Finite Group Databasebeta

← #10594219

Subgroups — #10594219

Order 576.

519 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64090C211yesmaximal, characteristic, complemented
2C3×#885C211yesmaximal, characteristic, complemented
3C3×#883C211yesmaximal, characteristic, complemented
4#63979C211yesmaximal, characteristic
5#67⋊Dic3C211yesmaximal, characteristic, complemented
6#63962C211yesmaximal, characteristic, complemented
7#63953C211yesmaximal, characteristic, complemented
8#63915C211yesmaximal, characteristic, complemented
9#63907C211yesmaximal, characteristic, complemented
10C4×#626C211yesmaximal, characteristic, complemented
11#63890C211yesmaximal, characteristic, complemented
12#63879C211yesmaximal, characteristic, complemented
13#63853C211yesmaximal, characteristic, complemented
14#63843C211yesmaximal, characteristic, complemented
15#63837C211yesmaximal, characteristic, complemented
16#5681—31nomaximal
17#5664—31nomaximal
18C2×#638C2211yescharacteristic
19C22×C3⋊Dic3C2211yescharacteristic
20C2×C4×C3⋊D3C2211yescharacteristic
21C2×C3×#67C2211yescharacteristic
22C2×C3×#67C2211yescharacteristic
23C2×C3×C4×D3C2211yescharacteristic
24C2×C3×Dic6C2211yescharacteristic
25C2×#627C2211yescharacteristic
26C2×#626C2211yescharacteristic
27C2×#626C2211yescharacteristic
28C2×#625C2211yescharacteristic
29C2×D3×Dic3C2211yescharacteristic
30C32×C22⋊C4C2211yescharacteristic
31#3752C2211yescharacteristic
32#3750C2211yescharacteristic
33C4×C3⋊Dic3C2211yescharacteristic
34C3×#217C2211yescharacteristic
35C3×#217C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×(C2×D3)⋊C4C2211yescharacteristic
39C3×C4⋊Dic3C2211yescharacteristic
40C3×Dic3⋊C4C2211yescharacteristic
41C3×C4×Dic3C2211yescharacteristic
42Dic3⋊Dic3C2211yescharacteristic
43Dic3⋊Dic3C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45#3722C2211yescharacteristic
46#3722C2211yescharacteristic
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50Dic32C2211yescharacteristic
51Dic32C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×#124—31no—
54C3×#123—31no—
55#947—31no—
56Q8×Dic3—31no—
57#940—31no—
58#940—31no—
59#935—31no—
60D4×Dic3—31no—
61C4×#67—31no—
62C4×#67—31no—
63#894—31no—
64#894—31no—
65#887—31no—
66#887—31no—
67#886—31no—
68#886—31no—
69#885—31no—
70#885—31no—
71#885—31no—
72#885—31no—
73#885—31no—
74#885D311yescharacteristic, complemented
75#883—31no—
76#883D311yescharacteristic, complemented
77#67⋊C4—31no—
78#67⋊C4—31no—
79#67⋊C4—31no—
80#67⋊C4—31no—
81#67⋊C4—61no—
82C4×D12—31no—
83C4×Dic6—31no—
84C23×C32C2311yescharacteristic
85C22×C3⋊D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C2×C32×C4C2311yescharacteristic
89C2×C32×C4C2311yescharacteristic
90#638—22no—
91C2×C3⋊Dic3—41no—
92C2×C3⋊Dic3C2311yescharacteristic
93C2×C3⋊Dic3C2311yescharacteristic
94C2×C3⋊Dic3C2311yescharacteristic
95C4×C3⋊D3—41no—
96C3×#67—22no—
97C3×#67—41no—
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C3×C4×D3—41no—
105C3×Dic6—22no—
106#627—22no—
107#626—22no—
108#626—41no—
109#625—22no—
110D3×Dic3—41no—
111#559—91noSylow
112C2×C3×Q8—31no—
113C2×C3×D4—31no—
114C2×C3×D4—31no—
115C22×C3×C4—31no—
116C2×#67—31no—
117C2×#67—31no—
118C2×#67—31no—
119C2×#67—31no—
120C2×#67—31no—
121C2×#67—31no—
122C2×#67—61no—
123C2×#67C2×D311yescharacteristic
124C2×#67C2×D311yescharacteristic
125C22×Dic3—31no—
126C22×Dic3—31no—
127C22×Dic3—31no—
128C22×Dic3—61no—
129C2×D12—31no—
130C2×D12—31no—
131C2×C4×D3—31no—
132C2×C4×D3—31no—
133C2×C4×D3—31no—
134C2×C4×D3—61no—
135C2×C4×D3C2×D311yescharacteristic
136C2×Dic6—31no—
137C2×Dic6—31no—
138C2×Dic6C2×D311yescharacteristic
139C3×C4⋊C4—31no—
140C3×C4⋊C4—31no—
141C3×C22⋊C4—21no—
142C3×C22⋊C4—31no—
143C3×C22⋊C4—31no—
144C3×C22⋊C4—31no—
145C3×C22⋊C4—31no—
146C3×C22⋊C4—31no—
147C3×C22⋊C4C2×D311yescharacteristic
148C3×C22⋊C4C2×D311yescharacteristic
149C3×C42—31no—
150#217—31no—
151#217—31no—
152#217—31no—
153#217C2×D311yescharacteristic
154#217C2×D311yescharacteristic
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)⋊C4—31no—
161(C2×D3)⋊C4—31no—
162(C2×D3)⋊C4—31no—
163(C2×D3)⋊C4—31no—
164(C2×D3)⋊C4—61no—
165(C2×D3)⋊C4C2×D311yescharacteristic
166(C2×D3)⋊C4C2×D311yescharacteristic
167(C2×D3)⋊C4C2×D311yescharacteristic
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3C2×D311yescharacteristic
172Dic3⋊C4—31no—
173Dic3⋊C4—31no—
174Dic3⋊C4—31no—
175Dic3⋊C4—31no—
176Dic3⋊C4—31no—
177Dic3⋊C4—61no—
178Dic3⋊C4C2×D311yescharacteristic
179C4×Dic3—31no—
180C4×Dic3—31no—
181C4×Dic3—31no—
182C4×Dic3—31no—
183C4×Dic3—31no—
184C4×Dic3—31no—
185C4×Dic3—31no—
186C4×Dic3—31no—
187C4×Dic3—61no—
188C4×Dic3C2×D311yescharacteristic
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32C2411yescharacteristic
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—
204C3⋊Dic3—21no—
205C3⋊Dic3—21no—
206C3⋊Dic3#4112yes—
207C3×Dic3—21no—
208C3×Dic3—21no—
209C3×Dic3—21no—
210C3×Dic3—21no—
211C3×Dic3—21no—
212C3×Dic3#4112yes—
213#127—91no—
214#124—31no—
215#124—91no—
216#124—91no—
217#124—91no—
218#123—31no—
219#123—91no—
220#123—91no—
221#123—91no—
222C4×Q8—91no—
223C4×Q8—91no—
224C4×D4—91no—
225C4×D4—91no—
226C4×D4—91no—
227C4×D4—91no—
228C23×C3—21no—
229C23×C3—31no—
230C23×C3—31no—
231C23×C3C22×D311yescharacteristic
232C23×C3C22×D311yescharacteristic
233C22×D3—31no—
234C22×D3—31no—
235C22×D3—61no—
236C22×D3C22×D311yescharacteristic
237C22×D3C22×D311yescharacteristic
238C3×Q8—62no—
239C3×D4—62no—
240C3×D4—121no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—121no—
250C2×C3×C4C22×D311yescharacteristic
251C2×C3×C4C22×D311yescharacteristic
252C2×C3×C4C22×D311yescharacteristic
253C2×C3×C4C22×D311yescharacteristic
254#67—22no—
255#67—41no—
256#67—62no—
257#67—62no—
258#67—62no—
259#67—62no—
260#67—64no—
261#67—62no—
262#67—121no—
263C2×Dic3—31no—
264C2×Dic3—31no—
265C2×Dic3—31no—
266C2×Dic3—31no—
267C2×Dic3—31no—
268C2×Dic3—31no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—121no—
273C2×Dic3—121no—
274C2×Dic3—121no—
275C2×Dic3—122no—
276C2×Dic3C22×D311yescharacteristic
277C2×Dic3C22×D311yescharacteristic
278C2×Dic3C22×D311yescharacteristic
279C2×Dic3C22×D311yescharacteristic
280C2×Dic3C22×D311yescharacteristic
281C2×Dic3C22×D311yescharacteristic
282D12—62no—
283D12—121no—
284C4×D3—41no—
285C4×D3—121no—
286C4×D3—121no—
287C4×D3—122no—
288C4×D3—121no—
289Dic6—22no—
290Dic6—62no—
291Dic6—62no—
292C2×C32—41no—
293C2×C32C2×#4111yescharacteristic
294C2×C32C2×#4111yescharacteristic
295C2×C32H(5,𝔽2)11yescharacteristic
296C3⋊D3—41no—
297C3×D3—41no—
298C3×D3—41no—
299C2×Q8—31no—
300C2×Q8—91no—
301C2×D4—31no—
302C2×D4—31no—
303C2×D4—91no—
304C2×D4—91no—
305C2×D4—91no—
306C2×D4—91no—
307C22×C4—31no—
308C22×C4—91no—
309C22×C4—91no—
310C22×C4—91no—
311C4⋊C4—31no—
312C4⋊C4—31no—
313C4⋊C4—91no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4—91no—
317C22⋊C4—31no—
318C22⋊C4—31no—
319C22⋊C4—31no—
320C22⋊C4—31no—
321C22⋊C4—31no—
322C22⋊C4—91no—
323C22⋊C4—91no—
324C22⋊C4—91no—
325C22⋊C4—91no—
326C22⋊C4—91no—
327C22⋊C4—91no—
328C22⋊C4D3211yescharacteristic
329C42—31no—
330C42—91no—
331C42—91no—
332C42—91no—
333C42—91no—
334C22×C3—22no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—61no—
345C22×C3—61no—
346C22×C3—61no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—61no—
350C22×C3C23×D311yescharacteristic
351C22×C3C23×D311yescharacteristic
352C2×D3—21no—
353C2×D3—21no—
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—21no—
357C2×D3—21no—
358C2×D3—62no—
359C2×D3—61no—
360C2×D3—61no—
361C2×D3—61no—
362C2×D3—61no—
363C2×D3—61no—
364C2×D3—61no—
365C2×D3—121no—
366C2×D3—121no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—32no—
372C3×C4—41no—
373C3×C4—41no—
374C3×C4—61no—
375C3×C4—61no—
376C3×C4—61no—
377C3×C4—61no—
378C3×C4—61no—
379Dic3—21no—
380Dic3—21no—
381Dic3—21no—
382Dic3—21no—
383Dic3—21no—
384Dic3—32no—
385Dic3—32no—
386Dic3—62no—
387Dic3—62no—
388Dic3—61no—
389Dic3—61no—
390Dic3—61no—
391Dic3—61no—
392Dic3—121no—
393Dic3#23712yes—
394C32#55911yescharacteristic, complemented, Sylow
395C23—31no—
396C23—31no—
397C23—91no—
398C23C2×D3211yescharacteristic
399Q8—62no—
400Q8—182no—
401D4—62no—
402D4—121no—
403D4—182no—
404D4—182no—
405D4—182no—
406D4—361no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—31no—
412C2×C4—31no—
413C2×C4—91no—
414C2×C4—91no—
415C2×C4—91no—
416C2×C4—121no—
417C2×C4—361no—
418C2×C4—361no—
419C2×C4—361no—
420C2×C4C2×D3211yescharacteristic
421C2×C4C2×D3211yescharacteristic
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—42no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—121no—
429C2×C3—121no—
430C2×C3C2×#23711yescharacteristic
431C2×C3#100411yescharacteristic
432C2×C3D3×#4111yescharacteristic
433C2×C3D3×#4111yescharacteristic
434C2×C3D3×#4111yescharacteristic
435C2×C3#100911yescharacteristic
436D3—41no—
437D3—41no—
438D3—122no—
439D3—121no—
440D3—121no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—61no—
445C22—61no—
446C22—61no—
447C22—61no—
448C22—61no—
449C22—61no—
450C22—181no—
451C22—181no—
452C22—181no—
453C22C22×D3211yescharacteristic
454C4—21no—
455C4—21no—
456C4—32no—
457C4—61no—
458C4—61no—
459C4—61no—
460C4—61no—
461C4—61no—
462C4—92no—
463C4—181no—
464C4—181no—
465C3—21no—
466C3#566411yescharacteristic, complemented
467C3#568111yescharacteristic, complemented
468C2—41no—
469C2—121no—
470C2—121no—
471C2—361no—
472C2#6430611yescharacteristic
473C2D3×#23711yescharacteristic
474C2#6431411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
prime radical#63979C2
hypercenterC22⋊C4D32
hypocenterC32#559
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #64090C2semidirect
2. C3×#885C2semidirect
3. C3×#883C2semidirect
5. #67⋊Dic3C2semidirect
6. #63962C2semidirect
7. #63953C2semidirect
8. #63915C2semidirect
9. #63907C2semidirect
10. C4×#626C2semidirect
11. #63890C2semidirect
12. #63879C2semidirect
13. #63853C2semidirect
14. #63843C2semidirect
15. #63837C2semidirect
74. #885D3semidirect
76. #883D3semidirect
394. C32#559semidirect
466. C3#5664semidirect
467. C3#5681semidirect

Maximal subgroups

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

subgroupindex
#640902
C3×#8852
C3×#8832
#639792
#67⋊Dic32
#639622
#639532
#639152
#639072
C4×#6262
#638902
#638792
#638532
#638432
#638372
#56813
#56643