Finite Group Databasebeta

← #29730133

Subgroups — #29730133

Order 15616.

958 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840053C211yesmaximal, characteristic, complemented
2C2×#16839827C211yesmaximal, characteristic, complemented
3#21450988C211yesmaximal, characteristic, complemented
4#21450735C211yesmaximal, characteristic
5#21447254C212yesmaximal, complemented
6#21447010C212yesmaximal, complemented
7#21446779C211yesmaximal, characteristic, complemented
8#21446246C212yesmaximal, complemented
9#21445841C212yesmaximal, complemented
10C4×#14519697C211yesmaximal, characteristic, complemented
11C61×#1948C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519752—21no—
15C2×#14519752C2212yes—
16C2×C4×D244C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18#16840053—24no—
19#16839827—24no—
20C2×#14519709C2212yes—
21C2×#14519699C2212yes—
22C2×#14519697C2211yescharacteristic
23C2×#14519697C2211yescharacteristic
24C2×#14519689C2212yes—
25C2×C4×#92418C2211yescharacteristic
26#16839679—22no—
27#16839675—22no—
28#16839664—21no—
29#16839628—21no—
30#16839619—21no—
31#16839619C2212yes—
32#16839619C2212yes—
33#16839619C2212yes—
34#16839524C2211yescharacteristic
35#16839524C2211yescharacteristic
36#16839524C2212yes—
37#16839524C2212yes—
38C2×#106×C61C2212yes—
39C2×C4⋊C8×C61C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2212yes—
43C22×Dic122—21no—
44C22×D244—22no—
45C22×D244D411yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61C2312yes—
49C22×#92418C2311yescharacteristic
50C22×#92418C2311yescharacteristic
51C2×C8×D61—24no—
52C2×C8×D61—24no—
53C2×C8×D61—42no—
54#14519777—22no—
55#14519777—22no—
56#14519752—24no—
57#14519752—42no—
58#14519752—44no—
59C4×D244—22no—
60C4×D244—22no—
61C4×D244—22no—
62C4×D244—22no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61D411yescharacteristic
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2312yes—
67C2×#11758198—22no—
68C2×#11758198—21no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×#11758198D412yes—
73#14519709—24no—
74#14519699—24no—
75#14519697—22no—
76#14519697—22no—
77#14519689—24no—
78C4×#92418—22no—
79#14519684—22no—
80#14519684—22no—
81#14519684—22no—
82C22×C8×C61C2312yes—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61C2312yes—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C42×C61—21no—
89C2×C42×C61C2311yescharacteristic
90#106×C61—42no—
91C4⋊C8×C61D414yescomplemented
92#95×C61—21no—
93C23×D61—21no—
94C23×D61C2×D412yes—
95C22×C4×C61—22no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2411yescharacteristic
101C22×Dic61—21no—
102C22×Dic61—22no—
103C22×Dic61C2×D412yes—
104C2×D244—22no—
105C2×D244—22no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244—44no—
112C2×D244—44no—
113C2×D244—82no—
114C2×D244SD1614yes—
115C2×D244#4112yes—
116C2×C4×D61—22no—
117C2×C4×D61—22no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—42no—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×Dic122—42no—
126C2×Dic122—81no—
127C2×Dic122—81no—
128C2×C8×C61—22no—
129C2×C8×C61—22no—
130C2×C8×C61C2×D414yes—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—42no—
138C4⋊C4×C61—81no—
139C42×C61—42no—
140C42×C61C2×D412yes—
141C42×C61#4112yes—
142#11758198—24no—
143#11758198—44no—
144#11758198—44no—
145#11758198—42no—
146#11758198—82no—
147#11758198—82no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—22no—
153C4⋊Dic61—42no—
154C2×#92418—21no—
155C2×#92418—21no—
156C2×#92418—21no—
157C2×#92418—21no—
158C2×#92418—21no—
159C2×#92418#4112yes—
160C8×D61—48no—
161C23×C61C22×D411yescharacteristic
162C22×D61—22no—
163C22×D61—22no—
164C22×D61—22no—
165C22×D61—22no—
166C22×D61—22no—
167C22×D61—22no—
168C22×D61—22no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61C2×SD1614yes—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—42no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—42no—
186C2×C4×C61—41no—
187C2×C4×C61#12112yes—
188C2×C4×C61#12112yes—
189C2×C4×C61C2×SD1612yes—
190C2×C4×C61#13512yes—
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61H(5,𝔽2)11yescharacteristic
196C2×C4×C612−1+411yescharacteristic
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—42no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206D244—22no—
207D244—22no—
208D244—28no—
209D244—41no—
210D244—41no—
211D244—88no—
212C4×D61—42no—
213C4×D61—42no—
214C4×D61—42no—
215C4×D61—42no—
216C4×D61—84no—
217Dic122—84no—
218C8×C61—24no—
219#92418—22no—
220#92418—41no—
221#25339—611nomaximal, Sylow
222C22×C61C2×#12111yescharacteristic
223C22×C61#54611yescharacteristic
224C22×C61#54711yescharacteristic
225C22×C61C22×SD1611yescharacteristic
226C22×C61C2×#13511yescharacteristic
227C22×C61#57611yescharacteristic
228C22×C61#57711yescharacteristic
229C2×D61—24no—
230C2×D61—24no—
231C2×D61—24no—
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—42no—
235C2×D61—42no—
236C2×D61—42no—
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—81no—
240C2×D61—81no—
241C2×D61—81no—
242C2×D61—81no—
243C4×C61—22no—
244C4×C61—81no—
245C4×C61—82no—
246C4×C61—81no—
247C4×C61#46514yes—
248C4×C61#52911yescharacteristic
249C4×C61#54211yescharacteristic
250C4×C61#54611yescharacteristic
251C4×C61#54811yescharacteristic
252Dic61—42no—
253Dic61—81no—
254Dic61—82no—
255C2×#476—611no—
256C2×C8×D4—611no—
257#2563—611no—
258#2537—611no—
259#2077—612no—
260#2034—612no—
261#1990—611no—
262#1948D6111yescharacteristic, complemented
263#1893—612no—
264#1807—612no—
265C4×#102—611no—
266C2×C61#242711yescharacteristic
267C2×C61C2×#46511yescharacteristic
268C2×C61#306211yescharacteristic
269C2×C61#329511yescharacteristic
270C2×C61#330311yescharacteristic
271C2×C61#333511yescharacteristic
272C2×C61#334111yescharacteristic
273D61—44no—
274D61—82no—
275C23×C8—612no—
276C2×C4⋊Q8—611no—
277C2×#121—612no—
278C2×#121—1221no—
279C2×C4×D4—611no—
280C2×C4×D4—611no—
281#476—1224no—
282C8×D4—1224no—
283C2×#106—612no—
284C2×#106C2×D6112yes—
285C2×C4⋊C8C2×D6111yescharacteristic
286C2×#102—612no—
287C2×#102—611no—
288C2×#102—611no—
289C2×C22⋊C8—612no—
290C2×C4×C8—611no—
291#394—1222no—
292#393—1222no—
293#390—1221no—
294#384C2×D6111yescharacteristic
295#382—1221no—
296#380—612no—
297#380—612no—
298#380—612no—
299#380—1221no—
300C4×C4⋊C4C2×D6111yescharacteristic
301#340—611no—
302#340—612no—
303#340—611no—
304#340—612no—
305#340C2×D6112yes—
306C61#2533911yescharacteristic, complemented, Sylow
307C22×Q8—1221no—
308C22×D4—611no—
309C22×D4—611no—
310C22×D4—1222no—
311C23×C4—612no—
312C23×C4—1221no—
313C22×C8—611no—
314C22×C8—611no—
315C22×C8—1224no—
316C22×C8—1224no—
317C22×C8—2442no—
318C22×C8C22×D6112yes—
319C4⋊Q8—1222no—
320C4⋊Q8—1222no—
321#121—1224no—
322#121—2442no—
323#121—2444no—
324C4×D4—1222no—
325C4×D4—1222no—
326C4×D4—1222no—
327C4×D4—1222no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—611no—
331C2×C4⋊C4—611no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4C22×D6112yes—
335C2×C4⋊C4C22×D6111yescharacteristic
336C2×C4⋊C4C22×D6111yescharacteristic
337C2×C22⋊C4—612no—
338C2×C22⋊C4—1222no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1221no—
343C2×C42—21no—
344C2×C42C22×D6111yescharacteristic
345#106—42no—
346#106—1224no—
347C4⋊C8D24414yescomplemented
348#102—1222no—
349#102—1222no—
350#102—1224no—
351C22⋊C8—1224no—
352C4×C8—1222no—
353#95—21no—
354#95—1222no—
355#95—1222no—
356#95—1222no—
357C24—612no—
358C24—1221no—
359C2×Q8—4881no—
360C2×Q8—4881no—
361C2×Q8—2442no—
362C2×D4—612no—
363C2×D4—614no—
364C2×D4—1221no—
365C2×D4—1221no—
366C2×D4—1221no—
367C2×D4—1221no—
368C2×D4—1221no—
369C2×D4—1222no—
370C2×D4—1222no—
371C2×D4—4882no—
372C2×D4—2444no—
373C2×D4—2444no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—612no—
378C22×C4—1222no—
379C22×C4—1222no—
380C22×C4—1222no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1221no—
385C22×C4—1222no—
386C22×C4—2442no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4D4×D6111yescharacteristic
390C22×C4D4×D6111yescharacteristic
391C22×C4C23×D6111yescharacteristic
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—612no—
395C2×C8—1221no—
396C2×C8—1221no—
397C2×C8—1221no—
398C2×C8—1221no—
399C2×C8—1221no—
400C2×C8—2448no—
401C2×C8C2×D24414yes—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—42no—
409C4⋊C4—81no—
410C4⋊C4—1222no—
411C4⋊C4—1222no—
412C4⋊C4—1221no—
413C4⋊C4—1221no—
414C4⋊C4—1222no—
415C4⋊C4—2442no—
416C22⋊C4—1224no—
417C22⋊C4—4882no—
418C22⋊C4—4882no—
419C22⋊C4—2444no—
420C22⋊C4—2444no—
421C22⋊C4—2442no—
422C42—42no—
423C42C2×D24412yes—
424C42#1175822612yes—
425C23—614no—
426C23—1222no—
427C23—1222no—
428C23—1222no—
429C23—1222no—
430C23—1222no—
431C23—1222no—
432C23—1222no—
433C23—2441no—
434C23—2441no—
435C23—2441no—
436C23—2441no—
437C23—2441no—
438C23—2441no—
439C23C2×D4×D6111yescharacteristic
440Q8—4884no—
441D4—1228no—
442D4—1222no—
443D4—1222no—
444D4—4888no—
445D4—2441no—
446D4—2441no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—4884no—
462C2×C4—2441no—
463C2×C4—2442no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4#1451975212yes—
473C2×C4#1451975212yes—
474C2×C4SD16×D6112yes—
475C2×C4#1451979812yes—
476C2×C4C22×D24411yescharacteristic
477C2×C4#41×D6111yescharacteristic
478C2×C4#41×D6111yescharacteristic
479C2×C4C2×#1175822611yescharacteristic
480C2×C4#1451982211yescharacteristic
481C2×C4#1451982511yescharacteristic
482C8—24no—
483C8—1222no—
484C8—2441no—
485C22—1224no—
486C22—1224no—
487C22—1224no—
488C22—4881no—
489C22—4881no—
490C22—4881no—
491C22—4881no—
492C22—2442no—
493C22—2442no—
494C22—2442no—
495C22—2442no—
496C22—2442no—
497C22—2442no—
498C22—2442no—
499C22C2×#1451975211yescharacteristic
500C22#1684040911yescharacteristic
501C22#1684041411yescharacteristic
502C22C2×SD16×D6111yescharacteristic
503C22C2×#1451979811yescharacteristic
504C22#1684056611yescharacteristic
505C22#1684057111yescharacteristic
506C4—22no—
507C4—81no—
508C4—81no—
509C4—82no—
510C4—4881no—
511C4—4882no—
512C4—2442no—
513C4#1683999214yes—
514C4#1684029711yescharacteristic
515C4#1684038611yescharacteristic
516C4#1684040711yescharacteristic
517C4#1684042411yescharacteristic
518C2—4882no—
519C2—2444no—
520C2#2144939411yescharacteristic
521C2C2×#1683999211yescharacteristic
522C2#2145550011yescharacteristic
523C2#2145764611yescharacteristic
524C2#2145775011yescharacteristic
525C2#2145817911yescharacteristic
526C2#2145824011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1948C2
socleC23×C61C22×D4
prime radical#21450735C2
hypercenter#1948D61
hypocenterC61#25339
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
1. C2×#16840053C2semidirect
2. C2×#16839827C2semidirect
3. #21450988C2semidirect
5. #21447254C2semidirect
6. #21447010C2semidirect
7. #21446779C2semidirect
8. #21446246C2semidirect
9. #21445841C2semidirect
10. C4×#14519697C2semidirect
11. C61×#1948C2semidirect
91. C4⋊C8×C61D4semidirect
262. #1948D61semidirect
306. C61#25339semidirect
347. C4⋊C8D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168400532
C2×#168398272
#214509882
#214507352
#214472542
#214470102
#214467792
#214462462
#214458412
C4×#145196972
C61×#19482
#2533961