Finite Group Databasebeta

← #10594239

Subgroups — #10594239

Order 576.

477 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64087C211yesmaximal, characteristic, complemented
2C3×#884C211yesmaximal, characteristic, complemented
3C3×#883C211yesmaximal, characteristic, complemented
4#63980C211yesmaximal, characteristic, complemented
5#63970C211yesmaximal, characteristic, complemented
6#63962C211yesmaximal, characteristic, complemented
7#63959C211yesmaximal, characteristic, complemented
8#63913C211yesmaximal, characteristic, complemented
9#63908C211yesmaximal, characteristic, complemented
10#63905C211yesmaximal, characteristic, complemented
11#63899C211yesmaximal, characteristic, complemented
12#63884C211yesmaximal, characteristic, complemented
13#63857C211yesmaximal, characteristic, complemented
14#63851C211yesmaximal, characteristic, complemented
15#63845C211yesmaximal, characteristic, complemented
16#5667—31nomaximal
17#5621—31nomaximal
18C22×C3⋊Dic3C2211yescharacteristic
19C2×#634C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C2×C3×#67C2211yescharacteristic
22C2×C3×#67C2211yescharacteristic
23C2×C3×D12C2211yescharacteristic
24C2×C3×C4×D3C2211yescharacteristic
25C2×C3×C4×D3C2211yescharacteristic
26C2×#627C2211yescharacteristic
27C2×#625C2211yescharacteristic
28C2×#625C2211yescharacteristic
29C2×D3×Dic3C2211yescharacteristic
30C32×C22⋊C4C2211yescharacteristic
31#3757C2211yescharacteristic
32C4⋊(C3⋊Dic3)C2211yescharacteristic
33#3750C2211yescharacteristic
34#3750C2211yescharacteristic
35C3×#217C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×C4⋊Dic3C2211yescharacteristic
39C3×Dic3⋊C4C2211yescharacteristic
40C3×Dic3⋊C4C2211yescharacteristic
41#3724C2211yescharacteristic
42Dic3⋊Dic3C2211yescharacteristic
43Dic3⋊Dic3C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45(C2×D3)⋊Dic3C2211yescharacteristic
46(C2×D3)⋊Dic3C2211yescharacteristic
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×#123—31no—
54C3×#121—31no—
55#939—31no—
56#939—31no—
57#939—31no—
58#938—31no—
59#938—31no—
60#936—31no—
61#935—31no—
62#935—31no—
63#930—31no—
64#929—31no—
65#924—31no—
66#924—31no—
67#897—31no—
68#896—31no—
69#896—31no—
70#894—31no—
71#890—31no—
72#889—31no—
73#886—31no—
74#885—31no—
75#885—31no—
76#885—31no—
77#884D311yescharacteristic, complemented
78#883—31no—
79#883D311yescharacteristic, complemented
80#878—31no—
81#878—31no—
82#878—31no—
83#878—61no—
84C23×C32C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C22×C3×D3C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C2×C32×C4C2311yescharacteristic
89C2×C32×C4C2311yescharacteristic
90C2×C3⋊Dic3—41no—
91C2×C3⋊Dic3C2311yescharacteristic
92C2×C3⋊Dic3C2311yescharacteristic
93C2×C3⋊Dic3C2311yescharacteristic
94C2×C3⋊Dic3C2311yescharacteristic
95#634—41no—
96C3×#67—41no—
97C3×#67—41no—
98C3×#67—41no—
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C3×D12—41no—
105C3×C4×D3—41no—
106C3×C4×D3—41no—
107#627—41no—
108#625—41no—
109#625—41no—
110D3×Dic3—41no—
111#562—91noSylow
112C2×C3×D4—31no—
113C2×C3×D4—31no—
114C2×C3×D4—31no—
115C2×C3×D4—31no—
116C22×C3×C4—31no—
117C22×C3×C4—31no—
118C2×#67—31no—
119C2×#67—31no—
120C2×#67—31no—
121C2×#67—31no—
122C2×#67C2×D311yescharacteristic
123C2×#67C2×D311yescharacteristic
124C2×#67C2×D311yescharacteristic
125C22×Dic3—31no—
126C22×Dic3—31no—
127C22×Dic3—31no—
128C22×Dic3—61no—
129C2×D12C2×D311yescharacteristic
130C2×C4×D3—31no—
131C2×C4×D3C2×D311yescharacteristic
132C2×C4×D3C2×D311yescharacteristic
133C2×Dic6—31no—
134C2×Dic6—31no—
135C2×Dic6—31no—
136C2×Dic6—31no—
137C2×Dic6—61no—
138C3×C4⋊C4—31no—
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⋊C4C2×D311yescharacteristic
146C3×C22⋊C4C2×D311yescharacteristic
147#217—31no—
148#217—31no—
149#217—31no—
150#217—31no—
151#217—31no—
152#217—31no—
153#217—31no—
154#217—31no—
155#217—31no—
156#217—61no—
157#217C2×D311yescharacteristic
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—31no—
165(C2×D3)⋊C4C2×D311yescharacteristic
166(C2×D3)⋊C4C2×D311yescharacteristic
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3—61no—
173C4⋊Dic3C2×D311yescharacteristic
174Dic3⋊C4—31no—
175Dic3⋊C4—31no—
176Dic3⋊C4—31no—
177Dic3⋊C4—31no—
178Dic3⋊C4—31no—
179Dic3⋊C4—31no—
180Dic3⋊C4—31no—
181Dic3⋊C4—31no—
182Dic3⋊C4—31no—
183Dic3⋊C4—61no—
184Dic3⋊C4—61no—
185Dic3⋊C4C2×D311yescharacteristic
186Dic3⋊C4C2×D311yescharacteristic
187C4×Dic3—31no—
188C4×Dic3—31no—
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—21no—
207C3⋊Dic3—21no—
208C3×Dic3—21no—
209C3×Dic3—21no—
210C3×Dic3—21no—
211C3×Dic3—21no—
212C3×Dic3—21no—
213#125—91no—
214#124—91no—
215#124—91no—
216#123—31no—
217#123—91no—
218#123—91no—
219#123—91no—
220C22⋊Q8—91no—
221C22⋊Q8—91no—
222C22⋊Q8—91no—
223C22⋊Q8—91no—
224#121—31no—
225#121—91no—
226#121—91no—
227#121—91no—
228C23×C3—21no—
229C23×C3—31no—
230C23×C3—31no—
231C23×C3—31no—
232C23×C3C22×D311yescharacteristic
233C23×C3C22×D311yescharacteristic
234C22×D3C22×D311yescharacteristic
235C22×D3C22×D311yescharacteristic
236C22×D3C22×D311yescharacteristic
237C3×D4—121no—
238C3×D4—121no—
239C3×D4—121no—
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—121no—
249C2×C3×C4—121no—
250C2×C3×C4C22×D311yescharacteristic
251C2×C3×C4C22×D311yescharacteristic
252C2×C3×C4C22×D311yescharacteristic
253C2×C3×C4C22×D311yescharacteristic
254#67—41no—
255#67—41no—
256#67—41no—
257#67—121no—
258#67—121no—
259#67—121no—
260#67—121no—
261C2×Dic3—31no—
262C2×Dic3—31no—
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—61no—
273C2×Dic3—121no—
274C2×Dic3—121no—
275C2×Dic3—121no—
276C2×Dic3—122no—
277C2×Dic3C22×D311yescharacteristic
278C2×Dic3C22×D311yescharacteristic
279C2×Dic3C22×D311yescharacteristic
280C2×Dic3C22×D311yescharacteristic
281C2×Dic3C22×D311yescharacteristic
282D12—41no—
283C4×D3—41no—
284C4×D3—41no—
285C4×D3—121no—
286Dic6—121no—
287Dic6—121no—
288Dic6—121no—
289Dic6—121no—
290Dic6—122no—
291C2×C32—41no—
292C2×C32H(5,𝔽2)11yescharacteristic
293C2×C32H(5,𝔽2)11yescharacteristic
294C2×C322−1+411yescharacteristic
295C3×D3—41no—
296C3×D3—41no—
297C3×D3—41no—
298C2×Q8—91no—
299C2×Q8—91no—
300C2×D4—31no—
301C2×D4—31no—
302C2×D4—31no—
303C2×D4—31no—
304C2×D4—91no—
305C2×D4—91no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—91no—
309C22×C4—91no—
310C4⋊C4—31no—
311C4⋊C4—31no—
312C4⋊C4—31no—
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—91no—
324C22⋊C4—91no—
325C22⋊C4—91no—
326C22⋊C4—91no—
327C22⋊C4—91no—
328C22⋊C4—91no—
329C22⋊C4—91no—
330C22⋊C4—91no—
331C22⋊C4D3211yescharacteristic
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—41no—
342C22×C3—41no—
343C22×C3—61no—
344C22×C3—61no—
345C22×C3—61no—
346C22×C3—61no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—61no—
350C22×C3—61no—
351C22×C3—61no—
352C22×C3C23×D311yescharacteristic
353C22×C3C23×D311yescharacteristic
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—21no—
357C2×D3—21no—
358C2×D3—21no—
359C2×D3—21no—
360C2×D3—21no—
361C2×D3—21no—
362C2×D3—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—41no—
368C3×C4—41no—
369C3×C4—61no—
370C3×C4—61no—
371C3×C4—61no—
372C3×C4—61no—
373C3×C4—61no—
374Dic3—21no—
375Dic3—21no—
376Dic3—21no—
377Dic3—21no—
378Dic3—21no—
379Dic3—62no—
380Dic3—61no—
381Dic3—61no—
382Dic3—61no—
383Dic3—61no—
384Dic3—61no—
385Dic3—61no—
386Dic3—61no—
387Dic3—61no—
388Dic3—121no—
389Dic3—121no—
390Dic3—121no—
391C32#56211yescharacteristic, complemented, Sylow
392C23—31no—
393C23—31no—
394C23—31no—
395C23C2×D3211yescharacteristic
396Q8—361no—
397Q8—361no—
398D4—121no—
399D4—121no—
400D4—121no—
401D4—121no—
402D4—361no—
403D4—361no—
404C2×C4—31no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—91no—
410C2×C4—91no—
411C2×C4—91no—
412C2×C4—91no—
413C2×C4—121no—
414C2×C4—121no—
415C2×C4—361no—
416C2×C4—361no—
417C2×C4C2×D3211yescharacteristic
418C2×C4C2×D3211yescharacteristic
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—42no—
423C2×C3—41no—
424C2×C3—41no—
425C2×C3—121no—
426C2×C3—121no—
427C2×C3—121no—
428C2×C3#100411yescharacteristic
429C2×C3#100411yescharacteristic
430C2×C3#100411yescharacteristic
431C2×C3#100911yescharacteristic
432C2×C3#101011yescharacteristic
433C2×C3#101011yescharacteristic
434D3—41no—
435D3—41no—
436D3—41no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—61no—
441C22—61no—
442C22—61no—
443C22—61no—
444C22—61no—
445C22—61no—
446C22—61no—
447C22—61no—
448C22—61no—
449C22C22×D3211yescharacteristic
450C4—21no—
451C4—21no—
452C4—61no—
453C4—61no—
454C4—61no—
455C4—61no—
456C4—61no—
457C4—181no—
458C4—181no—
459C4—181no—
460C4—181no—
461C3—21no—
462C3#562111yescharacteristic, complemented
463C3#566711yescharacteristic, complemented
464C2—41no—
465C2—121no—
466C2—121no—
467C2—121no—
468C2#6429811yescharacteristic
469C2#6431311yescharacteristic
470C2#6431411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
hypercenterC22⋊C4D32
hypocenterC32#562
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #64087C2semidirect
2. C3×#884C2semidirect
3. C3×#883C2semidirect
4. #63980C2semidirect
5. #63970C2semidirect
6. #63962C2semidirect
7. #63959C2semidirect
8. #63913C2semidirect
9. #63908C2semidirect
10. #63905C2semidirect
11. #63899C2semidirect
12. #63884C2semidirect
13. #63857C2semidirect
14. #63851C2semidirect
15. #63845C2semidirect
77. #884D3semidirect
79. #883D3semidirect
391. C32#562semidirect
462. C3#5621semidirect
463. C3#5667semidirect

Maximal subgroups

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

subgroupindex
#640872
C3×#8842
C3×#8832
#639802
#639702
#639622
#639592
#639132
#639082
#639052
#638992
#638842
#638572
#638512
#638452
#56673
#56213