Finite Group Databasebeta

← #29656178

Subgroups — #29656178

Order 15616.

940 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
1#21456215C211yesmaximal, characteristic, complemented
2#21455941C211yesmaximal, characteristic, complemented
3#21455910C211yesmaximal, characteristic
4#21444991C214yesmaximal, complemented
5#21444929C214yesmaximal, complemented
6#21444536C211yesmaximal, characteristic, complemented
7#21444266C211yesmaximal, characteristic, complemented
8#21444262C211yesmaximal, characteristic, complemented
9C2×C61×#556C211yesmaximal, characteristic, complemented
10#16840312C2211yescharacteristic
11#16840124—22no—
12#16840114C2212yescomplemented
13#16840110C2212yescomplemented
14#16840080—21no—
15#16840075—21no—
16#16840057C2212yescomplemented
17#16840052C2212yes—
18#16839962C2212yescomplemented
19#16839957C2212yes—
20#16839901—21no—
21#16839896—21no—
22#16839872—22no—
23#16839860—22no—
24#16839819C2211yescharacteristic
25#16839819C2211yescharacteristic
26#16839761—22no—
27C4×#11758194C2211yescharacteristic
28#16839508C2214yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2214yes—
31C61×#556—22no—
32C61×#556—22no—
33C61×#556—22no—
34C61×#556C2214yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×#126×C61—22no—
37C2×#124×C61—21no—
38C2×C22⋊Q8×C61—21no—
39C2×C4×Q8×C61—21no—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C2×#117×C61—21no—
43C2×#11758222—21no—
44C2×#11758201—22no—
45C2×#11758199—22no—
46C2×#11758194—21no—
47#14519777C2311yescharacteristic
48#14519774C2311yescharacteristic
49#14519765—21no—
50#14519751—21no—
51#14519751—21no—
52C4×D244—21no—
53C4×D244—21no—
54#14519710—22no—
55#14519708—22no—
56C4⋊#92418C2312yes—
57C4⋊#92418C2312yes—
58#14519701—22no—
59#14519700—24no—
60#14519698—22no—
61#14519697—22no—
62#14519697—22no—
63#14519697—24no—
64C22⋊#92418—21no—
65C22⋊#92418—21no—
66#14519688—21no—
67#14519688—21no—
68#14519688—21no—
69C4×#92418C2312yes—
70C22×Q8×C61—21no—
71C22×Q8×C61D411yescharacteristic
72C22×D4×C61D411yescharacteristic
73C23×C4×C61—21no—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61D412yes—
77C4⋊Q8×C61C2312yes—
78#126×C61—28no—
79#126×C61—28no—
80#124×C61—24no—
81#124×C61—24no—
82C22⋊Q8×C61—24no—
83C22⋊Q8×C61—24no—
84C4×Q8×C61—24no—
85C4×Q8×C61—24no—
86C4×Q8×C61—21no—
87C4×Q8×C61—21no—
88C4×Q8×C61D412yes—
89C4×Q8×C61C2312yes—
90C4×D4×C61—21no—
91C4×D4×C61—21no—
92C4×D4×C61D412yes—
93C4×D4×C61C2312yes—
94#117×C61—24no—
95#117×C61—24no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61D412yes—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61—21no—
105C2×C42×C61—22no—
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C24×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—42no—
113C2×Q8×C61—44no—
114C2×Q8×C61C2×D412yes—
115C2×D4×C61—21no—
116C2×D4×C61—42no—
117C2×D4×C61C2×D412yes—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—44no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C2×C22⋊D61—41no—
129#11758222—44no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×C4×D61—41no—
134C2×Dic122—21no—
135C4⋊C4×C61—28no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61C2×D412yes—
144C4⋊C4×C61C2×D412yes—
145C4⋊C4×C61C2×D414yes—
146C22⋊C4×C61—28no—
147C22⋊C4×C61—28no—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—22no—
150C42×C61—28no—
151C42×C61—22no—
152C42×C61—22no—
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61C2411yescharacteristic
157#11758201—44no—
158#11758199—44no—
159#11758198—41no—
160#11758198—41no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163#11758194—42no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167C23×C61—22no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61#12011yescharacteristic
172C22×D61—41no—
173Q8×C61—22no—
174Q8×C61—22no—
175Q8×C61—42no—
176Q8×C61—44no—
177Q8×C61—44no—
178D4×C61—22no—
179D4×C61—22no—
180D4×C61—42no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—24no—
188C2×C4×C61—44no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—44no—
192C2×C4×C61—42no—
193C2×C4×C61—44no—
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C22⋊D61—82no—
201C2×Dic61—41no—
202D244—42no—
203D244—42no—
204D244—42no—
205C4×D61—82no—
206Dic122—42no—
207#92418—42no—
208#92418—42no—
209#92418—42no—
210#18536—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—42no—
214C22×C61—42no—
215C22×C61—42no—
216C22×C61—84no—
217C22×C61#45412yes—
218C22×C61C2×#12011yescharacteristic
219C2×D61—81no—
220C2×D61—81no—
221C2×D61—81no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—42no—
227C4×C61—42no—
228C4×C61—44no—
229C4×C61—42no—
230C4×C61#57512yes—
231C4×C61#57512yes—
232Dic61—81no—
233C2×#556D6111yescharacteristic, complemented
234#3141—611no—
235#3112—611no—
236#3109—611no—
237#1638—614no—
238#1628—614no—
239#1556—611no—
240#1491—611no—
241#1490—611no—
242C2×C61—22no—
243C2×C61—82no—
244C2×C61#300511yescharacteristic
245C2×C61#300511yescharacteristic
246C2×C61C2×#45411yescharacteristic
247D61—161no—
248#556—22no—
249#556—22no—
250#556—22no—
251#556C2×D6114yes—
252#532—611no—
253C2×C4⋊Q8C2×D6111yescharacteristic
254C2×#126—22no—
255C2×#124—21no—
256C2×C22⋊Q8—21no—
257C2×C4×Q8—21no—
258C2×C4×Q8C2×D6111yescharacteristic
259C2×C4×D4C2×D6111yescharacteristic
260C2×#117—21no—
261#489—1222no—
262#487—612no—
263#486—612no—
264#481—1221no—
265#480—1221no—
266#477—612no—
267#476—612no—
268#460—612no—
269#459—612no—
270#448—1221no—
271#447—1221no—
272#442—1222no—
273#440—1222no—
274#432—611no—
275#432—611no—
276#417—1222no—
277C4×M16—611no—
278#332—614no—
279Q8⋊C8—614no—
280D4⋊C8—614no—
281C61#1853611yescharacteristic, complemented, Sylow
282C2×#41—1221no—
283C22×Q8—21no—
284C22×Q8C22⋊D6111yescharacteristic
285C22×D4C22⋊D6111yescharacteristic
286C23×C4—21no—
287C2×SD16—1222no—
288C2×D8—1222no—
289C2×M16—1221no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—611no—
293C4⋊Q8C22⋊D6112yes—
294C4⋊Q8C22×D6112yes—
295#127—611no—
296#126—28no—
297#126—28no—
298#124—24no—
299#124—24no—
300#124—1221no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303#121—1221no—
304#121—1221no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8C22⋊D6112yes—
310C4×Q8C22×D6112yes—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—1221no—
314C4×D4—1221no—
315C4×D4C22⋊D6112yes—
316C4×D4C22×D6112yes—
317#117—24no—
318#117—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4C22⋊D6111yescharacteristic
323C2×C4⋊C4C22⋊D6111yescharacteristic
324C2×C4⋊C4C22⋊D6112yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42C22×D6111yescharacteristic
331#107—1222no—
332#106—1222no—
333C4⋊C8—612no—
334C4⋊C8—612no—
335#103—1222no—
336#103—1224no—
337#102—1222no—
338#102—1224no—
339#102—1222no—
340#102—1222no—
341C22⋊C8—1221no—
342C22⋊C8—1221no—
343#97—1221no—
344#97—1221no—
345#97—1221no—
346C4×C8—612no—
347C24—21no—
348#41—2444no—
349C2×Q8—22no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—42no—
353C2×Q8—44no—
354C2×Q8—1221no—
355C2×Q8C2×C22⋊D6112yes—
356C2×D4—21no—
357C2×D4—42no—
358C2×D4—1221no—
359C2×D4—1221no—
360C2×D4—1221no—
361C2×D4—2441no—
362C2×D4C2×C22⋊D6112yes—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—44no—
370C22×C4—2441no—
371C22×C4C2×C22⋊D6111yescharacteristic
372C22×C4C2×C22⋊D6111yescharacteristic
373C22×C4C2×C22⋊D6111yescharacteristic
374SD16—2444no—
375D8—2444no—
376M16—2442no—
377C2×C8—1222no—
378C2×C8—1222no—
379C2×C8—1222no—
380C4⋊C4—28no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—1221no—
389C4⋊C4—1221no—
390C4⋊C4C2×C22⋊D6114yes—
391C4⋊C4C2×C22⋊D6112yes—
392C4⋊C4C2×C22⋊D6112yes—
393C22⋊C4—28no—
394C22⋊C4—28no—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—2441no—
398C22⋊C4—2441no—
399C42—28no—
400C42—22no—
401C42—22no—
402C42C2×C22⋊D6111yescharacteristic
403C42C2×C22⋊D6111yescharacteristic
404C42C2×C22⋊D6111yescharacteristic
405C42C23×D6111yescharacteristic
406C23—22no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—2441no—
411C23#1451974611yescharacteristic
412Q8—22no—
413Q8—22no—
414Q8—42no—
415Q8—44no—
416Q8—44no—
417Q8—2442no—
418D4—22no—
419D4—22no—
420D4—42no—
421D4—4882no—
422D4—2442no—
423D4—2442no—
424D4—2442no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—4882no—
439C2×C4—2441no—
440C2×C4#1451974611yescharacteristic
441C2×C4#1451974611yescharacteristic
442C2×C4#1451974611yescharacteristic
443C2×C4C22×C22⋊D6111yescharacteristic
444C2×C4C22×C22⋊D6111yescharacteristic
445C2×C4C22×C22⋊D6111yescharacteristic
446C8—2442no—
447C8—2442no—
448C8—2442no—
449C22—21no—
450C22—21no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—84no—
455C22—4881no—
456C22—4881no—
457C22—4881no—
458C22#1683993512yes—
459C22C2×#1451974611yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—44no—
468C4—4881no—
469C4#1684056012yes—
470C4#1684056012yes—
471C2—22no—
472C2—82no—
473C2—9761no—
474C2#2145506111yescharacteristic
475C2#2145506111yescharacteristic
476C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#556C2
socleC22×C61C2×#120
prime radical#21455910C2
hypercenterC2×#556D61
hypocenterC61#18536
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. #21456215C2semidirect
2. #21455941C2semidirect
4. #21444991C2semidirect
5. #21444929C2semidirect
6. #21444536C2semidirect
7. #21444266C2semidirect
8. #21444262C2semidirect
9. C2×C61×#556C2semidirect
12. #16840114C22semidirect
13. #16840110C22semidirect
16. #16840057C22semidirect
18. #16839962C22semidirect
28. #16839508C22semidirect
29. Q8⋊#92418C22semidirect
233. C2×#556D61semidirect
281. C61#18536semidirect

Maximal subgroups

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

subgroupindex
#214562152
#214559412
#214559102
#214449912
#214449292
#214445362
#214442662
#214442622
C2×C61×#5562
#1853661