Finite Group Databasebeta

← #29657191

Subgroups — #29657191

Order 15616.

1088 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456734C211yesmaximal, characteristic, complemented
2C2×#16840136C211yesmaximal, characteristic, complemented
3#21455299C211yesmaximal, characteristic, complemented
4#21444902C214yesmaximal, complemented
5#21444891C214yesmaximal, complemented
6#21444500C211yesmaximal, characteristic, complemented
7#21444270C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3511C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C22×#11758201—22no—
12C2×#14519774C2211yescharacteristic
13#16840138—21no—
14#16840136—22no—
15#16840136C2214yes—
16#16840110—21no—
17#16839962—21no—
18#16839962C2212yes—
19#16839957—21no—
20#16839957C2214yes—
21#16839957C2212yes—
22#16839895—22no—
23#16839815C2211yescharacteristic
24#16839786C2211yescharacteristic
25#16839766—21no—
26C2×C4×#92418C2211yescharacteristic
27#16839511C2214yescomplemented
28Q8⋊#92418C2212yescomplemented
29D4⋊#92418C2214yescomplemented
30D4⋊#92418C2212yescomplemented
31C61×#556—21no—
32C61×#555C2212yescomplemented
33C61×#554C2212yescomplemented
34C61×#553—22no—
35C61×#548—22no—
36C61×#547—21no—
37C61×#542—22no—
38C61×#537—21no—
39C61×#533—21no—
40C61×#532—21no—
41C61×#532C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#520—21no—
44C4×#41×C61C2211yescharacteristic
45C22×D244—21no—
46C22×D244—21no—
47C22×D244—21no—
48#14519802—44no—
49C2×#11758201—22no—
50C2×#11758201—28no—
51C2×#11758201—48no—
52C2×#11758199—22no—
53C2×#11758199—24no—
54C2×#11758194—21no—
55C22×#92418—21no—
56#14519774—21no—
57#14519774—21no—
58#14519774—21no—
59#14519774C2312yes—
60C4⋊#92418—21no—
61C4⋊#92418C2312yes—
62C4⋊#92418C2312yes—
63#14519697—22no—
64#14519697—22no—
65#14519697—24no—
66C22⋊#92418—21no—
67C4×#92418—21no—
68C4×#92418—21no—
69C4×#92418C2312yes—
70C2×#41×C61—22no—
71C2×#41×C61—21no—
72C2×#41×C61D411yescharacteristic
73C22×Q8×C61D412yes—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61D412yes—
77C4⋊Q8×C61D411yescharacteristic
78C4⋊Q8×C61C2312yes—
79#127×C61D411yescharacteristic
80#126×C61—24no—
81#125×C61—22no—
82#124×C61—22no—
83#124×C61—24no—
84#124×C61—21no—
85#123×C61—24no—
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61—24no—
90#121×C61—22no—
91#121×C61—24no—
92#121×C61—22no—
93#121×C61—22no—
94#121×C61—21no—
95#121×C61D412yes—
96C4×Q8×C61—22no—
97C4×Q8×C61—22no—
98C4×Q8×C61D411yescharacteristic
99C4×Q8×C61C2311yescharacteristic
100C4×D4×C61—22no—
101C4×D4×C61—24no—
102C4×D4×C61—22no—
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106C4×D4×C61D411yescharacteristic
107C4×D4×C61C2312yes—
108C4×D4×C61C2311yescharacteristic
109#117×C61—22no—
110#117×C61—22no—
111#117×C61—21no—
112#117×C61D411yescharacteristic
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—22no—
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118C23×D61—41no—
119#41×C61—22no—
120#41×C61—22no—
121#41×C61—42no—
122#41×C61—42no—
123#41×C61—84no—
124C2×Q8×C61—21no—
125C2×Q8×C61—22no—
126C2×Q8×C61—44no—
127C2×Q8×C61C2×D411yescharacteristic
128C2×Q8×C61C2×D414yes—
129C2×D4×C61—21no—
130C2×D4×C61—22no—
131C2×D4×C61—24no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61C2×D412yes—
135C2×D4×C61C2×D411yescharacteristic
136C22×C4×C61—24no—
137C22×C4×C61—22no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—22no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—22no—
148C2×D244—44no—
149C2×D244—42no—
150C2×D244—42no—
151C2×D244—41no—
152C2×D244—41no—
153C2×D244—41no—
154C2×D244—82no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61C2×D411yescharacteristic
165C4⋊C4×C61C2×D411yescharacteristic
166C4⋊C4×C61C2×D412yes—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—21no—
171C22⋊C4×C61—21no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—22no—
175C42×C61—24no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61C2×D411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2411yescharacteristic
183#11758201—416no—
184#11758201—44no—
185#11758199—44no—
186#11758199—84no—
187#11758194—42no—
188C2×#92418—24no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C23×C61—21no—
193C23×C61—22no—
194C23×C61#12011yescharacteristic
195C22×D61—42no—
196C22×D61—82no—
197C22×D61—82no—
198C22×D61—82no—
199Q8×C61—21no—
200Q8×C61—21no—
201Q8×C61—44no—
202Q8×C61—42no—
203Q8×C61—44no—
204D4×C61—21no—
205D4×C61—21no—
206D4×C61—24no—
207D4×C61—42no—
208D4×C61—44no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—84no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—21no—
218C2×C4×C61—24no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—82no—
225C2×C4×C61—84no—
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61#13512yes—
230C2×C4×C61AGL(1,ℤ/8)12yes—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234D244—44no—
235D244—42no—
236D244—42no—
237D244—44no—
238D244—82no—
239D244—82no—
240D244—82no—
241#92418—24no—
242#92418—42no—
243#92418—42no—
244#18626—611nomaximal, Sylow
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—41no—
248C22×C61—42no—
249C22×C61—41no—
250C22×C61—42no—
251C22×C61—41no—
252C22×C61—42no—
253C22×C61#44912yes—
254C22×C61C2×#12011yescharacteristic
255C2×D61—82no—
256C2×D61—82no—
257C2×D61—82no—
258C2×D61—164no—
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—41no—
262C4×C61—42no—
263C4×C61—41no—
264C4×C61—41no—
265C4×C61—44no—
266C4×C61#44912yes—
267C4×C61#45312yes—
268C4×C61C2×#13511yescharacteristic
269C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
270C4×C61#57712yes—
271#3511D6111yescharacteristic, complemented
272#3205—611no—
273C2×#492—611no—
274#3038—611no—
275#1624—614no—
276#1622—614no—
277#1548—611no—
278#1492—611no—
279#1476—611no—
280C2×C61—22no—
281C2×C61—81no—
282C2×C61—82no—
283C2×C61C2×#44911yescharacteristic
284C2×C61#301011yescharacteristic
285C2×C61C2×#45311yescharacteristic
286D61—162no—
287C2×AGL(1,ℤ/8)—1221no—
288C22×SD16—1222no—
289#556—21no—
290#555C2×D6112yescomplemented
291#554C2×D6112yescomplemented
292#553—22no—
293#548—22no—
294#547—21no—
295#542—22no—
296#537—21no—
297#533—21no—
298#532—21no—
299#532C2×D6111yescharacteristic
300C2×C4⋊Q8C2×D6111yescharacteristic
301C2×#127—611no—
302#520—21no—
303C4×#41C2×D6111yescharacteristic
304#493—1221no—
305#492—614no—
306#492—1222no—
307#486—1221no—
308#460—612no—
309#460—1221no—
310#459—614no—
311#459—612no—
312#459—1221no—
313#447—1222no—
314#431—611no—
315#423—611no—
316#418—1221no—
317C2×C4×C8—611no—
318#333—614no—
319Q8⋊C8—612no—
320D4⋊C8—614no—
321D4⋊C8—612no—
322C61#1862611yescharacteristic, complemented, Sylow
323C2×#41—22no—
324C2×#41—21no—
325C2×#41C22⋊D6111yescharacteristic
326C22×Q8C22⋊D6112yes—
327C22×D4—1221no—
328C22×D4—1221no—
329C22×D4—1221no—
330AGL(1,ℤ/8)—2444no—
331C2×SD16—1228no—
332C2×SD16—1222no—
333C2×SD16—2448no—
334C2×D8—1224no—
335C2×D8—1222no—
336C2×M16—1221no—
337C22×C8—1221no—
338C4⋊Q8—21no—
339C4⋊Q8—21no—
340C4⋊Q8C22⋊D6112yes—
341C4⋊Q8C22⋊D6111yescharacteristic
342C4⋊Q8C22×D6112yes—
343#127—612no—
344#127—1221no—
345#127—1221no—
346#127—1221no—
347#127C22⋊D6111yescharacteristic
348#126—24no—
349#125—22no—
350#124—22no—
351#124—24no—
352#124—21no—
353#123—24no—
354C22⋊Q8—22no—
355C22⋊Q8—22no—
356C22⋊Q8—22no—
357C22⋊Q8—24no—
358#121—22no—
359#121—24no—
360#121—22no—
361#121—22no—
362#121—21no—
363#121C22⋊D6112yes—
364C4×Q8—22no—
365C4×Q8—22no—
366C4×Q8C22⋊D6111yescharacteristic
367C4×Q8C22×D6111yescharacteristic
368C4×D4—22no—
369C4×D4—24no—
370C4×D4—22no—
371C4×D4—21no—
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C22⋊D6111yescharacteristic
375C4×D4C22×D6111yescharacteristic
376C4×D4C22×D6112yes—
377#117—22no—
378#117—22no—
379#117—21no—
380#117C22⋊D6111yescharacteristic
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—22no—
384C2×C42—21no—
385C2×C42C22×D6111yescharacteristic
386C4⋊C8—612no—
387C4⋊C8—612no—
388C4⋊C8—1221no—
389#102—1222no—
390#102—1222no—
391#102—1224no—
392C22⋊C8—1221no—
393C4×C8—612no—
394C4×C8—1221no—
395C4×C8—1221no—
396C24—2441no—
397#41—22no—
398#41—22no—
399#41—42no—
400#41—42no—
401#41—84no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8C2×C22⋊D6111yescharacteristic
406C2×Q8C2×C22⋊D6114yes—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—1222no—
413C2×D4—1222no—
414C2×D4—1222no—
415C2×D4—4882no—
416C2×D4—2444no—
417C2×D4—2441no—
418C2×D4—2441no—
419C2×D4—2441no—
420C2×D4—2442no—
421C2×D4—2442no—
422C2×D4C2×C22⋊D6112yes—
423C2×D4C2×C22⋊D6111yescharacteristic
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4C2×C22⋊D6111yescharacteristic
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C2×C22⋊D6111yescharacteristic
433SD16—24416no—
434SD16—2444no—
435D8—4884no—
436D8—2444no—
437M16—2442no—
438C2×C8—1224no—
439C2×C8—1222no—
440C2×C8—1222no—
441C2×C8—1222no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4C2×C22⋊D6111yescharacteristic
452C4⋊C4C2×C22⋊D6111yescharacteristic
453C4⋊C4C2×C22⋊D6112yes—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—22no—
460C22⋊C4—21no—
461C22⋊C4—22no—
462C42—24no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42C2×C22⋊D6111yescharacteristic
467C42C2×C22⋊D6111yescharacteristic
468C42C2×C22⋊D6111yescharacteristic
469C42C23×D6111yescharacteristic
470C23—21no—
471C23—22no—
472C23—4882no—
473C23—4882no—
474C23—4882no—
475C23—2442no—
476C23#1451974611yescharacteristic
477Q8—21no—
478Q8—21no—
479Q8—44no—
480Q8—42no—
481Q8—44no—
482D4—21no—
483D4—21no—
484D4—24no—
485D4—42no—
486D4—44no—
487D4—42no—
488D4—42no—
489D4—84no—
490D4—4882no—
491D4—4882no—
492D4—4882no—
493D4—2444no—
494D4—2442no—
495D4—2442no—
496D4—2444no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—82no—
510C2×C4—84no—
511C2×C4#1451974611yescharacteristic
512C2×C4#1451974611yescharacteristic
513C2×C4#1451974611yescharacteristic
514C2×C4#1451979712yes—
515C2×C4#1451980212yes—
516C2×C4C22×C22⋊D6111yescharacteristic
517C2×C4C22×C22⋊D6111yescharacteristic
518C2×C4C22×C22⋊D6111yescharacteristic
519C8—1224no—
520C8—2442no—
521C8—2442no—
522C22—21no—
523C22—21no—
524C22—41no—
525C22—42no—
526C22—41no—
527C22—42no—
528C22—41no—
529C22—42no—
530C22—9764no—
531C22—4882no—
532C22—4882no—
533C22—4882no—
534C22#1683990712yes—
535C22C2×#1451974611yescharacteristic
536C4—21no—
537C4—21no—
538C4—41no—
539C4—41no—
540C4—44no—
541C4—41no—
542C4—42no—
543C4#1683990712yes—
544C4#1683993012yes—
545C4C2×#1451979711yescharacteristic
546C4C2×#1451980211yescharacteristic
547C4#1684057012yes—
548C2—22no—
549C2—81no—
550C2—82no—
551C2—9762no—
552C2C2×#1683990711yescharacteristic
553C2#2145510511yescharacteristic
554C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3511C2
socleC22×C61C2×#120
hypercenter#3511D61
hypocenterC61#18626
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #21456734C2semidirect
2. C2×#16840136C2semidirect
3. #21455299C2semidirect
4. #21444902C2semidirect
5. #21444891C2semidirect
6. #21444500C2semidirect
7. #21444270C2semidirect
8. #21444201C2semidirect
9. C61×#3511C2semidirect
27. #16839511C22semidirect
28. Q8⋊#92418C22semidirect
29. D4⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
32. C61×#555C22semidirect
33. C61×#554C22semidirect
271. #3511D61semidirect
290. #555C2×D61semidirect
291. #554C2×D61semidirect
322. C61#18626semidirect

Maximal subgroups

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

subgroupindex
#214567342
C2×#168401362
#214552992
#214449022
#214448912
#214445002
#214442702
#214442012
C61×#35112
#1862661