Finite Group Databasebeta

← #30624020

Subgroups — #30624020

Order 15616.

1220 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840748C212yesmaximal, complemented
2#21462121C212yesmaximal, complemented
3#21462118C212yesmaximal, complemented
4#21462114C214yesmaximal, complemented
5#21462110C214yesmaximal, complemented
6#21459148C211yesmaximal, characteristic
7C2×D61⋊M16—22no—
8#16840748—22no—
9#16840748—22no—
10#16840748C2214yes—
11#16840748C2214yescomplemented
12C2×#14519838C2214yescomplemented
13C2×#14519837—22no—
14#16840672C2218yescomplemented
15#16840670C2218yescomplemented
16C22×#11758222C2211yescharacteristic
17#16840340—22no—
18#16840336C411yescharacteristic
19#16840336C2211yescharacteristic
20#16840333C2212yes—
21#16840329C411yescharacteristic
22#16840329C2211yescharacteristic
23#16840326C412yes—
24#16840321—22no—
25#16840308C412yes—
26#16840308C2212yes—
27#16840273—22no—
28#16840269—22no—
29C2×#14519736C411yescharacteristic
30C22×#41×C61C411yescharacteristic
31C2×#11758213—42no—
32D61⋊M16—24no—
33D61⋊M16—24no—
34D61⋊M16—42no—
35D61⋊M16—42no—
36C2×#11758217—24no—
37C2×D61⋊C8—42no—
38#14519838—24no—
39#14519838C2318yes—
40#14519837—24no—
41#14519837—24no—
42C2×#11758222—42no—
43C2×#11758222—42no—
44C2×#11758222C2×C411yescharacteristic
45C2×#11758222C2×C411yescharacteristic
46C2×#11758222D412yes—
47C2×#11758222D412yes—
48C2×#11758222C2311yescharacteristic
49C2×#11758222C2311yescharacteristic
50C22×Dic122C2311yescharacteristic
51C22×D244C2×C411yescharacteristic
52C22×C22⋊D61D412yes—
53C22×C4×D61D412yes—
54#14519775C2×C412yes—
55#14519775C2312yes—
56#14519773C2×C412yes—
57#14519773C2×C412yes—
58#14519766C2×C412yes—
59#14519766C2×C412yes—
60#14519764C2×C412yes—
61#14519764C2312yes—
62#14519760—22no—
63#14519760—22no—
64#14519758—24no—
65#14519756—24no—
66#14519753—22no—
67#14519753—22no—
68#14519751—24no—
69#14519749—24no—
70#14519749—24no—
71#14519748—22no—
72#14519748—22no—
73#14519746—22no—
74#14519746—22no—
75#14519745—24no—
76C4×C22⋊D61—24no—
77C4×C22⋊D61—24no—
78#14519736—21no—
79#14519736—21no—
80#14519736C2×C412yes—
81#14519736C2×C411yescharacteristic
82#14519736C2×C411yescharacteristic
83C2×C4⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C4×Dic61C2×C412yes—
86C2×#41×C61—21no—
87C2×#41×C61—21no—
88C2×#41×C61—42no—
89C2×#41×C61—42no—
90C2×#41×C61C2×C412yes—
91C2×#41×C61C2×C412yes—
92C22×Q8×C61C2×C411yescharacteristic
93C22×D4×C61—21no—
94C22×D4×C61C2×C411yescharacteristic
95C23×C4×C61—21no—
96C23×C4×C61C2×C411yescharacteristic
97C24×C61—21no—
98C24×C61C22⋊C411yescharacteristic
99C23×D61C22⋊C412yes—
100#41×C61—44no—
101#41×C61—44no—
102#41×C61—42no—
103#41×C61—42no—
104#41×C61—42no—
105#41×C61—42no—
106C2×Q8×C61—21no—
107C2×Q8×C61—42no—
108C2×Q8×C61C22×C412yes—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—21no—
112C2×D4×C61—42no—
113C2×D4×C61—44no—
114C2×D4×C61C22×C412yes—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C22⋊C411yescharacteristic
123C22×C4×C61C22⋊C411yescharacteristic
124C22×C4×C61C22⋊C411yescharacteristic
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C22×C411yescharacteristic
128C2×C22⋊D61—24no—
129C2×C22⋊D61—24no—
130C2×C22⋊D61—42no—
131C2×C22⋊D61—42no—
132C2×C22⋊D61C22⋊C412yes—
133C2×C22⋊D61C22⋊C412yes—
134C2×C22⋊D61C2×D412yes—
135C2×C22⋊D61C2×D412yes—
136C22×Dic61—22no—
137C22×Dic61C2×D412yes—
138#11758222—42no—
139#11758222—41no—
140#11758222—41no—
141#11758222—42no—
142#11758222—42no—
143#11758222—42no—
144#11758222—42no—
145#11758222—41no—
146#11758222—42no—
147#11758222—41no—
148C2×D244—42no—
149C2×D244C22⋊C412yes—
150C2×D244C22×C411yescharacteristic
151C2×D244C22×C411yescharacteristic
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61C22⋊C412yes—
157C2×C4×D61C22⋊C412yes—
158C2×C4×D61C2×D412yes—
159C2×C4×D61C2×D412yes—
160C2×Dic122—42no—
161C2×Dic122C22×C411yescharacteristic
162C2×Dic122C2×D412yes—
163C2×Dic122C2411yescharacteristic
164#11758217—28no—
165#11758217—44no—
166C2×#92420—44no—
167#11758213—44no—
168#11758213—44no—
169D61⋊C8—44no—
170D61⋊C8—44no—
171C22⋊Dic61—22no—
172C22⋊Dic61—21no—
173C22⋊Dic61—21no—
174C22⋊Dic61—24no—
175C22⋊Dic61—24no—
176#11758198—24no—
177#11758198—24no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—22no—
181#11758196—24no—
182#11758196—24no—
183C4×Dic61—22no—
184C4×Dic61C22×C412yes—
185C4×Dic61C22×C412yes—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—42no—
189C23×C61—82no—
190C23×C61—81no—
191C23×C61C2×C22⋊C411yescharacteristic
192C23×C61C2×C22⋊C411yescharacteristic
193C23×C61C2×C22⋊C411yescharacteristic
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—82no—
197C22×D61C2×C22⋊C412yes—
198C22×D61C2×C22⋊C412yes—
199Q8×C61—42no—
200Q8×C61—42no—
201D4×C61—42no—
202D4×C61—44no—
203D4×C61—44no—
204D4×C61—42no—
205C2×C4×C61—22no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—44no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61#9912yes—
218C2×C4×C61#9912yes—
219C2×C4×C61C2×C22⋊C411yescharacteristic
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C23×C411yescharacteristic
223C22⋊D61—42no—
224C22⋊D61—42no—
225C22⋊D61—42no—
226C22⋊D61—42no—
227C22⋊D61—84no—
228C22⋊D61—84no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—82no—
234C2×Dic61C2×C22⋊C412yes—
235C2×Dic61C22×D412yes—
236D244—42no—
237D244—41no—
238D244—41no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—22no—
242C4×D61—22no—
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247Dic122—41no—
248Dic122—42no—
249Dic122—41no—
250#92420—48no—
251#33369—611nomaximal, Sylow
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—41no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—44no—
258C22×C61—42no—
259C22×C61—82no—
260C22×C61—84no—
261C22×C61C2×#9912yes—
262C22×C61C22×C22⋊C411yescharacteristic
263C2×D61—22no—
264C2×D61—22no—
265C2×D61—44no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—82no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—42no—
272C4×C61C2×#9911yescharacteristic
273C4×C61C2×#9912yes—
274C4×C61C2×#9911yescharacteristic
275Dic61—22no—
276Dic61—22no—
277Dic61—42no—
278Dic61—42no—
279#3449—611no—
280C2×#413—612no—
281#2135—612no—
282#2134—612no—
283#2133—614no—
284#2132—614no—
285C2×C61—21no—
286C2×C61—22no—
287C2×C61—41no—
288C2×C61—41no—
289C2×C61—82no—
290C2×C61C22×#9911yescharacteristic
291D61—42no—
292D61—42no—
293C22×#41—611no—
294C22×#41#692311yescharacteristic
295C22×M16—1222no—
296#537—1222no—
297#536—612no—
298#536—611no—
299#536—611no—
300#535—612no—
301#535—611no—
302#535—611no—
303#534—1222no—
304#532—612no—
305#532—612no—
306#525—1222no—
307#525—1222no—
308C2×#117—611no—
309#413—614no—
310#413—614no—
311#413—1222no—
312#413—1222no—
313C2×#101—614no—
314C2×#100—1222no—
315#356—618no—
316#355—618no—
317C61#3336911yescharacteristic, complemented, Sylow
318C2×#41—21no—
319C2×#41—21no—
320C2×#41—42no—
321C2×#41—42no—
322C2×#41—612no—
323C2×#41—612no—
324C2×#41—611no—
325C2×#41—611no—
326C2×#41—611no—
327C2×#41—611no—
328C2×#41—2442no—
329C2×#41—2442no—
330C2×#41C2×#692312yes—
331C2×#41C2×#692312yes—
332C22×Q8—611no—
333C22×Q8C2×#692311yescharacteristic
334C22×D4—21no—
335C22×D4—612no—
336C22×D4—611no—
337C22×D4C2×#692311yescharacteristic
338C23×C4—21no—
339C23×C4—612no—
340C23×C4C2×#692311yescharacteristic
341C2×M16—1224no—
342C2×M16—1224no—
343C2×M16—1224no—
344C2×M16—2442no—
345C2×M16—2442no—
346C2×M16—2442no—
347C22×C8—2442no—
348C4⋊Q8—612no—
349C4⋊Q8—612no—
350#127—612no—
351#127—612no—
352#124—612no—
353#124—612no—
354#124—612no—
355#124—612no—
356#123—1222no—
357#123—1222no—
358#123—1224no—
359C22⋊Q8—1224no—
360C22⋊Q8—1222no—
361C22⋊Q8—1222no—
362#121—1224no—
363#121—1222no—
364#121—1222no—
365#121—1224no—
366#121—1224no—
367#120—1224no—
368#120—1222no—
369#120—1222no—
370C4×D4—1224no—
371C4×D4—1224no—
372#117—612no—
373#117—611no—
374#117—611no—
375#117—1221no—
376#117—1221no—
377C2×C4⋊C4—1221no—
378C2×C22⋊C4—1221no—
379C2×C42—612no—
380#101—618no—
381#101—1224no—
382#100—1224no—
383#100—1224no—
384C24—21no—
385C24—612no—
386C24C22⋊#692311yescharacteristic
387#41—44no—
388#41—44no—
389#41—42no—
390#41—42no—
391#41—42no—
392#41—42no—
393#41—2442no—
394#41—2442no—
395#41—2442no—
396#41—2442no—
397#41—2442no—
398#41—2441no—
399#41—2441no—
400#41—2442no—
401#41—2441no—
402#41—2441no—
403C2×Q8—21no—
404C2×Q8—42no—
405C2×Q8—612no—
406C2×Q8—611no—
407C2×Q8—611no—
408C2×Q8—2442no—
409C2×Q8C22×#692312yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—612no—
416C2×D4—612no—
417C2×D4—612no—
418C2×D4—612no—
419C2×D4—612no—
420C2×D4—611no—
421C2×D4—611no—
422C2×D4—1224no—
423C2×D4—1224no—
424C2×D4—2442no—
425C2×D4—2442no—
426C2×D4—2442no—
427C2×D4C22×#692312yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—612no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4C22⋊#692311yescharacteristic
446C22×C4C22⋊#692311yescharacteristic
447C22×C4C22⋊#692311yescharacteristic
448C22×C4C22×#692311yescharacteristic
449C22×C4C22×#692311yescharacteristic
450C22×C4C22×#692311yescharacteristic
451M16—1228no—
452M16—2444no—
453M16—2444no—
454M16—2444no—
455C2×C8—2444no—
456C2×C8—2444no—
457C2×C8—2444no—
458C4⋊C4—1224no—
459C4⋊C4—1224no—
460C4⋊C4—1222no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C22⋊C4—1224no—
464C22⋊C4—1224no—
465C22⋊C4—1224no—
466C22⋊C4—1224no—
467C22⋊C4—1221no—
468C22⋊C4—1221no—
469C22⋊C4—1222no—
470C42—612no—
471C42—612no—
472C42—1222no—
473C23—22no—
474C23—22no—
475C23—42no—
476C23—81no—
477C23—82no—
478C23—612no—
479C23—612no—
480C23—1222no—
481C23—1222no—
482C23—4882no—
483C23C2×C22⋊#692311yescharacteristic
484C23C2×C22⋊#692311yescharacteristic
485C23C2×C22⋊#692311yescharacteristic
486Q8—42no—
487Q8—42no—
488Q8—2442no—
489Q8—2441no—
490Q8—2441no—
491D4—42no—
492D4—44no—
493D4—44no—
494D4—42no—
495D4—4884no—
496D4—4884no—
497D4—2442no—
498D4—2441no—
499D4—2441no—
500D4—2442no—
501D4—2442no—
502D4—2442no—
503D4—2442no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—612no—
517C2×C4—612no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—4882no—
531C2×C4#1451983512yes—
532C2×C4#1451983512yes—
533C2×C4C2×C22⋊#692311yescharacteristic
534C2×C4C2×C22⋊#692311yescharacteristic
535C2×C4C2×C22⋊#692311yescharacteristic
536C2×C4C23×#692311yescharacteristic
537C8—2448no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—44no—
544C22—42no—
545C22—82no—
546C22—84no—
547C22—1222no—
548C22—1222no—
549C22—4882no—
550C22—2442no—
551C22—2442no—
552C22—2444no—
553C22C2×#1451983512yes—
554C22C22×C22⋊#692311yescharacteristic
555C4—21no—
556C4—21no—
557C4—42no—
558C4—1222no—
559C4—1222no—
560C4—2442no—
561C4—2442no—
562C4C2×#1451983511yescharacteristic
563C4C2×#1451983512yes—
564C4C2×#1451983511yescharacteristic
565C2—21no—
566C2—22no—
567C2—41no—
568C2—41no—
569C2—82no—
570C2—2442no—
571C2—2442no—
572C2C22×#1451983511yescharacteristic
subgroupG/H
centerC4C2×#14519835
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC22×#41×C61C4
socleC2×C61C22×#99
prime radical#21459148C2
cosocleC2×Dic122C24
hypercenterC22×#41#6923
hypocenterC61#33369
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16840748C2semidirect
2. #21462121C2semidirect
3. #21462118C2semidirect
4. #21462114C2semidirect
5. #21462110C2semidirect
11. #16840748C22semidirect
12. C2×#14519838C22semidirect
14. #16840672C22semidirect
15. #16840670C22semidirect
317. C61#33369semidirect

Maximal subgroups

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

subgroupindex
C2×#168407482
#214621212
#214621182
#214621142
#214621102
#214591482
#3336961