Finite Group Databasebeta

← #29740129

Subgroups — #29740129

Order 15616.

792 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839972C211yesmaximal, characteristic
2C2×#16839854C211yesmaximal, characteristic
3#21449597C211yesmaximal, characteristic
4#21448737C211yesmaximal, characteristic
5#21447368C211yesmaximal, characteristic
6#21447310C211yesmaximal, characteristic
7#21447241C211yesmaximal, characteristic
8#21447066C211yesmaximal, characteristic
9#21446493C211yesmaximal, characteristic
10#21446379C211yesmaximal, characteristic
11#21446158C211yesmaximal, characteristic
12#21446158C211yesmaximal, characteristic
13#21446158C211yesmaximal, characteristic
14#21446084C211yesmaximal, characteristic
15C61×#1865C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C2×#14519777C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22#16839972—24no—
23#16839854—24no—
24C2×#14519710C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839704—21no—
34#16839695—21no—
35#16839667C2211yescharacteristic
36#16839667C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839660—21no—
39#16839660C2211yescharacteristic
40#16839658—21no—
41#16839658C2211yescharacteristic
42#16839649—21no—
43#16839648—21no—
44#16839627—21no—
45#16839627C2211yescharacteristic
46C4×#11758196—21no—
47C4×#11758196C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C61×#389C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758202—22no—
66C2×#11758202—22no—
67C2×#11758202—22no—
68C2×#11758202D414yes—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519777—22no—
72#14519777—41no—
73Q8×Dic61—22no—
74Q8×Dic61—22no—
75Q8×Dic61—22no—
76Q8×Dic61—41no—
77#14519742—22no—
78#14519742—41no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196D411yescharacteristic
89C2×#11758196D411yescharacteristic
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519710—22no—
96C4⋊#92418—22no—
97#14519702—22no—
98#14519702—22no—
99#14519702—22no—
100#14519702—41no—
101#14519700—22no—
102#14519700—22no—
103C8×Dic61—22no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684D411yescharacteristic
115#14519684D411yescharacteristic
116C22×Q8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C4×Q8×C61—22no—
120C4×Q8×C61—41no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#106×C61—41no—
129#103×C61—22no—
130#103×C61—22no—
131#95×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61C2×D412yes—
136C2×Q8×C61#4112yes—
137C2×Q8×C61#4112yes—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122C2×D412yes—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61—42no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—42no—
182#11758202—28no—
183#11758202—44no—
184C4⋊Dic61—24no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—42no—
189C4⋊Dic61—42no—
190C4⋊Dic61—42no—
191C4⋊Dic61#4112yes—
192#11758196—22no—
193#11758196—22no—
194#11758196—42no—
195#11758196—42no—
196#11758196—42no—
197C4×Dic61—24no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—42no—
203C4×Dic61—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C2×D412yes—
210C23×C61C22×D411yescharacteristic
211Q8×C61—22no—
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61#12314yes—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61#13512yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—22no—
244C2×Dic61—21no—
245C2×Dic61—22no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61#13712yes—
263Dic122—22no—
264Dic122—22no—
265Dic122—41no—
266Dic122—41no—
267Dic122—41no—
268Dic122—41no—
269Dic122—41no—
270Dic122—41no—
271C8×C61—41no—
272C8×C61—41no—
273#92418—22no—
274#92418—41no—
275#26167—611nomaximal, Sylow
276C22×C61#52511yescharacteristic
277C22×C61D4211yescharacteristic
278C22×C61#54811yescharacteristic
279C22×C61C2×#13511yescharacteristic
280C22×C61C2×#13711yescharacteristic
281C22×C61#57811yescharacteristic
282C22×C61#57811yescharacteristic
283C4×C61—24no—
284C4×C61—22no—
285C4×C61—41no—
286C4×C61—81no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61C2×#12311yescharacteristic
290C4×C61#53911yescharacteristic
291C4×C61#54711yescharacteristic
292C4×C61#55311yescharacteristic
293Dic61—22no—
294Dic61—41no—
295Dic61—42no—
296Dic61—41no—
297Dic61—42no—
298Dic61—41no—
299Dic61—81no—
300Dic61—81no—
301C2×#462—611no—
302C2×C4×Q16—611no—
303#2446—611no—
304#2351—611no—
305#2097—611no—
306#2086—611no—
307#2074—611no—
308#2043—611no—
309#1933—611no—
310#1916—611no—
311#1878—611no—
312#1878—611no—
313#1878—611no—
314#1865—611no—
315#1865D6111yescharacteristic
316C2×C61#244311yescharacteristic
317C2×C61#310711yescharacteristic
318C2×C61#312711yescharacteristic
319C2×C61#327011yescharacteristic
320C2×C61#328511yescharacteristic
321C2×C61#336411yescharacteristic
322C2×C61#337411yescharacteristic
323C22×Q16—611no—
324C22×Q16—611no—
325C2×C4⋊Q8—611no—
326C2×C4×Q8—611no—
327C2×C4×Q8—611no—
328C2×C4×Q8—611no—
329C2×C4×Q8C2×D6111yescharacteristic
330#462—1224no—
331C4×Q16—1224no—
332C2×#107—611no—
333C2×#106C2×D6111yescharacteristic
334C2×C4⋊C8—611no—
335C2×#103—611no—
336C2×#103—611no—
337C2×#103—611no—
338C2×#103—611no—
339C2×#103—611no—
340C2×#103—611no—
341C2×#103C2×D6111yescharacteristic
342C2×#103C2×D6111yescharacteristic
343C2×C4×C8—611no—
344#400—1221no—
345#398—1221no—
346#391—611no—
347#391—611no—
348#391—611no—
349#389—611no—
350#389—611no—
351#389—1221no—
352#389—1221no—
353#389C2×D6111yescharacteristic
354#387—1221no—
355#387—1221no—
356#382—611no—
357#382—1221no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—1221no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340C2×D6111yescharacteristic
364#336—611no—
365#336—611no—
366#336—611no—
367#336C2×D6111yescharacteristic
368C61#2616711yescharacteristic, complemented, Sylow
369C22×Q8—611no—
370C22×Q8—611no—
371C22×Q8C22×D6111yescharacteristic
372C2×Q16—614no—
373C2×Q16—1222no—
374C2×Q16—1222no—
375C2×Q16—1222no—
376C2×Q16—1222no—
377C2×Q16—1222no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8C22×D6111yescharacteristic
381C22×C8C22×D6111yescharacteristic
382C4⋊Q8—1222no—
383C4⋊Q8—2441no—
384C4×Q8—22no—
385C4×Q8—41no—
386C4×Q8—1222no—
387C4×Q8—1222no—
388C4×Q8—1222no—
389C4×Q8—1222no—
390C4×Q8—2441no—
391C4×Q8—2441no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C42—21no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—1221no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42C22×D6111yescharacteristic
415#107—1222no—
416#106—41no—
417C4⋊C8—1222no—
418#103—22no—
419#103—22no—
420#103—1222no—
421#103—1222no—
422#103—1222no—
423#103—1222no—
424#103—1222no—
425#103—2441no—
426C4×C8—1222no—
427#95—21no—
428#95—611no—
429#95—611no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—612no—
444C2×Q8—1221no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8#1175822212yes—
456C2×Q8D4×D6112yes—
457C2×Q8#1175822612yes—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4D4×D6111yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4C23×D6111yescharacteristic
473Q16—1228no—
474Q16—2444no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—612no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—612no—
496C4⋊C4—1222no—
497C4⋊C4—1224no—
498C4⋊C4—1221no—
499C4⋊C4—1222no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—2442no—
503C4⋊C4—2442no—
504C4⋊C4—2442no—
505C4⋊C4—2442no—
506C4⋊C4—2442no—
507C4⋊C4—2442no—
508C42—21no—
509C42—21no—
510C42—42no—
511C42—1224no—
512C42—1221no—
513C42—1221no—
514C42—1221no—
515C42—1221no—
516C42—2442no—
517C42—2442no—
518C23C2×D4×D6111yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—22no—
522Q8—1222no—
523Q8—1222no—
524Q8—2441no—
525Q8—2441no—
526Q8—2441no—
527Q8—2441no—
528Q8—2441no—
529Q8—2441no—
530Q8#1451976214yes—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—612no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1222no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1222no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4#1451979812yes—
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4#41×D6111yescharacteristic
575C2×C4C2×#1175822211yescharacteristic
576C2×C4C2×#1175822611yescharacteristic
577C2×C4#1451982211yescharacteristic
578C2×C4#1451982411yescharacteristic
579C8—41no—
580C8—41no—
581C8—1222no—
582C8—2441no—
583C22#1684027511yescharacteristic
584C22#1684039411yescharacteristic
585C22#1684042111yescharacteristic
586C22C2×#1451979811yescharacteristic
587C22#137×D6111yescharacteristic
588C22#1684057611yescharacteristic
589C22#1684057811yescharacteristic
590C4—24no—
591C4—22no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—1222no—
597C4—4881no—
598C4—4881no—
599C4—2441no—
600C4—2442no—
601C4—2442no—
602C4—2441no—
603C4—2441no—
604C4C2×#1451976211yescharacteristic
605C4#1684036111yescharacteristic
606C4#1684041011yescharacteristic
607C4#1684046111yescharacteristic
608C2#2144955811yescharacteristic
609C2#2145589311yescharacteristic
610C2#2145612111yescharacteristic
611C2#2145738711yescharacteristic
612C2#2145755311yescharacteristic
613C2#2145850211yescharacteristic
614C2#2145862311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1865C2
socle, prime radicalC23×C61C22×D4
hypercenter#1865D61
hypocenterC61#26167
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
368. C61#26167semidirect

Maximal subgroups

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

subgroupindex
C2×#168399722
C2×#168398542
#214495972
#214487372
#214473682
#214473102
#214472412
#214470662
#214464932
#214463792
#214461582
#214460842
C61×#18652
#2616761