Finite Group Databasebeta

← #29658638

Subgroups — #29658638

Order 15616.

964 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456947C211yesmaximal, characteristic, complemented
2#21455819C211yesmaximal, characteristic
3#21455291C211yesmaximal, characteristic, complemented
4#21445089C214yesmaximal, complemented
5#21444929C214yesmaximal, complemented
6#21444504C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic, complemented
8#21444214C211yesmaximal, characteristic, complemented
9C2×C61×#556C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11#16840312C2211yescharacteristic
12#16840158C2212yescomplemented
13#16840110C2212yescomplemented
14#16840080—22no—
15#16840075—21no—
16#16840049—22no—
17#16840042C2212yes—
18#16839984—22no—
19#16839972C2212yescomplemented
20#16839962C2212yescomplemented
21#16839957C2212yes—
22#16839896—21no—
23C4×#11758199—22no—
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839761—21no—
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—
44#14519808—44no—
45C2×#11758202—22no—
46C2×#11758201—22no—
47C2×#11758199—22no—
48C2×#11758194—21no—
49#14519777C2311yescharacteristic
50#14519774C2311yescharacteristic
51#14519765—21no—
52#14519751—21no—
53#14519751—21no—
54C4×D244—21no—
55C4×D244—21no—
56#14519710—22no—
57#14519708—24no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519700—22no—
63#14519698—22no—
64#14519697—22no—
65#14519697—22no—
66#14519697—24no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C4×#92418—21no—
70C4×#92418C2312yes—
71C22×Q8×C61—21no—
72C22×Q8×C61D411yescharacteristic
73C22×D4×C61D411yescharacteristic
74C23×C4×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—21no—
77C4⋊Q8×C61D412yes—
78C4⋊Q8×C61C2312yes—
79#126×C61—28no—
80#126×C61—28no—
81#124×C61—24no—
82#124×C61—24no—
83C22⋊Q8×C61—24no—
84C22⋊Q8×C61—24no—
85C4×Q8×C61—24no—
86C4×Q8×C61—24no—
87C4×Q8×C61—21no—
88C4×Q8×C61—21no—
89C4×Q8×C61D412yes—
90C4×Q8×C61C2312yes—
91C4×D4×C61—21no—
92C4×D4×C61—21no—
93C4×D4×C61D412yes—
94C4×D4×C61C2312yes—
95#117×C61—24no—
96#117×C61—24no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61D412yes—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61—22no—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—22no—
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C24×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—44no—
116C2×Q8×C61C2×D412yes—
117C2×D4×C61—21no—
118C2×D4×C61—42no—
119C2×D4×C61C2×D412yes—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C2×C22⋊D61—41no—
132#11758222—42no—
133#11758222—42no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×C4×D61—41no—
138C2×Dic122—21no—
139C4⋊C4×C61—28no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61C2×D412yes—
148C4⋊C4×C61C2×D412yes—
149C4⋊C4×C61C2×D414yes—
150C22⋊C4×C61—28no—
151C22⋊C4×C61—28no—
152C22⋊C4×C61—22no—
153C22⋊C4×C61—22no—
154C42×C61—28no—
155C42×C61—22no—
156C42×C61—22no—
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C42×C61C2411yescharacteristic
161#11758202—44no—
162#11758201—44no—
163#11758199—44no—
164#11758198—41no—
165#11758198—41no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168#11758194—42no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172C23×C61—22no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61#12011yescharacteristic
177C22×D61—41no—
178Q8×C61—22no—
179Q8×C61—22no—
180Q8×C61—44no—
181Q8×C61—44no—
182Q8×C61—42no—
183D4×C61—22no—
184D4×C61—22no—
185D4×C61—42no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—24no—
191C2×C4×C61—41no—
192C2×C4×C61—42no—
193C2×C4×C61—44no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—44no—
199C2×C4×C61#12011yescharacteristic
200C2×C4×C61#12011yescharacteristic
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61#13512yes—
203C2×C4×C61AGL(1,ℤ/8)12yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C22⋊D61—82no—
208C2×Dic61—41no—
209D244—42no—
210D244—41no—
211D244—41no—
212D244—42no—
213C4×D61—82no—
214Dic122—41no—
215Dic122—41no—
216#92418—42no—
217#92418—42no—
218#92418—42no—
219#18762—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—42no—
223C22×C61—42no—
224C22×C61—42no—
225C22×C61—84no—
226C22×C61#44912yes—
227C22×C61C2×#12011yescharacteristic
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—42no—
235C4×C61—42no—
236C4×C61—44no—
237C4×C61—42no—
238C4×C61#44912yes—
239C4×C61C2×#13511yescharacteristic
240C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
241C4×C61#57512yes—
242Dic61—81no—
243C2×#556D6111yescharacteristic, complemented
244#3227—611no—
245#3100—611no—
246#3037—611no—
247#1652—614no—
248#1628—614no—
249#1549—611no—
250#1486—611no—
251#1479—611no—
252C2×C61—22no—
253C2×C61—82no—
254C2×C61C2×#44911yescharacteristic
255C2×C61#300511yescharacteristic
256C2×C61#302011yescharacteristic
257D61—161no—
258C2×#137—1221no—
259#556—22no—
260#556—22no—
261#556—22no—
262#556C2×D6114yes—
263#532—611no—
264C2×C4⋊Q8C2×D6111yescharacteristic
265C2×#126—22no—
266C2×#124—21no—
267C2×C22⋊Q8—21no—
268C2×C4×Q8—21no—
269C2×C4×Q8C2×D6111yescharacteristic
270C2×C4×D4C2×D6111yescharacteristic
271C2×#117—21no—
272#498—612no—
273#486—612no—
274#481—1222no—
275#480—1221no—
276#476—1222no—
277#474—612no—
278#464—1222no—
279#462—612no—
280#460—612no—
281#459—612no—
282#447—1221no—
283C4×D8—1222no—
284#431—611no—
285#431—611no—
286#423—611no—
287#417—1221no—
288#332—614no—
289Q8⋊C8—614no—
290D4⋊C8—614no—
291C61#1876211yescharacteristic, complemented, Sylow
292C2×#41—1221no—
293C22×Q8—21no—
294C22×Q8C22⋊D6111yescharacteristic
295C22×D4C22⋊D6111yescharacteristic
296C23×C4—21no—
297#137—2444no—
298C2×Q16—1222no—
299C2×SD16—1222no—
300C2×D8—1222no—
301C2×M16—1221no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—611no—
305C4⋊Q8C22⋊D6112yes—
306C4⋊Q8C22×D6112yes—
307#127—611no—
308#126—28no—
309#126—28no—
310#124—24no—
311#124—24no—
312#124—1221no—
313C22⋊Q8—24no—
314C22⋊Q8—24no—
315#121—1221no—
316#121—1221no—
317C4×Q8—24no—
318C4×Q8—24no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×Q8C22⋊D6112yes—
322C4×Q8C22×D6112yes—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—1221no—
326C4×D4—1221no—
327C4×D4C22⋊D6112yes—
328C4×D4C22×D6112yes—
329#117—24no—
330#117—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4C22⋊D6111yescharacteristic
335C2×C4⋊C4C22⋊D6111yescharacteristic
336C2×C4⋊C4C22⋊D6112yes—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42C22×D6111yescharacteristic
343#107—1222no—
344#106—1224no—
345C4⋊C8—612no—
346C4⋊C8—612no—
347C4⋊C8—1221no—
348C4⋊C8—1221no—
349#103—1222no—
350#102—1222no—
351#102—1222no—
352#102—1222no—
353#102—1224no—
354C22⋊C8—1221no—
355C22⋊C8—1221no—
356C4×C8—612no—
357C4×C8—1221no—
358C24—21no—
359#41—2442no—
360#41—2442no—
361C2×Q8—22no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—44no—
367C2×Q8—1221no—
368C2×Q8C2×C22⋊D6112yes—
369C2×D4—21no—
370C2×D4—42no—
371C2×D4—1221no—
372C2×D4—1221no—
373C2×D4—1221no—
374C2×D4—2441no—
375C2×D4C2×C22⋊D6112yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—2441no—
385C22×C4C2×C22⋊D6111yescharacteristic
386C22×C4C2×C22⋊D6111yescharacteristic
387C22×C4C2×C22⋊D6111yescharacteristic
388Q16—2444no—
389SD16—2444no—
390D8—2444no—
391M16—2442no—
392C2×C8—1222no—
393C2×C8—1222no—
394C2×C8—1222no—
395C4⋊C4—28no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—1221no—
404C4⋊C4—1221no—
405C4⋊C4C2×C22⋊D6112yes—
406C4⋊C4C2×C22⋊D6114yes—
407C4⋊C4C2×C22⋊D6112yes—
408C22⋊C4—28no—
409C22⋊C4—28no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C42—28no—
415C42—22no—
416C42—22no—
417C42C2×C22⋊D6111yescharacteristic
418C42C2×C22⋊D6111yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—22no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—2441no—
426C23#1451974611yescharacteristic
427Q8—22no—
428Q8—22no—
429Q8—44no—
430Q8—44no—
431Q8—42no—
432Q8—2441no—
433Q8—2441no—
434D4—22no—
435D4—22no—
436D4—42no—
437D4—4882no—
438D4—2442no—
439D4—2441no—
440D4—2441no—
441D4—2442no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—4882no—
456C2×C4—2441no—
457C2×C4#1451974611yescharacteristic
458C2×C4#1451974611yescharacteristic
459C2×C4#1451974611yescharacteristic
460C2×C4#1451979712yes—
461C2×C4#1451980212yes—
462C2×C4C22×C22⋊D6111yescharacteristic
463C2×C4C22×C22⋊D6111yescharacteristic
464C2×C4C22×C22⋊D6111yescharacteristic
465C8—2442no—
466C8—2442no—
467C8—2442no—
468C22—21no—
469C22—21no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—84no—
474C22—4881no—
475C22—4881no—
476C22—4881no—
477C22#1683990712yes—
478C22C2×#1451974611yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—44no—
486C4—4881no—
487C4#1683990712yes—
488C4C2×#1451979711yescharacteristic
489C4C2×#1451980211yescharacteristic
490C4#1684056012yes—
491C2—22no—
492C2—82no—
493C2—9761no—
494C2C2×#1683990711yescharacteristic
495C2#2145506111yescharacteristic
496C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#556C2
socleC22×C61C2×#120
prime radical#21455819C2
hypercenterC2×#556D61
hypocenterC61#18762
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. #21456947C2semidirect
3. #21455291C2semidirect
4. #21445089C2semidirect
5. #21444929C2semidirect
6. #21444504C2semidirect
7. #21444244C2semidirect
8. #21444214C2semidirect
9. C2×C61×#556C2semidirect
12. #16840158C22semidirect
13. #16840110C22semidirect
19. #16839972C22semidirect
20. #16839962C22semidirect
28. #16839508C22semidirect
29. Q8⋊#92418C22semidirect
243. C2×#556D61semidirect
291. C61#18762semidirect

Maximal subgroups

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

subgroupindex
#214569472
#214558192
#214552912
#214450892
#214449292
#214445042
#214442442
#214442142
C2×C61×#5562
#1876261