋊D3—41no—
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C3×D12—41no—
103C3×C4×D3—41no—
104#626—22no—
105#626—22no—
106#626—22no—
107#625—22no—
108#625—41no—
109D3×Dic3—22no—
110D3×Dic3—41no—
111#553—91noSylow
112C2×C3×D4—31no—
113C22×C3×C4—31no—
114C2×#67—31no—
115C2×#67—31no—
116C2×#67—31no—
117C2×#67—31no—
118C2×#67—31no—
119C2×#67—31no—
120C2×#67—31no—
121C22×Dic3—31no—
122C22×Dic3—31no—
123C2×D12—31no—
124C2×D12—31no—
125C2×D12—31no—
126C2×D12C2×D311yescharacteristic
127C2×C4×D3—31no—
128C2×C4×D3—31no—
129C2×C4×D3—31no—
130C2×C4×D3—31no—
131C2×C4×D3—31no—
132C2×C4×D3—31no—
133C2×C4×D3—31no—
134C2×C4×D3—31no—
135C2×C4×D3—61no—
136C2×C4×D3—61no—
137C2×C4×D3—61no—
138C2×C4×D3C2×D311yescharacteristic
139C3×C4⋊C4—21no—
140C3×C4⋊C4—31no—
141C3×C4⋊C4—31no—
142C3×C4⋊C4—31no—
143C3×C4⋊C4C2×D311yescharacteristic
144C3×C4⋊C4C2×D311yescharacteristic
145C3×C22⋊C4—31no—
146C3×C22⋊C4—31no—
147C3×C22⋊C4—31no—
148C3×C22⋊C4—31no—
149C3×C22⋊C4—31no—
150C3×C22⋊C4—31no—
151C3×C42—31no—
152#217—31no—
153#217—31no—
154#217—31no—
155(C2×D3)⋊C4—31no—
156(C2×D3)⋊C4—31no—
157(C2×D3)⋊C4—31no—
158(C2×D3)⋊C4—31no—
159(C2×D3)⋊C4—31no—
160(C2×D3)⋊C4C2×D311yescharacteristic
161(C2×D3)⋊C4C2×D311yescharacteristic
162(C2×D3)⋊C4C2×D311yescharacteristic
163(C2×D3)⋊C4C2×D311yescharacteristic
164(C2×D3)⋊C4C2×D311yescharacteristic
165(C2×D3)⋊C4C2×D311yescharacteristic
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—61no—
170C4⋊Dic3C2×D311yescharacteristic
171Dic3⋊C4—31no—
172Dic3⋊C4—31no—
173Dic3⋊C4—31no—
174Dic3⋊C4—31no—
175Dic3⋊C4—31no—
176Dic3⋊C4—31no—
177Dic3⋊C4—31no—
178Dic3⋊C4—31no—
179Dic3⋊C4—31no—
180Dic3⋊C4—61no—
181Dic3⋊C4—61no—
182Dic3⋊C4C2×D311yescharacteristic
183Dic3⋊C4C2×D311yescharacteristic
184C4×Dic3—31no—
185C4×Dic3—31no—
186C4×Dic3—31no—
187C4×Dic3—31no—
188C4×Dic3C2×D311yescharacteristic
189C22×C32C2411yescharacteristic
190C2×C3⋊D3—21no—
191C2×C3⋊D3—21no—
192C2×C3⋊D3#4112yes—
193C2×C3×D3—21no—
194C2×C3×D3—21no—
195C2×C3×D3—21no—
196C2×C3×D3—21no—
197C2×C3×D3—21no—
198C2×C3×D3—21no—
199C2×C3×D3—21no—
200C2×C3×D3—21no—
201C2×C3×D3—21no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—21no—
205C3⋊Dic3—21no—
206C3⋊Dic3—21no—
207C3⋊Dic3#4112yes—
208C3×Dic3—21no—
209C3×Dic3—21no—
210C3×Dic3—21no—
211C3×Dic3—21no—
212C3×Dic3—21no—
213#126—31no—
214#126—91no—
215#125—91no—
216#123—31no—
217#123—91no—
218#121—91no—
219#121—91no—
220#121—91no—
221#121—91no—
222C4×D4—91no—
223C4×D4—91no—
224C4×D4—91no—
225C4×D4—91no—
226#117—91no—
227C2×C4⋊C4—91no—
228C23×C3—31no—
229C23×C3—31no—
230C23×C3—31no—
231C22×D3—31no—
232C22×D3—31no—
233C22×D3—61no—
234C22×D3C22×D311yescharacteristic
235C22×D3C22×D311yescharacteristic
236C22×D3C22×D311yescharacteristic
237C3×D4—121no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—121no—
247C2×C3×C4C22×D311yescharacteristic
248C2×C3×C4C22×D311yescharacteristic
249C2×C3×C4C22×D311yescharacteristic
250C2×C3×C4C22×D311yescharacteristic
251C2×C3×C4C22×D311yescharacteristic
252C2×C3×C4C22×D311yescharacteristic
253#67—62no—
254#67—62no—
255#67—62no—
256#67—62no—
257#67—62no—
258#67—121no—
259#67—121no—
260C2×Dic3—31no—
261C2×Dic3—31no—
262C2×Dic3—31no—
263C2×Dic3—31no—
264C2×Dic3—31no—
265C2×Dic3—31no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—62no—
270C2×Dic3—121no—
271C2×Dic3C22×D311yescharacteristic
272C2×Dic3C22×D311yescharacteristic
273C2×Dic3C22×D311yescharacteristic
274C2×Dic3C22×D311yescharacteristic
275C2×Dic3C22×D311yescharacteristic
276D12—41no—
277D12—62no—
278D12—62no—
279D12—62no—
280C4×D3—41no—
281C4×D3—62no—
282C4×D3—62no—
283C4×D3—62no—
284C4×D3—64no—
285C4×D3—121no—
286C4×D3—121no—
287C4×D3—121no—
288C4×D3—121no—
289C4×D3—121no—
290C4×D3—122no—
291C4×D3—122no—
292C2×C32C2×#4111yescharacteristic
293C2×C32C2×#4111yescharacteristic
294C2×C32H(5,𝔽2)11yescharacteristic
295C3⋊D3—22no—
296C3×D3—41no—
297C3×D3—41no—
298C3×D3—41no—
299C2×D4—31no—
300C2×D4—91no—
301C2×D4—91no—
302C2×D4—91no—
303C2×D4—91no—
304C2×D4—91no—
305C22×C4—31no—
306C22×C4—91no—
307C22×C4—91no—
308C22×C4—91no—
309C22×C4—91no—
310C22×C4—91no—
311C4⋊C4—31no—
312C4⋊C4—31no—
313C4⋊C4—31no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4—91no—
317C4⋊C4—91no—
318C4⋊C4—91no—
319C4⋊C4—91no—
320C4⋊C4D3211yescharacteristic
321C22⋊C4—31no—
322C22⋊C4—31no—
323C22⋊C4—31no—
324C22⋊C4—31no—
325C22⋊C4—31no—
326C22⋊C4—31no—
327C22⋊C4—91no—
328C22⋊C4—91no—
329C22⋊C4—91no—
330C22⋊C4—91no—
331C42—31no—
332C42—91no—
333C42—91no—
334C22×C3—21no—
335C22×C3—61no—
336C22×C3—61no—
337C22×C3—61no—
338C22×C3—61no—
339C22×C3—61no—
340C22×C3—61no—
341C22×C3—61no—
342C22×C3—61no—
343C22×C3—61no—
344C22×C3C23×D311yescharacteristic
345C22×C3C23×D311yescharacteristic
346C2×D3—21no—
347C2×D3—21no—
348C2×D3—21no—
349C2×D3—21no—
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—21no—
354C2×D3—21no—
355C2×D3—32no—
356C2×D3—32no—
357C2×D3—62no—
358C2×D3—62no—
359C2×D3—62no—
360C2×D3—61no—
361C2×D3—61no—
362C2×D3—61no—
363C2×D3—61no—
364C3×C4—22no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—41no—
372C3×C4—41no—
373C3×C4—61no—
374C3×C4—61no—
375C3×C4—61no—
376C3×C4—61no—
377C3×C4—61no—
378Dic3—21no—
379Dic3—21no—
380Dic3—21no—
381Dic3—21no—
382Dic3—21no—
383Dic3—32no—
384Dic3—32no—
385Dic3—62no—
386Dic3—61no—
387Dic3—61no—
388Dic3—61no—
389Dic3—61no—
390Dic3—121no—
391Dic3—121no—
392C32#55311yescharacteristic, complemented, Sylow
393C23—31no—
394C23—31no—
395C23—31no—
396C23—91no—
397D4—121no—
398D4—182no—
399D4—182no—
400D4—182no—
401D4—182no—
402D4—361no—
403C2×C4—31no—
404C2×C4—31no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—91no—
409C2×C4—91no—
410C2×C4—91no—
411C2×C4—121no—
412C2×C4—182no—
413C2×C4—182no—
414C2×C4—361no—
415C2×C4—361no—
416C2×C4—361no—
417C2×C4C2×D3211yescharacteristic
418C2×C4C2×D3211yescharacteristic
419C2×C4C2×D3211yescharacteristic
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—121no—
424C2×C3—121no—
425C2×C3—121no—
426C2×C3#100411yescharacteristic
427C2×C3D3×#4111yescharacteristic
428C2×C3D3×#4111yescharacteristic
429C2×C3D3×#4111yescharacteristic
430C2×C3D3×#4111yescharacteristic
431C2×C3#100911yescharacteristic
432D3—41no—
433D3—41no—
434D3—41no—
435D3—64no—
436D3—62no—
437D3—62no—
438C22—61no—
439C22—61no—
440C22—61no—
441C22—61no—
442C22—61no—
443C22—61no—
444C22—61no—
445C22—61no—
446C22—61no—
447C22—92no—
448C22—181no—
449C22—181no—
450C22C22×D3211yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—61no—
455C4—61no—
456C4—61no—
457C4—61no—
458C4—61no—
459C4—92no—
460C4—181no—
461C4—181no—
462C3—21no—
463C3#566111yescharacteristic, complemented
464C3#570911yescharacteristic, complemented
465C2—121no—
466C2—121no—
467C2—121no—
468C2—182no—
469C2#6430611yescharacteristic
470C2#6431411yescharacteristic
471C2#6431911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#553
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C4⋊C4×C3⋊D3C2semidirect
2. C3×#898C2semidirect
3. C3×#894C2semidirect
4. #63916C2semidirect
5. #63912C2semidirect
6. #63912C2semidirect
7. C4×#626C2semidirect
8. #63891C2semidirect
9. #63885C2semidirect
10. #63881C2semidirect
11. #63881C2semidirect
12. #63879C2semidirect
13. #63877C2semidirect
14. #63861C2semidirect
15. #63847C2semidirect
60. #898D3semidirect
62. #894D3semidirect
392. C32#553semidirect
463. C3#5661semidirect
464. C3#5709semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×C3⋊D32
C3×#8982
C3×#8942
#639162
#639122
C4×#6262
#638912
#638852
#638812
#638792
#638772
#638612
#638472
#57093
#56613