Finite Group Databasebeta

← #29718398

Subgroups — #29718398

Order 15616.

986 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840092C211yesmaximal, characteristic
2#21449459C211yesmaximal, characteristic
3#21448688C211yesmaximal, characteristic
4#21447196C212yesmaximal, complemented
5#21447064C212yesmaximal, complemented
6#21446740C212yesmaximal, complemented
7#21446456C212yesmaximal, complemented
8#21446242C211yesmaximal, characteristic
9#21445997C211yesmaximal, characteristic
10#21445754C211yesmaximal, characteristic
11C22×#103×C61C211yesmaximal, characteristic
12C2×#14519775C2212yes—
13C2×#14519756C2211yescharacteristic
14C2×#14519736—21no—
15#16840092—24no—
16C2×#14519708C2211yescharacteristic
17C2×#14519708C2211yescharacteristic
18C2×#14519706C2212yes—
19C2×#14519702C2212yes—
20C2×#14519701C2212yes—
21C2×#14519701C2212yes—
22C2×#14519700C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×C22⋊#92418C2211yescharacteristic
25#16839681—22no—
26#16839669C2212yes—
27#16839661C2212yes—
28#16839648—22no—
29#16839635—21no—
30#16839635—21no—
31#16839625—21no—
32#16839622C2211yescharacteristic
33C4⋊C4×Dic61C2212yes—
34C4×C22⋊Dic61C2211yescharacteristic
35#16839525C2212yes—
36#16839516C2212yes—
37C23×Q8×C61C2211yescharacteristic
38C23×C8×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C2×#103×C61—24no—
41C2×#103×C61C2212yes—
42C2×#103×C61C2212yes—
43C22×Dic122C2311yescharacteristic
44C22×#92418C2311yescharacteristic
45C22×#92418C2311yescharacteristic
46#14519775—24no—
47#14519775—42no—
48#14519756—22no—
49#14519756—22no—
50#14519736—44no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61C2311yescharacteristic
53C2×C4⋊Dic61C2312yes—
54C2×#11758196—21no—
55C2×#11758196—22no—
56C2×#11758196—22no—
57C2×C22⋊Dic61—21no—
58C2×C22⋊Dic61—21no—
59C2×C22⋊Dic61D411yescharacteristic
60C2×C22⋊Dic61D411yescharacteristic
61C2×C4×Dic61—22no—
62C2×C4×Dic61C2312yes—
63#14519708—22no—
64#14519708—22no—
65#14519706—42no—
66#14519702—42no—
67#14519701—24no—
68#14519701—42no—
69#14519700—22no—
70#14519700—22no—
71C22⋊#92418—22no—
72#14519684—21no—
73#14519684—22no—
74#14519684—22no—
75#14519684—22no—
76#14519684—22no—
77#14519684D412yes—
78C22×Q8×C61—21no—
79C22×Q8×C61—24no—
80C22×Q8×C61—22no—
81C22×Q8×C61—22no—
82C22×Q8×C61—42no—
83C22×Q8×C61C2312yes—
84C23×C4×C61—21no—
85C23×C4×C61—21no—
86C23×C4×C61C2311yescharacteristic
87C22×C8×C61—22no—
88C22×C8×C61—42no—
89C22×C8×C61C2312yes—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61D414yes—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95#103×C61—24no—
96#103×C61—24no—
97C24×C61C2×D411yescharacteristic
98C2×Q8×C61—22no—
99C2×Q8×C61—28no—
100C2×Q8×C61—22no—
101C2×Q8×C61—22no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—44no—
105C2×Q8×C61—48no—
106C2×Q8×C61—44no—
107C2×Q8×C61—42no—
108C2×Q8×C61#4114yes—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—24no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—44no—
116C22×C4×C61—42no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D412yes—
121C22×C4×C61#4112yes—
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61C2×D412yes—
127C22×Dic61C2×D412yes—
128C2×Dic122—21no—
129C2×Dic122—21no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×C8×C61—22no—
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C2×C8×C61—44no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61C2×D412yes—
140C4⋊C4×C61C2×D412yes—
141C22⋊Dic61—22no—
142C22⋊Dic61—42no—
143C22⋊Dic61—42no—
144C22⋊Dic61SD1614yes—
145C4⋊Dic61—21no—
146C4⋊Dic61—22no—
147C4⋊Dic61—21no—
148C4⋊Dic61—22no—
149C4⋊Dic61—44no—
150#11758196—44no—
151#11758196—44no—
152#11758196—42no—
153C4×Dic61—22no—
154C4×Dic61—22no—
155C4×Dic61—44no—
156C2×#92418—21no—
157C2×#92418—21no—
158C2×#92418—22no—
159C2×#92418—22no—
160C23×C61—22no—
161C23×C61—22no—
162C23×C61#12112yes—
163C23×C61C2×SD1612yes—
164C23×C61#13712yes—
165C23×C61C22×D411yescharacteristic
166Q8×C61—24no—
167Q8×C61—24no—
168Q8×C61—48no—
169Q8×C61—42no—
170Q8×C61—42no—
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—24no—
178C2×C4×C61—48no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—48no—
186C2×C4×C61#12112yes—
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C2×#4112yes—
189C2×C4×C61C2×#4111yescharacteristic
190C2×C4×C61H(5,𝔽2)12yes—
191C2×Dic61—22no—
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61C2×SD1614yes—
204Dic122—41no—
205Dic122—41no—
206Dic122—42no—
207C8×C61—42no—
208#92418—41no—
209#92418—41no—
210#24480—611nomaximal, Sylow
211C22×C61—22no—
212C22×C61—24no—
213C22×C61—22no—
214C22×C61—24no—
215C22×C61#46114yes—
216C22×C61C2×#12111yescharacteristic
217C22×C61#54612yes—
218C22×C61C22×SD1611yescharacteristic
219C22×C61C2×#13711yescharacteristic
220C22×C61#57812yes—
221C4×C61—22no—
222C4×C61—44no—
223C4×C61—42no—
224C4×C61—42no—
225C4×C61#52511yescharacteristic
226C4×C61#53511yescharacteristic
227C4×C61#55112yes—
228Dic61—24no—
229Dic61—42no—
230Dic61—81no—
231Dic61—82no—
232Dic61—81no—
233C2×#483—611no—
234C22×#103D6111yescharacteristic
235#2434—611no—
236#2344—611no—
237#2066—612no—
238#2043—612no—
239#1981—612no—
240#1927—612no—
241#1892—611no—
242#1850—611no—
243#1787—611no—
244C2×C61—24no—
245C2×C61#241111yescharacteristic
246C2×C61C2×#46111yescharacteristic
247C2×C61#304411yescharacteristic
248C2×C61#329812yes—
249C2×C61#331212yes—
250C23×Q8C2×D6111yescharacteristic
251C23×C8C2×D6111yescharacteristic
252C2×C4⋊Q8—612no—
253C2×C22⋊Q8—611no—
254C2×#117—1221no—
255C22×C4⋊C4C2×D6111yescharacteristic
256#483—1224no—
257C2×#106—611no—
258C2×#106—611no—
259C2×C4⋊C8—612no—
260C2×#103—24no—
261C2×#103—612no—
262C2×#103—611no—
263C2×#103—611no—
264C2×#103—612no—
265C2×#103—612no—
266C2×#103C2×D6112yes—
267C2×#103C2×D6112yes—
268C2×C22⋊C8—611no—
269#395—1222no—
270#391—612no—
271#389—612no—
272#387—1222no—
273#385—1221no—
274#385—1221no—
275#381—1221no—
276#380—611no—
277C4×C4⋊C4—612no—
278C4×C22⋊C4—611no—
279#340—612no—
280#336—612no—
281C61#2448011yescharacteristic, complemented, Sylow
282C22×Q8—21no—
283C22×Q8—24no—
284C22×Q8—22no—
285C22×Q8—22no—
286C22×Q8—42no—
287C22×Q8—611no—
288C22×Q8C22×D6112yes—
289C23×C4—21no—
290C23×C4—21no—
291C23×C4C22×D6111yescharacteristic
292C22×C8—22no—
293C22×C8—42no—
294C22×C8—611no—
295C22×C8—611no—
296C22×C8C22×D6112yes—
297C4⋊Q8—1224no—
298C4⋊Q8—2442no—
299C22⋊Q8—1222no—
300C22⋊Q8—1222no—
301#117—2444no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—611no—
305C2×C4⋊C4—612no—
306C2×C4⋊C4—1222no—
307C2×C4⋊C4—1222no—
308C2×C4⋊C4—1222no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4C22⋊D6114yes—
311C2×C4⋊C4C22×D6111yescharacteristic
312C2×C4⋊C4C22×D6111yescharacteristic
313C2×C22⋊C4—611no—
314C2×C22⋊C4—611no—
315C2×C22⋊C4—1221no—
316C2×C22⋊C4—1221no—
317C2×C42—612no—
318C2×C42—1222no—
319#106—1222no—
320#106—1222no—
321C4⋊C8—2442no—
322#103—24no—
323#103—24no—
324#103—1224no—
325#103—1222no—
326#103—1222no—
327#103—2442no—
328#103—2442no—
329C22⋊C8—1222no—
330#95—612no—
331#95—1222no—
332#95—1222no—
333#95—1222no—
334#95—1222no—
335#95—1221no—
336C24D4×D6111yescharacteristic
337C2×Q8—22no—
338C2×Q8—28no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—44no—
344C2×Q8—48no—
345C2×Q8—44no—
346C2×Q8—42no—
347C2×Q8—1221no—
348C2×Q8—1221no—
349C2×Q8—1222no—
350C2×Q8—1222no—
351C2×Q8#1175822214yes—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—612no—
363C22×C4—612no—
364C22×C4—1221no—
365C22×C4—1222no—
366C22×C4—1221no—
367C22×C4D4×D6111yescharacteristic
368C22×C4#1175822412yes—
369C22×C4C2×C22⋊D6112yes—
370C22×C4C23×D6111yescharacteristic
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—44no—
375C2×C8—1221no—
376C2×C8—1221no—
377C2×C8—1222no—
378C2×C8—1222no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—24no—
382C4⋊C4—1221no—
383C4⋊C4—1221no—
384C4⋊C4—1222no—
385C4⋊C4—1222no—
386C4⋊C4—2444no—
387C4⋊C4—2444no—
388C4⋊C4—2444no—
389C4⋊C4—2442no—
390C4⋊C4C2×C22⋊D6112yes—
391C4⋊C4C2×C22⋊D6112yes—
392C22⋊C4—614no—
393C22⋊C4—1222no—
394C22⋊C4—2442no—
395C22⋊C4—2442no—
396C42—1222no—
397C42—1222no—
398C42—2444no—
399C23—22no—
400C23—22no—
401C23#1451974912yes—
402C23SD16×D6112yes—
403C23#1451980612yes—
404C23C2×D4×D6111yescharacteristic
405Q8—24no—
406Q8—24no—
407Q8—48no—
408Q8—42no—
409Q8—42no—
410Q8—2441no—
411Q8—2441no—
412Q8—2442no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—614no—
429C2×C4—1222no—
430C2×C4—1222no—
431C2×C4—1222no—
432C2×C4—1222no—
433C2×C4—1222no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2442no—
437C2×C4—2442no—
438C2×C4—2442no—
439C2×C4—2442no—
440C2×C4—2442no—
441C2×C4#1451974912yes—
442C2×C4C22×C22⋊D6111yescharacteristic
443C2×C4C2×#1175822411yescharacteristic
444C2×C4C2×#1175822212yes—
445C2×C4#1451982212yes—
446C8—42no—
447C8—2441no—
448C8—2441no—
449C22—24no—
450C22—24no—
451C22—22no—
452C22—22no—
453C22#1683996514yes—
454C22C2×#1451974911yescharacteristic
455C22#1684040212yes—
456C22C2×SD16×D6111yescharacteristic
457C22C2×#1451980611yescharacteristic
458C22#1684057512yes—
459C4—22no—
460C4—42no—
461C4—42no—
462C4—44no—
463C4—1224no—
464C4—4882no—
465C4—4881no—
466C4—4881no—
467C4—2442no—
468C4#1684027311yescharacteristic
469C4#1684032911yescharacteristic
470C4#1684043812yes—
471C2—24no—
472C2#2144920411yescharacteristic
473C2C2×#1683996511yescharacteristic
474C2#2145534611yescharacteristic
475C2#2145768012yes—
476C2#2145786612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#103×C61C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC22×#103D61
hypocenterC61#24480
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
4. #21447196C2semidirect
5. #21447064C2semidirect
6. #21446740C2semidirect
7. #21446456C2semidirect
281. C61#24480semidirect

Maximal subgroups

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

subgroupindex
C2×#168400922
#214494592
#214486882
#214471962
#214470642
#214467402
#214464562
#214462422
#214459972
#214457542
C22×#103×C612
#2448061