Finite Group Databasebeta

← #10594943

Subgroups — #10594943

Order 576.

515 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64138C211yesmaximal, characteristic, complemented
2C3×C4×#67C211yesmaximal, characteristic, complemented
3C3×C4×#67C211yesmaximal, characteristic, complemented
4#63971C211yesmaximal, characteristic, complemented
5#63970C211yesmaximal, characteristic, complemented
6#63958C211yesmaximal, characteristic, complemented
7#63956C211yesmaximal, characteristic, complemented
8#63906C211yesmaximal, characteristic, complemented
9#63880C211yesmaximal, characteristic, complemented
10#63879C211yesmaximal, characteristic, complemented
11D3×Dic3⋊C4C211yesmaximal, characteristic, complemented
12#63867C211yesmaximal, characteristic, complemented
13#63859C211yesmaximal, characteristic, complemented
14#63850C211yesmaximal, characteristic, complemented
15#63836C211yesmaximal, characteristic, complemented
16#5558—31nomaximal
17#5540—31nomaximal
18C22×C32×C4C2211yescharacteristic
19C2×#638C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C2×C3×#67C2211yescharacteristic
22C2×C3×C4×D3C2211yescharacteristic
23C2×C3×C4×D3C2211yescharacteristic
24C2×#626C2211yescharacteristic
25C2×#626C2211yescharacteristic
26C2×#625C2211yescharacteristic
27C2×#624C2211yescharacteristic
28C2×D3×Dic3C2211yescharacteristic
29C2×D3×Dic3C2211yescharacteristic
30#3757C2211yescharacteristic
31#3752C2211yescharacteristic
32#3752C2211yescharacteristic
33#3750C2211yescharacteristic
34#3750C2211yescharacteristic
35C3×#217C2211yescharacteristic
36C3×#217C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×(C2×D3)⋊C4C2211yescharacteristic
39C3×Dic3⋊C4C2211yescharacteristic
40C3×Dic3⋊C4C2211yescharacteristic
41C3×C4×Dic3C2211yescharacteristic
42C3×C4×Dic3C2211yescharacteristic
43#3724C2211yescharacteristic
44#3724C2211yescharacteristic
45Dic3⋊Dic3C2211yescharacteristic
46Dic3⋊Dic3C2211yescharacteristic
47Dic3⋊Dic3C2211yescharacteristic
48Dic3⋊Dic3C2211yescharacteristic
49#3722C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×C4×D4—31no—
54C3×C4×D4—31no—
55#939—31no—
56#939—31no—
57#938—31no—
58#938—31no—
59#930—31no—
60#929—31no—
61#929—31no—
62#929—31no—
63#929—61no—
64C4×#67—31no—
65C4×#67D311yescharacteristic, complemented
66C4×#67D311yescharacteristic, complemented
67C2×Dic3⋊C4—31no—
68#894—31no—
69#892—31no—
70D3×C4⋊C4—31no—
71#889—31no—
72#884—31no—
73#884—31no—
74#884—31no—
75#67⋊C4—31no—
76#67⋊C4—31no—
77#879—31no—
78#879—31no—
79#879—31no—
80#876—31no—
81C4×D12—31no—
82#872—31no—
83#870—31no—
84C23×C32C2311yescharacteristic
85C22×C3⋊D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C2×C32×C4—41no—
89C2×C32×C4C2311yescharacteristic
90C2×C32×C4C2311yescharacteristic
91#638—41no—
92C2×C3⋊Dic3C2311yescharacteristic
93C2×C3⋊Dic3C2311yescharacteristic
94C2×C3⋊Dic3C2311yescharacteristic
95C3×#67—22no—
96C3×#67—22no—
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C3×C4×D3—41no—
104C3×C4×D3—41no—
105#626—22no—
106#626—41no—
107#625—22no—
108#624—22no—
109D3×Dic3—22no—
110D3×Dic3—41no—
111#553—91noSylow
112C2×C3×D4—31no—
113C2×C3×D4—31no—
114C22×C3×C4—21no—
115C22×C3×C4—31no—
116C22×C3×C4—31no—
117C22×C3×C4C2×D311yescharacteristic
118C22×C3×C4C2×D311yescharacteristic
119C2×#67—31no—
120C2×#67—31no—
121C2×#67—31no—
122C2×#67—31no—
123C2×#67—31no—
124C2×#67—31no—
125C2×#67—61no—
126C2×#67C2×D311yescharacteristic
127C2×#67C2×D311yescharacteristic
128C22×Dic3—31no—
129C22×Dic3—31no—
130C2×D12—31no—
131C2×D12—31no—
132C2×C4×D3—31no—
133C2×C4×D3—31no—
134C2×C4×D3—31no—
135C2×C4×D3—31no—
136C2×C4×D3C2×D311yescharacteristic
137C2×C4×D3C2×D311yescharacteristic
138C3×C4⋊C4—31no—
139C3×C4⋊C4—31no—
140C3×C22⋊C4—31no—
141C3×C22⋊C4—31no—
142C3×C22⋊C4—31no—
143C3×C22⋊C4—31no—
144C3×C42—31no—
145C3×C42—31no—
146#217—31no—
147#217—31no—
148#217—31no—
149#217—31no—
150#217—61no—
151#217C2×D311yescharacteristic
152#217C2×D311yescharacteristic
153(C2×D3)⋊C4—31no—
154(C2×D3)⋊C4—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)⋊C4—31no—
161(C2×D3)⋊C4—61no—
162(C2×D3)⋊C4—61no—
163(C2×D3)⋊C4C2×D311yescharacteristic
164(C2×D3)⋊C4C2×D311yescharacteristic
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—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—31no—
181Dic3⋊C4—61no—
182Dic3⋊C4—61no—
183Dic3⋊C4C2×D311yescharacteristic
184Dic3⋊C4C2×D311yescharacteristic
185C4×Dic3—31no—
186C4×Dic3—31no—
187C4×Dic3C2×D311yescharacteristic
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#4112yes—
202C32×C4—21no—
203C32×C4—21no—
204C3⋊Dic3—21no—
205C3⋊Dic3—21no—
206C3⋊Dic3—21no—
207C3×Dic3—21no—
208C3×Dic3—21no—
209C3×Dic3—21no—
210C3×Dic3—21no—
211C3×Dic3—21no—
212C3×Dic3#4112yes—
213#126—91no—
214#126—91no—
215#125—91no—
216#123—91no—
217#123—91no—
218#121—91no—
219#121—91no—
220#121—91no—
221#121—91no—
222C4×D4—31no—
223C4×D4—31no—
224C4×D4—91no—
225C4×D4—91no—
226#117—91no—
227C2×C4⋊C4—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×D4—62no—
239C3×D4—62no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4—121no—
252C2×C3×C4—121no—
253C2×C3×C4C22×D311yescharacteristic
254C2×C3×C4C22×D311yescharacteristic
255C2×C3×C4C22×D311yescharacteristic
256C2×C3×C4C22×D311yescharacteristic
257#67—22no—
258#67—22no—
259#67—62no—
260#67—62no—
261#67—62no—
262#67—121no—
263#67—121no—
264#67—121no—
265#67—122no—
266C2×Dic3—31no—
267C2×Dic3—31no—
268C2×Dic3—31no—
269C2×Dic3—31no—
270C2×Dic3—31no—
271C2×Dic3—31no—
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—61no—
275C2×Dic3—62no—
276C2×Dic3—121no—
277C2×Dic3C22×D311yescharacteristic
278C2×Dic3C22×D311yescharacteristic
279C2×Dic3C22×D311yescharacteristic
280C2×Dic3C22×D311yescharacteristic
281C2×Dic3C22×D311yescharacteristic
282C2×Dic3C22×D311yescharacteristic
283D12—62no—
284D12—121no—
285C4×D3—41no—
286C4×D3—41no—
287C4×D3—62no—
288C4×D3—62no—
289C4×D3—62no—
290C4×D3—121no—
291C2×C32—41no—
292C2×C32C2×#4111yescharacteristic
293C2×C32C2×#4111yescharacteristic
294C2×C32H(5,𝔽2)11yescharacteristic
295C3⋊D3—41no—
296C3×D3—22no—
297C3×D3—41no—
298C2×D4—31no—
299C2×D4—31no—
300C2×D4—91no—
301C2×D4—91no—
302C2×D4—91no—
303C2×D4—91no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—91no—
307C22×C4—91no—
308C22×C4—91no—
309C22×C4D3211yescharacteristic
310C4⋊C4—31no—
311C4⋊C4—31no—
312C4⋊C4—91no—
313C4⋊C4—91no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4—91no—
317C4⋊C4—91no—
318C4⋊C4—91no—
319C4⋊C4—91no—
320C22⋊C4—31no—
321C22⋊C4—31no—
322C22⋊C4—31no—
323C22⋊C4—31no—
324C22⋊C4—91no—
325C22⋊C4—91no—
326C22⋊C4—91no—
327C22⋊C4—91no—
328C22⋊C4—91no—
329C22⋊C4—91no—
330C42—31no—
331C42—31no—
332C42—91no—
333C22×C3—22no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—32no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—61no—
345C22×C3—61no—
346C22×C3—61no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3C23×D311yescharacteristic
350C22×C3C23×D311yescharacteristic
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—21no—
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—61no—
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—61no—
360C2×D3—61no—
361C2×D3—61no—
362C2×D3—121no—
363C2×D3—121no—
364C2×D3—121no—
365C2×D3#23512yes—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—32no—
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—61no—
384Dic3—61no—
385Dic3—61no—
386Dic3—61no—
387Dic3—61no—
388Dic3—61no—
389Dic3—121no—
390Dic3—121no—
391Dic3—121no—
392Dic3#23512yes—
393C32#55311yescharacteristic, complemented, Sylow
394C23—31no—
395C23—31no—
396C23—91no—
397C23C2×D3211yescharacteristic
398D4—62no—
399D4—62no—
400D4—182no—
401D4—182no—
402D4—361no—
403D4—361no—
404C2×C4—31no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—41no—
411C2×C4—91no—
412C2×C4—91no—
413C2×C4—91no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—182no—
417C2×C4—182no—
418C2×C4—361no—
419C2×C4C2×D3211yescharacteristic
420C2×C4C2×D3211yescharacteristic
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—42no—
425C2×C3—41no—
426C2×C3—41no—
427C2×C3—62no—
428C2×C3—121no—
429C2×C3C2×#23511yescharacteristic
430C2×C3C2×#23511yescharacteristic
431C2×C3#100411yescharacteristic
432C2×C3#100411yescharacteristic
433C2×C3D3×#4111yescharacteristic
434C2×C3D3×#4111yescharacteristic
435D3—22no—
436D3—41no—
437D3—121no—
438D3—121no—
439D3—241no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—32no—
444C22—61no—
445C22—61no—
446C22—61no—
447C22—61no—
448C22—61no—
449C22—181no—
450C22—181no—
451C22—181no—
452C22C22×D3211yescharacteristic
453C4—21no—
454C4—21no—
455C4—32no—
456C4—61no—
457C4—61no—
458C4—61no—
459C4—61no—
460C4—61no—
461C4—181no—
462C4—181no—
463C4—181no—
464C3—21no—
465C3#554011yescharacteristic, complemented
466C3#555811yescharacteristic, complemented
467C2—41no—
468C2—62no—
469C2—121no—
470C2—361no—
471C2D3×#23511yescharacteristic
472C2D3×#23511yescharacteristic
473C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#553
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #64138C2semidirect
2. C3×C4×#67C2semidirect
3. C3×C4×#67C2semidirect
4. #63971C2semidirect
5. #63970C2semidirect
6. #63958C2semidirect
7. #63956C2semidirect
8. #63906C2semidirect
9. #63880C2semidirect
10. #63879C2semidirect
11. D3×Dic3⋊C4C2semidirect
12. #63867C2semidirect
13. #63859C2semidirect
14. #63850C2semidirect
15. #63836C2semidirect
65. C4×#67D3semidirect
66. C4×#67D3semidirect
393. C32#553semidirect
465. C3#5540semidirect
466. C3#5558semidirect

Maximal subgroups

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

subgroupindex
#641382
C3×C4×#672
#639712
#639702
#639582
#639562
#639062
#638802
#638792
D3×Dic3⋊C42
#638672
#638592
#638502
#638362
#55583
#55403