Finite Group Databasebeta

← #29736079

Subgroups — #29736079

Order 15616.

1040 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840168C211yesmaximal, characteristic, complemented
2C2×#16840062C211yesmaximal, characteristic
3#21451207C211yesmaximal, characteristic, complemented
4#21447154C212yesmaximal, complemented
5#21447154C212yesmaximal, complemented
6#21446772C211yesmaximal, characteristic, complemented
7#21446760C211yesmaximal, characteristic, complemented
8#21446080C212yesmaximal
9#21446066C212yesmaximal
10#21445644C211yesmaximal, characteristic
11C2×C61×#554C211yesmaximal, characteristic
12C2×#14519769C2211yescharacteristic
13#16840168—24no—
14#16840062—24no—
15C2×#14519710C2212yes—
16C2×#14519710C2211yescharacteristic
17C2×#14519708C2212yes—
18C2×#14519708C2211yescharacteristic
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519698C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519698C2212yes—
23C2×#14519688C2211yescharacteristic
24#16839693—21no—
25#16839681—21no—
26#16839660C2212yes—
27#16839660C2212yes—
28#16839657—22no—
29C4×C4⋊Dic61C2211yescharacteristic
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34#16839517C2212yes—
35C61×#554—24no—
36C61×#554—24no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#125×C61C2212yes—
39C2×C22⋊Q8×C61—21no—
40C2×C22⋊Q8×C61—22no—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4×D4×C61C2212yes—
43C2×C4×D4×C61C2211yescharacteristic
44C22×C4⋊C4×C61—21no—
45C22×#92418C2311yescharacteristic
46C22×#92418C2311yescharacteristic
47C22×#92418C2312yes—
48#14519769—22no—
49#14519769—22no—
50C2×C4⋊Dic61C2311yescharacteristic
51C2×C4⋊Dic61C2311yescharacteristic
52C2×C4⋊Dic61C2312yes—
53C2×#11758196—21no—
54C2×#11758196—22no—
55C2×#11758196—21no—
56C2×#11758196—22no—
57C2×C4×Dic61—21no—
58#14519710—22no—
59#14519710—24no—
60#14519708—22no—
61#14519708—24no—
62C4⋊#92418—22no—
63#14519698—22no—
64#14519698—22no—
65#14519698Q818yescomplemented
66#14519688—22no—
67#14519684—21no—
68#14519684—22no—
69#14519684—22no—
70C22×Q8×C61—21no—
71C22×Q8×C61D411yescharacteristic
72C22×D4×C61C2311yescharacteristic
73C23×C4×C61—21no—
74C23×C4×C61—22no—
75C4⋊Q8×C61—22no—
76C4⋊Q8×C61D414yes—
77#125×C61—24no—
78#125×C61—24no—
79C22⋊Q8×C61—28no—
80C22⋊Q8×C61—216no—
81C4×Q8×C61—22no—
82C4×Q8×C61—22no—
83C4×D4×C61—24no—
84C4×D4×C61—24no—
85C4×D4×C61—22no—
86C4×D4×C61D414yes—
87C2×C4⋊C4×C61—28no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61D411yescharacteristic
92C2×C4⋊C4×C61D412yes—
93C2×C4⋊C4×C61C2312yes—
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C22⋊C4×C61—22no—
96C2×C22⋊C4×C61—21no—
97C2×C42×C61C2312yes—
98C2×C42×C61C2311yescharacteristic
99C24×C61—21no—
100C2×Q8×C61—22no—
101C2×Q8×C61—24no—
102C2×Q8×C61—42no—
103C2×Q8×C61—81no—
104C2×Q8×C61—81no—
105C2×D4×C61—21no—
106C2×D4×C61—22no—
107C2×D4×C61C2×D412yes—
108C2×D4×C61C2×Q814yes—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—28no—
112C22×C4×C61—24no—
113C22×C4×C61—44no—
114C22×C4×C61—82no—
115C22×C4×C61—82no—
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—22no—
122C22×Dic61—21no—
123C22×Dic61—21no—
124C4⋊C4×C61—28no—
125C4⋊C4×C61—28no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—24no—
128C4⋊C4×C61—24no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61C2×D414yes—
132C4⋊C4×C61C2×D412yes—
133C22⋊C4×C61—28no—
134C22⋊C4×C61—24no—
135C42×C61—22no—
136C42×C61—22no—
137C42×C61C2×D412yes—
138C42×C61C2×D412yes—
139C4⋊Dic61—22no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61C2×Q814yes—
145#11758196—44no—
146#11758196—42no—
147#11758196—42no—
148#11758196—82no—
149C4×Dic61—81no—
150C2×#92418—22no—
151C2×#92418—22no—
152C2×#92418—21no—
153C2×#92418—21no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418C2×Q814yes—
157C23×C61—22no—
158C23×C61—24no—
159C23×C61—42no—
160C23×C61—41no—
161C23×C61—41no—
162C23×C61C22×D411yescharacteristic
163Q8×C61—44no—
164Q8×C61—84no—
165D4×C61—22no—
166D4×C61—24no—
167D4×C61C4⋊Q814yes—
168C2×C4×C61—22no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—24no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—48no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—88no—
184C2×C4×C61#12012yes—
185C2×C4×C61#12012yes—
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C22×Q812yes—
189C2×C4×C612−1+412yes—
190C2×Dic61—24no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—42no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198#92418—24no—
199#92418—41no—
200#92418—41no—
201#25845—611nomaximal, Sylow
202C22×C61—24no—
203C22×C61—44no—
204C22×C61—44no—
205C22×C61—42no—
206C22×C61—42no—
207C22×C61C2×#12011yescharacteristic
208C22×C61D4×Q812yes—
209C22×C61#57512yes—
210C22×C61#57611yescharacteristic
211C22×C61#57711yescharacteristic
212C4×C61—22no—
213C4×C61—22no—
214C4×C61—42no—
215C4×C61—42no—
216C4×C61—42no—
217C4×C61—44no—
218C4×C61C2×C4⋊Q811yescharacteristic
219C4×C61#53611yescharacteristic
220C4×C61D4×Q812yes—
221Dic61—44no—
222Dic61—81no—
223Dic61—81no—
224C2×#554D6111yescharacteristic
225C2×#501—611no—
226C2×#478—611no—
227#2584—611no—
228#2059—612no—
229#2059—612no—
230#1988—611no—
231#1985—611no—
232#1864—612no—
233#1862—612no—
234#1762—611no—
235C2×C61—44no—
236C2×C61#243411yescharacteristic
237C2×C61#300511yescharacteristic
238C2×C61#301211yescharacteristic
239C2×C61#338812yes—
240C2×C61#338912yes—
241#554—24no—
242#554—24no—
243C2×C4⋊Q8C2×D6111yescharacteristic
244C2×#125—611no—
245C2×#125C2×D6112yes—
246C2×C22⋊Q8—21no—
247C2×C22⋊Q8—22no—
248C2×C4×Q8C2×D6111yescharacteristic
249C2×C4×D4C2×D6112yes—
250C2×C4×D4C2×D6111yescharacteristic
251C22×C4⋊C4—21no—
252#501—1224no—
253#478—1224no—
254C2×#107—611no—
255C2×#107—612no—
256C2×#106—611no—
257C2×#106—612no—
258C2×C4⋊C8—611no—
259C2×#102—612no—
260C2×#102—611no—
261C2×#102—611no—
262C2×#97—611no—
263#398—1221no—
264#395—1221no—
265#389—612no—
266#389—612no—
267#389—1222no—
268C4×C4⋊C4—611no—
269#340—612no—
270#340—612no—
271#340—612no—
272#336—612no—
273#336—612no—
274C61#2584511yescharacteristic, complemented, Sylow
275C22×Q8—21no—
276C22×Q8C22⋊D6111yescharacteristic
277C22×D4C22×D6111yescharacteristic
278C23×C4—21no—
279C23×C4—22no—
280C22×C8—612no—
281C22×C8—611no—
282C22×C8—611no—
283C4⋊Q8—22no—
284C4⋊Q8C22⋊D6114yes—
285#125—24no—
286#125—24no—
287#125—1222no—
288#125—1222no—
289C22⋊Q8—28no—
290C22⋊Q8—216no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—22no—
296C4×D4C22⋊D6114yes—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—612no—
302C2×C4⋊C4—611no—
303C2×C4⋊C4—611no—
304C2×C4⋊C4—1222no—
305C2×C4⋊C4—1221no—
306C2×C4⋊C4—1221no—
307C2×C4⋊C4—1222no—
308C2×C4⋊C4C22⋊D6111yescharacteristic
309C2×C4⋊C4C22⋊D6112yes—
310C2×C4⋊C4C22×D6111yescharacteristic
311C2×C4⋊C4C22×D6112yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C42—1221no—
315C2×C42C22×D6112yes—
316C2×C42C22×D6111yescharacteristic
317#107—1222no—
318#107—1224no—
319#106—1222no—
320#106—1224no—
321C4⋊C8—1222no—
322#102—618no—
323#102—1222no—
324#102—1222no—
325#97—1222no—
326#95—1221no—
327#95—1222no—
328#95—1222no—
329C24—21no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×Q8—42no—
333C2×Q8—81no—
334C2×Q8—81no—
335C2×D4—21no—
336C2×D4—22no—
337C2×D4Q8×D6114yes—
338C2×D4C2×C22⋊D6112yes—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—28no—
342C22×C4—24no—
343C22×C4—44no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—1222no—
347C22×C4—1221no—
348C22×C4—1221no—
349C22×C4C2×C22⋊D6112yes—
350C22×C4C2×C22⋊D6112yes—
351C22×C4C2×C22⋊D6111yescharacteristic
352C22×C4C2×C22⋊D6111yescharacteristic
353C22×C4C23×D6111yescharacteristic
354C2×C8—614no—
355C2×C8—1222no—
356C2×C8—1222no—
357C2×C8—1221no—
358C2×C8—1221no—
359C2×C8—1222no—
360C2×C8—1222no—
361C4⋊C4—28no—
362C4⋊C4—28no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—614no—
369C4⋊C4—1222no—
370C4⋊C4—1221no—
371C4⋊C4—1221no—
372C4⋊C4—1221no—
373C4⋊C4—1221no—
374C4⋊C4—4882no—
375C4⋊C4—2444no—
376C4⋊C4—2442no—
377C4⋊C4—2442no—
378C4⋊C4C2×C22⋊D6114yes—
379C4⋊C4C2×C22⋊D6112yes—
380C22⋊C4—28no—
381C22⋊C4—24no—
382C42—22no—
383C42—22no—
384C42—4881no—
385C42C2×C22⋊D6112yes—
386C42C2×C22⋊D6112yes—
387C23—22no—
388C23—24no—
389C23—42no—
390C23—41no—
391C23—41no—
392C23C22×C22⋊D6111yescharacteristic
393Q8—44no—
394Q8—84no—
395D4—22no—
396D4—24no—
397D4#1451977514yes—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—48no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—88no—
414C2×C4—1224no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4—2442no—
418C2×C4—2441no—
419C2×C4—2441no—
420C2×C4—2442no—
421C2×C4—2442no—
422C2×C4#1451974612yes—
423C2×C4#1451974612yes—
424C2×C4C22×C22⋊D6111yescharacteristic
425C2×C4C22×C22⋊D6111yescharacteristic
426C2×C4C2×Q8×D6112yes—
427C2×C4#1451982512yes—
428C8—1224no—
429C8—2441no—
430C8—2441no—
431C22—24no—
432C22—42no—
433C22—44no—
434C22—44no—
435C22—42no—
436C22C2×#1451974611yescharacteristic
437C22Q8×C22⋊D6112yes—
438C22#1684056012yes—
439C22#1684056511yescharacteristic
440C22#1684057311yescharacteristic
441C4—22no—
442C4—22no—
443C4—42no—
444C4—44no—
445C4—42no—
446C4—42no—
447C4—4881no—
448C4—4881no—
449C4—2444no—
450C4C2×#1451977511yescharacteristic
451C4#1684033611yescharacteristic
452C4Q8×C22⋊D6112yes—
453C2—44no—
454C2#2144945911yescharacteristic
455C2#2145506111yescharacteristic
456C2#2145512311yescharacteristic
457C2#2145876412yes—
458C2#2145877312yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#554C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#554D61
hypocenterC61#25845
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16840168C2semidirect
3. #21451207C2semidirect
4. #21447154C2semidirect
5. #21447154C2semidirect
6. #21446772C2semidirect
7. #21446760C2semidirect
65. #14519698Q8semidirect
274. C61#25845semidirect

Maximal subgroups

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

subgroupindex
C2×#168401682
C2×#168400622
#214512072
#214471542
#214467722
#214467602
#214460802
#214460662
#214456442
C2×C61×#5542
#2584561