Finite Group Databasebeta

← #11754782

Subgroups — #11754782

Order 960.

580 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#1204C211yesmaximal, characteristic, complemented
2C2×#6780C211yesmaximal, characteristic, complemented
3D4×Dic15C211yesmaximal, characteristic, complemented
4C5×#939C211yesmaximal, characteristic, complemented
5C3×#4085C211yesmaximal, characteristic, complemented
6#91576C211yesmaximal, characteristic, complemented
7#91571C211yesmaximal, characteristic, complemented
8Dic3×C22⋊D5C211yesmaximal, characteristic, complemented
9Dic5×#67C211yesmaximal, characteristic, complemented
10#91534C211yesmaximal, characteristic, complemented
11#91525C211yesmaximal, characteristic, complemented
12#91464C211yesmaximal, characteristic, complemented
13#91422C211yesmaximal, characteristic, complemented
14#91417C211yesmaximal, characteristic, complemented
15#91325C211yesmaximal, characteristic, complemented
16#65932—31nomaximal, Hall
17C2×C3×C5×D4C2211yescharacteristic
18C22×Dic15C2211yescharacteristic
19C22×Dic15C2211yescharacteristic
20C2×C5×#67C2211yescharacteristic
21C2×C5×#67C2211yescharacteristic
22C22×C5×Dic3C2211yescharacteristic
23C2×C3×C22⋊D5C2211yescharacteristic
24C2×C3×C22⋊D5C2211yescharacteristic
25C22×C3×Dic5C2211yescharacteristic
26C2×#1204—24no—
27C2×#1204C2211yescharacteristic
28C2×#1204C2211yescharacteristic
29C2×#1204C2211yescharacteristic
30C2×#1204C2211yescharacteristic
31C2×#1201C2211yescharacteristic
32C2×D3×Dic5C2211yescharacteristic
33#6780—24no—
34C2×D5×Dic3C2211yescharacteristic
35#6719C2211yescharacteristic
36#6719C2211yescharacteristic
37C4⋊Dic15C2211yescharacteristic
38C4×Dic15C2211yescharacteristic
39C5×#217C2211yescharacteristic
40C5×(C2×D3)⋊C4C2211yescharacteristic
41C5×Dic3⋊C4C2211yescharacteristic
42C3×C22⋊Dic5C2211yescharacteristic
43C3×#685C2211yescharacteristic
44C3×#683C2211yescharacteristic
45Dic3⋊Dic5C2211yescharacteristic
46#6669C2211yescharacteristic
47#6668C2211yescharacteristic
48(C2×D3)⋊Dic5C2211yescharacteristic
49(C2×D3)⋊Dic5C2211yescharacteristic
50#6665C2211yescharacteristic
51#6665C2211yescharacteristic
52Dic3×Dic5C2211yescharacteristic
53Dic3×Dic5C2211yescharacteristic
54#5598—51nomaximal, Hall
55C2×#711—31no—
56C22×Dic10—31no—
57C5×#121—31no—
58#4085—31no—
59#4085D311yescharacteristic, complemented
60#4082—31no—
61D4×Dic5—31no—
62D4×Dic5—31no—
63C4×C22⋊D5—31no—
64#4043—31no—
65#4033—31no—
66#4031—31no—
67#4031—31no—
68#4024—31no—
69#4024—31no—
70C23×C3×C5C2311yescharacteristic
71C23×C3×C5C2311yescharacteristic
72C22×C5×D3C2311yescharacteristic
73C22×C3×D5C2311yescharacteristic
74C3×C5×D4—22no—
75C2×C3×C4×C5C2311yescharacteristic
76C2×Dic15—22no—
77C2×Dic15—22no—
78C2×Dic15C2311yescharacteristic
79C2×Dic15C2311yescharacteristic
80C2×Dic15C2311yescharacteristic
81C2×Dic15C2311yescharacteristic
82C5×#67—22no—
83C5×#67—22no—
84C2×C5×Dic3—22no—
85C2×C5×Dic3C2311yescharacteristic
86C2×C5×Dic3C2311yescharacteristic
87C2×C5×Dic3C2311yescharacteristic
88C3×C22⋊D5—22no—
89C3×C22⋊D5—22no—
90C2×C3×Dic5—22no—
91C2×C3×Dic5C2311yescharacteristic
92C2×C3×Dic5C2311yescharacteristic
93C2×C3×Dic5C2311yescharacteristic
94#1204—22no—
95#1204—22no—
96#1204—22no—
97#1204D414yescomplemented
98#1201—22no—
99D3×Dic5—22no—
100D5×Dic3—22no—
101C2×#237—51no—
102C22×Dic6—51no—
103C3×#121—51no—
104#939—51no—
105#939D511yescharacteristic, complemented
106#936—51no—
107D4×Dic3—51no—
108D4×Dic3—51no—
109C4×#67—51no—
110#897—51no—
111#887—51no—
112#885—51no—
113#885—51no—
114#878—51no—
115#878—51no—
116C2×C5×D4—31no—
117C2×C5×D4—31no—
118C2×C5×D4C2×D311yescharacteristic
119C22×C4×C5—31no—
120C2×C22⋊D5—31no—
121C2×C22⋊D5C2×D311yescharacteristic
122C2×C22⋊D5C2×D311yescharacteristic
123C22×Dic5—31no—
124C22×Dic5—31no—
125C22×Dic5—31no—
126C22×Dic5C2×D311yescharacteristic
127#711—64no—
128C2×C4×D5—31no—
129C2×Dic10—31no—
130C2×Dic10—31no—
131C2×Dic10—31no—
132C2×Dic10—31no—
133C2×Dic10—64no—
134C5×C4⋊C4—31no—
135C5×C22⋊C4—31no—
136C5×C22⋊C4—31no—
137C22⋊Dic5—31no—
138C22⋊Dic5—31no—
139C22⋊Dic5—31no—
140C22⋊Dic5—31no—
141C22⋊Dic5C2×D311yescharacteristic
142#685—31no—
143#685—31no—
144#685C2×D311yescharacteristic
145C4⋊Dic5—31no—
146C4⋊Dic5—31no—
147#683—31no—
148#683—31no—
149#683C2×D311yescharacteristic
150C4×Dic5—31no—
151C4×Dic5—31no—
152C4×Dic5—31no—
153#548—151noSylow
154C22×C3×C5—21no—
155C22×C3×C5—21no—
156C22×C3×C5—21no—
157C22×C3×C5—21no—
158C22×C3×C5—21no—
159C22×C3×C5#4112yescomplemented
160C22×C3×C5C2411yescharacteristic
161C2×C5×D3—21no—
162C2×C5×D3—21no—
163C2×C5×D3—21no—
164C2×C3×D5—21no—
165C2×C3×D5—21no—
166C2×C3×D5—21no—
167C3×C4×C5—21no—
168Dic15—21no—
169Dic15—21no—
170Dic15—21no—
171Dic15C2×D412yes—
172C3×Dic5—21no—
173C3×Dic5—21no—
174C3×Dic5C2×D412yes—
175C5×Dic3—21no—
176C5×Dic3—21no—
177C5×Dic3C2×D412yes—
178C2×C3×D4—51no—
179C2×C3×D4—51no—
180C2×C3×D4C2×D511yescharacteristic
181C22×C3×C4—51no—
182C2×#67—51no—
183C2×#67C2×D511yescharacteristic
184C2×#67C2×D511yescharacteristic
185C22×Dic3—51no—
186C22×Dic3—51no—
187C22×Dic3—51no—
188C22×Dic3C2×D511yescharacteristic
189#237—104no—
190C2×C4×D3—51no—
191C2×Dic6—51no—
192C2×Dic6—51no—
193C2×Dic6—51no—
194C2×Dic6—51no—
195C2×Dic6—104no—
196C3×C4⋊C4—51no—
197C3×C22⋊C4—51no—
198C3×C22⋊C4—51no—
199#217—51no—
200#217—51no—
201#217—51no—
202#217—51no—
203#217C2×D511yescharacteristic
204(C2×D3)⋊C4—51no—
205(C2×D3)⋊C4—51no—
206(C2×D3)⋊C4C2×D511yescharacteristic
207C4⋊Dic3—51no—
208C4⋊Dic3—51no—
209Dic3⋊C4—51no—
210Dic3⋊C4—51no—
211Dic3⋊C4C2×D511yescharacteristic
212C4×Dic3—51no—
213C4×Dic3—51no—
214C4×Dic3—51no—
215C23×C5—31no—
216C23×C5C22×D311yescharacteristic
217C23×C5C22×D311yescharacteristic
218C22×D5C22×D311yescharacteristic
219C5×D4—22no—
220C5×D4—62no—
221C5×D4—62no—
222C2×C4×C5—31no—
223C2×C4×C5—31no—
224C2×C4×C5—31no—
225C2×C4×C5—62no—
226C2×C4×C5C22×D311yescharacteristic
227C22⋊D5—22no—
228C22⋊D5—22no—
229C22⋊D5—62no—
230C2×Dic5—22no—
231C2×Dic5—31no—
232C2×Dic5—31no—
233C2×Dic5—31no—
234C2×Dic5—31no—
235C2×Dic5—62no—
236C2×Dic5—62no—
237C2×Dic5—62no—
238C2×Dic5C22×D311yescharacteristic
239C2×Dic5C22×D311yescharacteristic
240C2×Dic5C22×D311yescharacteristic
241C4×D5—62no—
242Dic10—34no—
243Dic10—62no—
244Dic10—62no—
245Dic10—62no—
246C2×#41—151no—
247C22×Q8—151no—
248#124—151no—
249#124—151no—
250#124—151no—
251C22⋊Q8—151no—
252C22⋊Q8—151no—
253C22⋊Q8—151no—
254#121—31no—
255#121—51no—
256#121—151no—
257C4×Q8—151no—
258C4×D4—151no—
259C4×D4—151no—
260C4×D4—151no—
261C2×C3×C5—22no—
262C2×C3×C5—41no—
263C2×C3×C5C22×D411yescharacteristic
264C2×C3×C5C2×#4111yescharacteristic
265C2×C3×C52−1+411yescharacteristic
266C3×D5—41no—
267C5×D3—41no—
268C23×C3—51no—
269C23×C3C22×D511yescharacteristic
270C23×C3C22×D511yescharacteristic
271C22×D3C22×D511yescharacteristic
272C3×D4—22no—
273C3×D4—102no—
274C3×D4—102no—
275C2×C3×C4—51no—
276C2×C3×C4—51no—
277C2×C3×C4—51no—
278C2×C3×C4—102no—
279C2×C3×C4C22×D511yescharacteristic
280#67—22no—
281#67—22no—
282#67—102no—
283C2×Dic3—22no—
284C2×Dic3—51no—
285C2×Dic3—51no—
286C2×Dic3—51no—
287C2×Dic3—51no—
288C2×Dic3—102no—
289C2×Dic3—102no—
290C2×Dic3—102no—
291C2×Dic3C22×D511yescharacteristic
292C2×Dic3C22×D511yescharacteristic
293C2×Dic3C22×D511yescharacteristic
294C4×D3—102no—
295Dic6—54no—
296Dic6—102no—
297Dic6—102no—
298Dic6—102no—
299C22×C5—21no—
300C22×C5—21no—
301C22×C5—21no—
302C22×C5—21no—
303C22×C5—21no—
304C22×C5—61no—
305C22×C5—61no—
306C22×C5—61no—
307C22×C5#23712yescomplemented
308C22×C5C23×D311yescharacteristic
309C2×D5—21no—
310C2×D5—21no—
311C2×D5—21no—
312C4×C5—21no—
313C4×C5—32no—
314C4×C5—61no—
315C4×C5—61no—
316Dic5—21no—
317Dic5—21no—
318Dic5—32no—
319Dic5—61no—
320Dic5—61no—
321Dic5—61no—
322Dic5D3×D412yes—
323#41—304no—
324C2×Q8—151no—
325C2×Q8—151no—
326C2×Q8—151no—
327C2×Q8—151no—
328C2×Q8—304no—
329C2×D4—31no—
330C2×D4—31no—
331C2×D4—51no—
332C2×D4—51no—
333C2×D4—151no—
334C2×D4D3×D511yescharacteristic
335C22×C4—31no—
336C22×C4—51no—
337C22×C4—151no—
338C22×C4—151no—
339C22×C4—151no—
340C22×C4—151no—
341C4⋊C4—31no—
342C4⋊C4—51no—
343C4⋊C4—151no—
344C4⋊C4—151no—
345C4⋊C4—151no—
346C4⋊C4—151no—
347C22⋊C4—31no—
348C22⋊C4—31no—
349C22⋊C4—51no—
350C22⋊C4—51no—
351C22⋊C4—151no—
352C22⋊C4—151no—
353C22⋊C4—151no—
354C22⋊C4—151no—
355C22⋊C4—151no—
356C22⋊C4—151no—
357C42—151no—
358C42—151no—
359C42—151no—
360C3×C5#54811yescharacteristic, complemented, Hall
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—101no—
367C22×C3—101no—
368C22×C3—101no—
369C22×C3#71112yescomplemented
370C22×C3C23×D511yescharacteristic
371C2×D3—21no—
372C2×D3—21no—
373C2×D3—21no—
374C3×C4—21no—
375C3×C4—52no—
376C3×C4—101no—
377C3×C4—101no—
378Dic3—21no—
379Dic3—21no—
380Dic3—52no—
381Dic3—101no—
382Dic3—101no—
383Dic3—101no—
384Dic3D4×D512yes—
385C2×C5—22no—
386C2×C5—41no—
387C2×C5—121no—
388C2×C5C2×D3×D411yescharacteristic
389C2×C5C2×#23711yescharacteristic
390C2×C5#101011yescharacteristic
391D5—41no—
392C23—31no—
393C23—51no—
394C23C2×D3×D511yescharacteristic
395C23C2×D3×D511yescharacteristic
396Q8—154no—
397Q8—302no—
398Q8—302no—
399Q8—302no—
400D4—22no—
401D4—62no—
402D4—62no—
403D4—102no—
404D4—102no—
405D4—302no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—51no—
410C2×C4—51no—
411C2×C4—51no—
412C2×C4—62no—
413C2×C4—102no—
414C2×C4—151no—
415C2×C4—151no—
416C2×C4—151no—
417C2×C4—151no—
418C2×C4—302no—
419C2×C4—302no—
420C2×C4—302no—
421C2×C4—302no—
422C2×C4C2×D3×D511yescharacteristic
423C2×C3—22no—
424C2×C3—41no—
425C2×C3—201no—
426C2×C3C2×D4×D511yescharacteristic
427C2×C3C2×#71111yescharacteristic
428C2×C3#415011yescharacteristic
429D3—41no—
430C5#559811yescharacteristic, complemented, Sylow
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—61no—
437C22—61no—
438C22—61no—
439C22—101no—
440C22—101no—
441C22—101no—
442C22#678012yescomplemented
443C22C22×D3×D511yescharacteristic
444C4—21no—
445C4—32no—
446C4—52no—
447C4—61no—
448C4—61no—
449C4—101no—
450C4—101no—
451C4—152no—
452C4—301no—
453C4—301no—
454C4—301no—
455C3#6593211yescharacteristic, complemented, Sylow
456C2—22no—
457C2—41no—
458C2—121no—
459C2—201no—
460C2#9202011yescharacteristic
461C2D3×D4×D511yescharacteristic
462C2C2×#678011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#548
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C22×#1204C2semidirect
2. C2×#6780C2semidirect
3. D4×Dic15C2semidirect
4. C5×#939C2semidirect
5. C3×#4085C2semidirect
6. #91576C2semidirect
7. #91571C2semidirect
8. Dic3×C22⋊D5C2semidirect
9. Dic5×#67C2semidirect
10. #91534C2semidirect
11. #91525C2semidirect
12. #91464C2semidirect
13. #91422C2semidirect
14. #91417C2semidirect
15. #91325C2semidirect
59. #4085D3semidirect
97. #1204D4semidirect
105. #939D5semidirect
159. C22×C3×C5#41semidirect
307. C22×C5#237semidirect
360. C3×C5#548semidirect
369. C22×C3#711semidirect
430. C5#5598semidirect
442. C22#6780semidirect
455. C3#65932semidirect

Maximal subgroups

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

subgroupindex
C22×#12042
C2×#67802
D4×Dic152
C5×#9392
C3×#40852
#915762
#915712
Dic3×C22⋊D52
Dic5×#672
#915342
#915252
#914642
#914222
#914172
#913252
#659323
#55985