Finite Group Databasebeta

← #29679803

Subgroups — #29679803

Order 15616.

1502 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519735C211yesmaximal, characteristic
2C2×C22⋊C8×D61C211yesmaximal, characteristic
3#21448201C211yesmaximal, characteristic
4#21445913C214yesmaximal, complemented
5#21445909C211yesmaximal, characteristic, complemented
6#21445906C211yesmaximal, characteristic, complemented
7#21445903C211yesmaximal, characteristic, complemented
8#21445534C214yesmaximal, complemented
9C61×#1824C211yesmaximal, characteristic, complemented
10C23×C4×D61C2211yescharacteristic
11C22×C8×D61C2212yes—
12C2×#14519735—21no—
13C2×#14519735—21no—
14C2×#14519735—24no—
15C2×#14519735C418yescomplemented
16C2×#14519735C2212yes—
17C2×#14519735C2212yes—
18C2×#14519735C2212yes—
19C22×C4⋊Dic61C2211yescharacteristic
20C22×#11758198—21no—
21C22×C4×Dic61—21no—
22C2×C22⋊#92418C2211yescharacteristic
23C2×#14519689C2212yes—
24C22⋊C8×D61—24no—
25#16839624—21no—
26#16839622C2211yescharacteristic
27#16839619C2212yes—
28#16839617—21no—
29C22⋊C4×Dic61—21no—
30#16839609—21no—
31#16839525C2214yescomplemented
32#16839524C2214yescomplemented
33#16839524C2214yescomplemented
34C22×C4⋊C4×C61C2211yescharacteristic
35C2×C22⋊C8×C61C2211yescharacteristic
36C61×#380C2211yescharacteristic
37C61×#340C2214yescomplemented
38C24×D61D411yescharacteristic
39C23×Dic61—21no—
40C23×Dic61D411yescharacteristic
41C22×C4×D61—28no—
42C22×C4×D61C2×C414yes—
43C22×C4×D61D411yescharacteristic
44C22×C4×D61D411yescharacteristic
45C22×C4×D61D414yes—
46C22×C4×D61C2311yescharacteristic
47C22×C4×D61C2311yescharacteristic
48C22×#92418C2312yes—
49C2×C8×D61—24no—
50C2×C8×D61—28no—
51#14519735—28no—
52#14519735—28no—
53#14519735—24no—
54#14519735—24no—
55#14519735—28no—
56#14519735C2×C418yes—
57#14519735C2×C418yes—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61C2×C414yes—
61C2×C4⋊Dic61C2312yes—
62C2×C4⋊Dic61C2312yes—
63C2×C22⋊Dic61—21no—
64C2×#11758198—28no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×C4×Dic61—28no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61—22no—
72C2×C4×Dic61—22no—
73C22⋊#92418—22no—
74#14519689—24no—
75#14519684—21no—
76#14519684—22no—
77C23×C4×C61—21no—
78C23×C4×C61C2311yescharacteristic
79C22×C8×C61C2312yes—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61—22no—
82C2×C4⋊C4×C61C2×C414yes—
83C2×C4⋊C4×C61C2312yes—
84C2×C4⋊C4×C61C2312yes—
85C2×C22⋊C4×C61—21no—
86C22⋊C8×C61—22no—
87#95×C61—21no—
88C24×C61C2×D411yescharacteristic
89C23×D61—22no—
90C23×D61—22no—
91C23×D61—24no—
92C23×D61—42no—
93C23×D61C22⋊C414yes—
94C23×D61C2×D411yescharacteristic
95C23×D61C2×D411yescharacteristic
96C22×C4×C61—24no—
97C22×C4×C61—24no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—44no—
101C22×C4×C61C22×C412yes—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D412yes—
104C22×C4×C61C2411yescharacteristic
105C22×Dic61—24no—
106C22×Dic61—24no—
107C22×Dic61—22no—
108C22×Dic61—22no—
109C22×Dic61—42no—
110C22×Dic61—42no—
111C22×Dic61C22⋊C414yes—
112C22×Dic61C2×D411yescharacteristic
113C22×Dic61C2×D411yescharacteristic
114C2×C4×D61—24no—
115C2×C4×D61—24no—
116C2×C4×D61—24no—
117C2×C4×D61—216no—
118C2×C4×D61—216no—
119C2×C4×D61C22⋊C414yes—
120C2×C4×D61C22⋊C414yes—
121C2×C4×D61C22×C412yes—
122C2×C4×D61C22×C412yes—
123C2×C4×D61C2×D412yes—
124C2×C4×D61C2×D412yes—
125C2×C4×D61#4118yes—
126C2×C8×C61—22no—
127C2×C8×C61—24no—
128C4⋊C4×C61—24no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—24no—
131C4⋊C4×C61C22×C414yes—
132C22⋊C4×C61—42no—
133C22⋊Dic61—42no—
134#11758198—28no—
135#11758198—28no—
136#11758198—44no—
137#11758198—44no—
138C4⋊Dic61—24no—
139C4⋊Dic61—22no—
140C4⋊Dic61—24no—
141C4⋊Dic61C22×C414yes—
142C4×Dic61—28no—
143C4×Dic61—28no—
144C4×Dic61—44no—
145C2×#92418—22no—
146C2×#92418—24no—
147C8×D61—48no—
148C23×C61—22no—
149C23×C61—22no—
150C23×C61C2×C22⋊C412yes—
151C23×C61#13512yes—
152C23×C61#13512yes—
153C23×C61C22×D411yescharacteristic
154C22×D61—21no—
155C22×D61—21no—
156C22×D61—22no—
157C22×D61—22no—
158C22×D61—21no—
159C22×D61—21no—
160C22×D61—22no—
161C22×D61—28no—
162C22×D61—28no—
163C22×D61—44no—
164C22×D61—48no—
165C22×D61—48no—
166C22×D61C2×C22⋊C412yes—
167C22×D61C2×C22⋊C412yes—
168C22×D61#12314yes—
169C2×C4×C61—22no—
170C2×C4×C61—24no—
171C2×C4×C61—24no—
172C2×C4×C61—24no—
173C2×C4×C61—48no—
174C2×C4×C61—44no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61C2×C22⋊C412yes—
179C2×C4×C61C23×C411yescharacteristic
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C2×#4114yes—
182C2×Dic61—22no—
183C2×Dic61—28no—
184C2×Dic61—24no—
185C2×Dic61—42no—
186C2×Dic61—42no—
187C2×Dic61—48no—
188C2×Dic61—42no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61C2×C22⋊C412yes—
192C2×Dic61C2×C22⋊C412yes—
193C2×Dic61#12314yes—
194C4×D61—28no—
195C4×D61—24no—
196C4×D61—24no—
197C4×D61—28no—
198C4×D61#11718yes—
199C4×D61#12318yes—
200C8×C61—42no—
201#92418—42no—
202#20699—611nomaximal, Sylow
203C22×C61—24no—
204C22×C61—24no—
205C22×C61—22no—
206C22×C61—22no—
207C22×C61#41614yes—
208C22×C61C22×C22⋊C411yescharacteristic
209C22×C61C2×#12312yes—
210C22×C61C2×#13511yescharacteristic
211C22×C61C2×#13511yescharacteristic
212C22×C61#57512yes—
213C2×D61—24no—
214C2×D61—24no—
215C2×D61—22no—
216C2×D61—22no—
217C2×D61—22no—
218C2×D61—22no—
219C2×D61—22no—
220C2×D61—416no—
221C2×D61—42no—
222C2×D61—42no—
223C2×D61#38114yes—
224C4×C61—22no—
225C4×C61—44no—
226C4×C61—82no—
227C4×C61C2×#11712yes—
228C4×C61C2×#12312yes—
229Dic61—22no—
230Dic61—44no—
231Dic61—82no—
232Dic61#38114yes—
233C22×#117—611no—
234C22×C22⋊C8—611no—
235#2280—611no—
236#1825—614no—
237#1824—611no—
238#1824D6111yescharacteristic, complemented
239#1823—611no—
240#1823—611no—
241#1735—614no—
242C2×C61—24no—
243C2×C61C2×#38111yescharacteristic
244C2×C61C2×#41611yescharacteristic
245C2×C61#289611yescharacteristic
246C2×C61#309112yes—
247C2×C61#309112yes—
248D61—24no—
249D61—42no—
250C24×C4—611no—
251C23×C8—612no—
252C2×#117—618no—
253C2×#117—612no—
254C2×#117—612no—
255C2×#117—612no—
256C2×#117—1221no—
257C2×#117—1221no—
258C2×#117—1224no—
259C22×C4⋊C4—611no—
260C22×C4⋊C4C2×D6111yescharacteristic
261C22×C22⋊C4—1221no—
262C22×C42—1221no—
263C2×C22⋊C8—611no—
264C2×C22⋊C8—612no—
265C2×C22⋊C8—1224no—
266C2×C22⋊C8C2×D6111yescharacteristic
267#381—1221no—
268#380—611no—
269#380—612no—
270#380C2×D6111yescharacteristic
271#379—1221no—
272C4×C22⋊C4—1221no—
273C4×C22⋊C4—1221no—
274#340—614no—
275#340—614no—
276#340—614no—
277#340C2×D6114yescomplemented
278C61#2069911yescharacteristic, complemented, Sylow
279C25—611no—
280C23×C4—21no—
281C23×C4—611no—
282C23×C4—611no—
283C23×C4—614no—
284C23×C4—611no—
285C23×C4—611no—
286C23×C4—614no—
287C23×C4—611no—
288C23×C4—1228no—
289C23×C4—1221no—
290C23×C4C22×D6111yescharacteristic
291C22×C8—612no—
292C22×C8—1224no—
293C22×C8—1228no—
294C22×C8C22×D6112yes—
295#117—618no—
296#117—618no—
297#117—1224no—
298#117—1224no—
299#117—1228no—
300#117—1228no—
301#117—1228no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—614no—
305C2×C4⋊C4—612no—
306C2×C4⋊C4—612no—
307C2×C4⋊C4—1222no—
308C2×C4⋊C4—1221no—
309C2×C4⋊C4C4×D6114yes—
310C2×C4⋊C4C22×D6112yes—
311C2×C4⋊C4C22×D6112yes—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—1222no—
314C2×C22⋊C4—1222no—
315C2×C22⋊C4—1228no—
316C2×C22⋊C4—1222no—
317C2×C22⋊C4—1222no—
318C2×C22⋊C4—1221no—
319C2×C42—1222no—
320C2×C42—1222no—
321C2×C42—1228no—
322C2×C42—1222no—
323C22⋊C8—22no—
324C22⋊C8—1224no—
325C22⋊C8—1222no—
326#95—21no—
327#95—1222no—
328#95—1221no—
329C24—611no—
330C24—611no—
331C24—614no—
332C24—1224no—
333C24—1222no—
334C24—1222no—
335C24—2442no—
336C24D4×D6111yescharacteristic
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—44no—
342C22×C4—611no—
343C22×C4—612no—
344C22×C4—612no—
345C22×C4—618no—
346C22×C4—611no—
347C22×C4—614no—
348C22×C4—612no—
349C22×C4—612no—
350C22×C4—614no—
351C22×C4—614no—
352C22×C4—1224no—
353C22×C4—12216no—
354C22×C4—12216no—
355C22×C4—1224no—
356C22×C4—1224no—
357C22×C4—1224no—
358C22×C4—1222no—
359C22×C4—1224no—
360C22×C4—1222no—
361C22×C4—2442no—
362C22×C4—2442no—
363C22×C4C2×C4×D6112yes—
364C22×C4D4×D6112yes—
365C22×C4D4×D6111yescharacteristic
366C22×C4C23×D6111yescharacteristic
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—1222no—
370C2×C8—1224no—
371C2×C8—2448no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—614no—
376C4⋊C4—1222no—
377C4⋊C4—1224no—
378C4⋊C4—1224no—
379C4⋊C4C2×C4×D6114yes—
380C22⋊C4—42no—
381C22⋊C4—1228no—
382C22⋊C4—1228no—
383C22⋊C4—2444no—
384C22⋊C4—2444no—
385C22⋊C4—2442no—
386C42—1228no—
387C42—1228no—
388C42—2444no—
389C23—22no—
390C23—22no—
391C23—612no—
392C23—614no—
393C23—612no—
394C23—1228no—
395C23—1221no—
396C23—1221no—
397C23—1222no—
398C23—1222no—
399C23—1228no—
400C23—1221no—
401C23—1221no—
402C23—1222no—
403C23—2444no—
404C23—2448no—
405C23—2448no—
406C23C22⋊C4×D6112yes—
407C23#1451979812yes—
408C23#1451980012yes—
409C23C2×D4×D6111yescharacteristic
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—48no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—612no—
420C2×C4—614no—
421C2×C4—618no—
422C2×C4—618no—
423C2×C4—612no—
424C2×C4—1228no—
425C2×C4—1228no—
426C2×C4—1228no—
427C2×C4—1222no—
428C2×C4—1224no—
429C2×C4—1224no—
430C2×C4—1224no—
431C2×C4—2442no—
432C2×C4—2442no—
433C2×C4—2442no—
434C2×C4—2448no—
435C2×C4—2442no—
436C2×C4—2442no—
437C2×C4C22⋊C4×D6112yes—
438C2×C4C22×C4×D6111yescharacteristic
439C2×C4C2×D4×D6111yescharacteristic
440C2×C4#41×D6114yes—
441C8—42no—
442C8—2442no—
443C22—22no—
444C22—24no—
445C22—24no—
446C22—22no—
447C22—614no—
448C22—1224no—
449C22—1224no—
450C22—1222no—
451C22—1222no—
452C22—1222no—
453C22—1222no—
454C22—1222no—
455C22—24416no—
456C22—2442no—
457C22—2442no—
458C22#1683975714yes—
459C22C2×C22⋊C4×D6111yescharacteristic
460C22#123×D6112yes—
461C22C2×#1451979811yescharacteristic
462C22C2×#1451980011yescharacteristic
463C22#1684055812yes—
464C4—22no—
465C4—44no—
466C4—82no—
467C4—614no—
468C4—1222no—
469C4—4882no—
470C4—2444no—
471C4#117×D6112yes—
472C4#123×D6112yes—
473C2—24no—
474C2—1224no—
475C2—2442no—
476C2#381×D6111yescharacteristic
477C2C2×#1683975711yescharacteristic
478C2#2145433311yescharacteristic
479C2#2145572912yes—
480C2#2145573512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1824C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1824D61
hypocenterC61#20699
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
4. #21445913C2semidirect
5. #21445909C2semidirect
6. #21445906C2semidirect
7. #21445903C2semidirect
8. #21445534C2semidirect
9. C61×#1824C2semidirect
15. C2×#14519735C4semidirect
31. #16839525C22semidirect
32. #16839524C22semidirect
33. #16839524C22semidirect
37. C61×#340C22semidirect
238. #1824D61semidirect
277. #340C2×D61semidirect
278. C61#20699semidirect

Maximal subgroups

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

subgroupindex
C22×#145197352
C2×C22⋊C8×D612
#214482012
#214459132
#214459092
#214459062
#214459032
#214455342
C61×#18242
#2069961