Finite Group Databasebeta

← #10595216

Subgroups — #10595216

Order 576.

475 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64144C211yesmaximal, characteristic, complemented
2C3×#939C211yesmaximal, characteristic
3C3×D4×Dic3C211yesmaximal, characteristic, complemented
4Dic3×#67C211yesmaximal, characteristic, complemented
5#63969C211yesmaximal, characteristic, complemented
6C2×#3724C211yesmaximal, characteristic, complemented
7#63964C211yesmaximal, characteristic, complemented
8#63964C211yesmaximal, characteristic, complemented
9#63962C211yesmaximal, characteristic, complemented
10#63961C211yesmaximal, characteristic, complemented
11#63956C211yesmaximal, characteristic, complemented
12#63899C211yesmaximal, characteristic, complemented
13#63867C211yesmaximal, characteristic, complemented
14#63859C211yesmaximal, characteristic, complemented
15#63847C211yesmaximal, characteristic, complemented
16#5602—31nomaximal
17#5564—31nomaximal
18C2×C32×D4C2211yescharacteristic
19C22×C3⋊Dic3C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C2×C3×#67C2211yescharacteristic
22C22×C3×Dic3C2211yescharacteristic
23C22×C3×Dic3C2211yescharacteristic
24C22×C3×Dic3C2211yescharacteristic
25C2×#627C2211yescharacteristic
26C2×D3×Dic3C2211yescharacteristic
27#3757C2211yescharacteristic
28#3757C2211yescharacteristic
29#3757C2211yescharacteristic
30#3750C2211yescharacteristic
31#3750C2211yescharacteristic
32C3×#217C2211yescharacteristic
33C3×#217C2211yescharacteristic
34C3×#217C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×C4⋊Dic3C2211yescharacteristic
37C3×Dic3⋊C4C2211yescharacteristic
38C3×C4×Dic3C2211yescharacteristic
39#3724C2211yescharacteristic
40#3724C2211yescharacteristic
41#3724C2211yescharacteristic
42#3724C2211yescharacteristic
43#3724C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45Dic3⋊Dic3C2211yescharacteristic
46Dic3⋊Dic3C2211yescharacteristic
47Dic3⋊Dic3C2211yescharacteristic
48Dic3⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×#121—31no—
54C3×C4×D4—31no—
55#939D311yescharacteristic
56#935—31no—
57#935—31no—
58#935—31no—
59#935—31no—
60#935—31no—
61#935—61no—
62D4×Dic3—31no—
63D4×Dic3D311yescharacteristic, complemented
64#929—31no—
65C4×#67—31no—
66#924—31no—
67#924—31no—
68C2×Dic3⋊C4—31no—
69C2×Dic3⋊C4—31no—
70#896—31no—
71#889—31no—
72#889—31no—
73#889—31no—
74#886—31no—
75#883—31no—
76#879—31no—
77#879—31no—
78#879—31no—
79#878—31no—
80#878—31no—
81#878—31no—
82#876—31no—
83#870—31no—
84C23×C32C2311yescharacteristic
85C23×C32C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C32×D4—22no—
88C2×C32×C4C2311yescharacteristic
89C2×C3⋊Dic3—22no—
90C2×C3⋊Dic3C2311yescharacteristic
91C2×C3⋊Dic3C2311yescharacteristic
92C2×C3⋊Dic3C2311yescharacteristic
93C2×C3⋊Dic3C2311yescharacteristic
94C3×#67—22no—
95C3×#67—41no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3—41no—
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106#627—41no—
107D3×Dic3—41no—
108#539—91noSylow
109C2×C3×D4—21no—
110C2×C3×D4—31no—
111C2×C3×D4—31no—
112C2×C3×D4C2×D311yescharacteristic
113C2×C3×D4C2×D311yescharacteristic
114C22×C3×C4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—31no—
117C2×#67C2×D311yescharacteristic
118C2×#67C2×D311yescharacteristic
119C22×Dic3—31no—
120C22×Dic3—31no—
121C22×Dic3—31no—
122C22×Dic3—61no—
123C22×Dic3C2×D311yescharacteristic
124C22×Dic3C2×D311yescharacteristic
125C22×Dic3C2×D311yescharacteristic
126C2×C4×D3—31no—
127C2×Dic6—31no—
128C2×Dic6—31no—
129C3×C4⋊C4—31no—
130C3×C4⋊C4—31no—
131C3×C22⋊C4—31no—
132C3×C22⋊C4—31no—
133C3×C22⋊C4—31no—
134C3×C22⋊C4—31no—
135C3×C42—31no—
136#217—31no—
137#217—31no—
138#217—31no—
139#217—31no—
140#217—31no—
141#217—31no—
142#217—31no—
143#217—31no—
144#217—31no—
145#217—61no—
146#217—61no—
147#217—61no—
148#217C2×D311yescharacteristic
149#217C2×D311yescharacteristic
150#217C2×D311yescharacteristic
151(C2×D3)⋊C4—31no—
152(C2×D3)⋊C4—31no—
153(C2×D3)⋊C4—31no—
154(C2×D3)⋊C4C2×D311yescharacteristic
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157C4⋊Dic3—31no—
158C4⋊Dic3—31no—
159C4⋊Dic3—31no—
160C4⋊Dic3C2×D311yescharacteristic
161Dic3⋊C4—31no—
162Dic3⋊C4—31no—
163Dic3⋊C4—31no—
164Dic3⋊C4—31no—
165Dic3⋊C4—31no—
166Dic3⋊C4—31no—
167Dic3⋊C4—31no—
168Dic3⋊C4—31no—
169Dic3⋊C4—31no—
170Dic3⋊C4—31no—
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
183C4×Dic3—31no—
184C4×Dic3—31no—
185C4×Dic3C2×D311yescharacteristic
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32#4112yes—
192C22×C32C2411yescharacteristic
193C2×C3×D3—21no—
194C2×C3×D3—21no—
195C2×C3×D3—21no—
196C32×C4—21no—
197C3⋊Dic3—21no—
198C3⋊Dic3—21no—
199C3⋊Dic3—21no—
200C3⋊Dic3—21no—
201C3×Dic3—21no—
202C3×Dic3—21no—
203C3×Dic3—21no—
204C3×Dic3—21no—
205C3×Dic3—21no—
206C3×Dic3—21no—
207C3×Dic3—21no—
208#126—91no—
209#126—91no—
210#125—91no—
211#125—91no—
212#123—91no—
213#123—91no—
214#123—91no—
215#123—91no—
216C22⋊Q8—91no—
217C22⋊Q8—91no—
218C22⋊Q8—91no—
219#121—31no—
220C4×D4—31no—
221C4×D4—91no—
222C2×C4⋊C4—91no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—31no—
226C23×C3C22×D311yescharacteristic
227C23×C3C22×D311yescharacteristic
228C23×C3C22×D311yescharacteristic
229C23×C3C22×D311yescharacteristic
230C22×D3C22×D311yescharacteristic
231C3×D4—22no—
232C3×D4—22no—
233C3×D4—42no—
234C3×D4—62no—
235C3×D4—121no—
236C2×C3×C4—21no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—121no—
245C2×C3×C4—121no—
246C2×C3×C4—121no—
247C2×C3×C4C22×D311yescharacteristic
248C2×C3×C4C22×D311yescharacteristic
249#67—22no—
250#67—41no—
251C2×Dic3—31no—
252C2×Dic3—31no—
253C2×Dic3—31no—
254C2×Dic3—31no—
255C2×Dic3—31no—
256C2×Dic3—31no—
257C2×Dic3—31no—
258C2×Dic3—31no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—62no—
267C2×Dic3—62no—
268C2×Dic3—122no—
269C2×Dic3—121no—
270C2×Dic3C22×D311yescharacteristic
271C2×Dic3C22×D311yescharacteristic
272C2×Dic3C22×D311yescharacteristic
273C2×Dic3C22×D311yescharacteristic
274C2×Dic3C22×D311yescharacteristic
275C2×Dic3C22×D311yescharacteristic
276C2×Dic3C22×D311yescharacteristic
277C4×D3—121no—
278Dic6—121no—
279Dic6—121no—
280C2×C32—22no—
281C2×C32—41no—
282C2×C32C2×#4111yescharacteristic
283C2×C32H(5,𝔽2)11yescharacteristic
284C2×C322−1+411yescharacteristic
285C3×D3—41no—
286C2×Q8—91no—
287C2×D4—31no—
288C2×D4—31no—
289C2×D4D3211yescharacteristic
290C22×C4—31no—
291C22×C4—31no—
292C22×C4—31no—
293C22×C4—91no—
294C22×C4—91no—
295C4⋊C4—31no—
296C4⋊C4—31no—
297C4⋊C4—91no—
298C4⋊C4—91no—
299C4⋊C4—91no—
300C4⋊C4—91no—
301C4⋊C4—91no—
302C4⋊C4—91no—
303C4⋊C4—91no—
304C4⋊C4—91no—
305C4⋊C4—91no—
306C4⋊C4—91no—
307C4⋊C4—91no—
308C4⋊C4—91no—
309C22⋊C4—31no—
310C22⋊C4—31no—
311C22⋊C4—31no—
312C22⋊C4—31no—
313C22⋊C4—91no—
314C22⋊C4—91no—
315C22⋊C4—91no—
316C22⋊C4—91no—
317C22⋊C4—91no—
318C22⋊C4—91no—
319C42—31no—
320C42—91no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—22no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—61no—
339C22×C3—61no—
340C22×C3—61no—
341C22×C3#23512yes—
342C22×C3#23712yes—
343C22×C3C23×D311yescharacteristic
344C22×C3C23×D311yescharacteristic
345C2×D3—21no—
346C2×D3—21no—
347C2×D3—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—41no—
351C3×C4—61no—
352C3×C4—61no—
353C3×C4—61no—
354C3×C4—61no—
355C3×C4—61no—
356C3×C4—61no—
357C3×C4—61no—
358Dic3—21no—
359Dic3—21no—
360Dic3—21no—
361Dic3—21no—
362Dic3—21no—
363Dic3—21no—
364Dic3—21no—
365Dic3—61no—
366Dic3—61no—
367Dic3—61no—
368Dic3—61no—
369Dic3—61no—
370Dic3—61no—
371Dic3—61no—
372Dic3—61no—
373Dic3—121no—
374Dic3—121no—
375Dic3—121no—
376Dic3—121no—
377C32#53911yescharacteristic, complemented, Sylow
378C23—31no—
379C23C2×D3211yescharacteristic
380C23C2×D3211yescharacteristic
381Q8—361no—
382D4—22no—
383D4—62no—
384D4—121no—
385C2×C4—31no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—31no—
390C2×C4—31no—
391C2×C4—31no—
392C2×C4—91no—
393C2×C4—91no—
394C2×C4—91no—
395C2×C4—91no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—182no—
400C2×C4—361no—
401C2×C4C2×D3211yescharacteristic
402C2×C3—22no—
403C2×C3—22no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—42no—
408C2×C3—41no—
409C2×C3—41no—
410C2×C3—81no—
411C2×C3—121no—
412C2×C3C2×#23511yescharacteristic
413C2×C3C2×#23711yescharacteristic
414C2×C3#100411yescharacteristic
415C2×C3#100411yescharacteristic
416C2×C3#101011yescharacteristic
417C2×C3#101011yescharacteristic
418D3—41no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—61no—
425C22—61no—
426C22—61no—
427C22#380412yes—
428C22C22×D3211yescharacteristic
429C4—21no—
430C4—61no—
431C4—61no—
432C4—61no—
433C4—61no—
434C4—61no—
435C4—61no—
436C4—61no—
437C4—181no—
438C4—181no—
439C4—181no—
440C4—181no—
441C3—21no—
442C3#556411yescharacteristic, complemented
443C3#560211yescharacteristic, complemented
444C2—22no—
445C2—41no—
446C2—121no—
447C2#6430911yescharacteristic
448C2#6431311yescharacteristic
449C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
prime radicalC3×#939C2
hypercenterC2×D4D32
hypocenterC32#539
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #64144C2semidirect
3. C3×D4×Dic3C2semidirect
4. Dic3×#67C2semidirect
5. #63969C2semidirect
6. C2×#3724C2semidirect
7. #63964C2semidirect
8. #63964C2semidirect
9. #63962C2semidirect
10. #63961C2semidirect
11. #63956C2semidirect
12. #63899C2semidirect
13. #63867C2semidirect
14. #63859C2semidirect
15. #63847C2semidirect
63. D4×Dic3D3semidirect
377. C32#539semidirect
442. C3#5564semidirect
443. C3#5602semidirect

Maximal subgroups

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

subgroupindex
#641442
C3×#9392
C3×D4×Dic32
Dic3×#672
#639692
C2×#37242
#639642
#639622
#639612
#639562
#638992
#638672
#638592
#638472
#56023
#55643