Finite Group Databasebeta

← #29580556

Subgroups — #29580556

Order 15616.

944 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839903C211yesmaximal, characteristic, complemented
2C2×#16839900C211yesmaximal, characteristic, complemented
3#21451923C211yesmaximal, characteristic
4C2×#16839495C211yesmaximal, characteristic, complemented
5#21444089C211yesmaximal, characteristic, complemented
6#21443661C211yesmaximal, characteristic, complemented
7C2×C61×#327C211yesmaximal, characteristic, complemented
8C23×Dic122C2211yescharacteristic
9C22×#11758190—21no—
10C22×#11758200—21no—
11C2×#14519751C2211yescharacteristic
12C2×#14519749—22no—
13#16839903—22no—
14#16839903—22no—
15#16839900—22no—
16#16839900—22no—
17C2×#14519702—21no—
18C2×#14519700—21no—
19C2×C22⋊#92418C2211yescharacteristic
20#16839698—21no—
21#16839697—21no—
22#16839671—21no—
23#16839670—21no—
24C2×#14519684—22no—
25C2×#14519684C2211yescharacteristic
26#16839495—22no—
27C2×#121×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2211yescharacteristic
29C2×#95×C61C2211yescharacteristic
30C61×#327—22no—
31C22×Dic122—21no—
32C22×Dic122—21no—
33C22×Dic122—21no—
34C22×Dic122—21no—
35C22×Dic122—21no—
36C22×Dic122—44no—
37C22×Dic122D412yescomplemented
38C22×D244D411yescharacteristic
39C22×C22⋊D61—21no—
40C22×C22⋊D61—22no—
41C23×Dic61—21no—
42C23×Dic61D412yes—
43C2×#11758190—24no—
44C2×#11758190—44no—
45C2×#11758200—24no—
46C2×#11758200—44no—
47C22×#92418—21no—
48#14519751—21no—
49#14519751—21no—
50#14519751—21no—
51#14519751—21no—
52#14519749—44no—
53#14519749—44no—
54C2×C4⋊Dic61D411yescharacteristic
55C2×#11758196—22no—
56C2×C22⋊Dic61—22no—
57C2×#11758198—22no—
58C2×#11758198—21no—
59#14519702—24no—
60#14519700—24no—
61C22⋊#92418—21no—
62C22⋊#92418D412yescomplemented
63#14519684—21no—
64#14519684—21no—
65#14519684—44no—
66C22×D4×C61—21no—
67C22×D4×C61D411yescharacteristic
68C23×C4×C61—21no—
69C23×C4×C61C2311yescharacteristic
70C22×C8×C61—21no—
71#121×C61—21no—
72#121×C61—21no—
73#121×C61—21no—
74#121×C61—21no—
75C2×C4⋊C4×C61D411yescharacteristic
76C2×C22⋊C4×C61—21no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61D412yescomplemented
79#95×C61—21no—
80#95×C61—21no—
81C24×C61—21no—
82C24×C61C2×D411yescharacteristic
83C23×D61—21no—
84C2×D4×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—42no—
87C2×D4×C61—41no—
88C2×D4×C61—41no—
89C2×D4×C61—42no—
90C2×D4×C61—42no—
91C22×C4×C61—22no—
92C22×C4×C61—21no—
93C22×C4×C61—42no—
94C22×C4×C61—41no—
95C22×C4×C61—42no—
96C22×C4×C61—41no—
97C22×C4×C61—42no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61#4111yescharacteristic
104C2×C22⋊D61—44no—
105C2×C22⋊D61—41no—
106C2×C22⋊D61—41no—
107C2×C22⋊D61—42no—
108C2×C22⋊D61—42no—
109C2×C22⋊D61—82no—
110C2×C22⋊D61—82no—
111C22×Dic61—24no—
112C22×Dic61—22no—
113C22×Dic61—22no—
114C22×Dic61—22no—
115C22×Dic61—21no—
116C22×Dic61—42no—
117C22×Dic61—41no—
118C22×Dic61—41no—
119C22×Dic61—42no—
120C22×Dic61—42no—
121C22×Dic61SD1614yescomplemented
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—42no—
125C2×Dic122—24no—
126C2×Dic122—24no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122—44no—
130C2×Dic122—44no—
131C2×Dic122—44no—
132C2×Dic122—44no—
133C2×Dic122—42no—
134C2×Dic122—42no—
135C2×Dic122—41no—
136C2×Dic122—41no—
137C2×C8×C61—22no—
138C2×C8×C61—42no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C22⋊C4×C61—41no—
142C22⋊C4×C61—41no—
143C22⋊Dic61—44no—
144#11758200—48no—
145#11758198—44no—
146#11758198—41no—
147#11758198—41no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150#11758196—44no—
151C2×#92418—22no—
152C2×#92418—42no—
153#11758190—48no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—22no—
157C23×C61—41no—
158C23×C61—42no—
159C23×C61—41no—
160C23×C61—41no—
161C23×C61—42no—
162C23×C61#12011yescharacteristic
163C23×C61#12111yescharacteristic
164C23×C61#12711yescharacteristic
165C23×C61C2×SD1612yes—
166C23×C61AGL(1,ℤ/8)12yes—
167C22×D61—41no—
168C22×D61—42no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—42no—
172D4×C61—42no—
173D4×C61—81no—
174D4×C61—82no—
175D4×C61—81no—
176D4×C61—81no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—44no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61—42no—
187C2×C4×C61—82no—
188C2×C4×C61—81no—
189C2×C4×C61—81no—
190C2×C4×C61#12011yescharacteristic
191C2×C4×C61#12011yescharacteristic
192C2×C4×C61#12111yescharacteristic
193C2×C4×C61#12111yescharacteristic
194C22⋊D61—88no—
195C22⋊D61—82no—
196C22⋊D61—81no—
197C22⋊D61—81no—
198C2×Dic61—24no—
199C2×Dic61—24no—
200C2×Dic61—42no—
201C2×Dic61—44no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—44no—
205C2×Dic61—42no—
206C2×Dic61—44no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—82no—
210D244—42no—
211D244—81no—
212Dic122—24no—
213Dic122—24no—
214Dic122—42no—
215Dic122—42no—
216Dic122—44no—
217Dic122—44no—
218C8×C61—42no—
219#92418—42no—
220#9964—611nomaximal, Sylow
221C22×C61—22no—
222C22×C61—22no—
223C22×C61—22no—
224C22×C61—22no—
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—82no—
230C22×C61—82no—
231C22×C61—81no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61#39211yescharacteristic
235C22×C61#45012yes—
236C22×C61#45511yescharacteristic
237C22×C61#45611yescharacteristic
238C22×C61B(4,𝔽2)12yes—
239C22×C61#46012yes—
240C22×C61#49211yescharacteristic
241C22×C61#49611yescharacteristic
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—82no—
252C4×C61—81no—
253Dic61—44no—
254Dic61—44no—
255Dic61—81no—
256C2×#448—611no—
257C2×#448—611no—
258#2656—611no—
259C2×#327—611no—
260C2×#327D6111yescharacteristic, complemented
261#1446—611no—
262#1280—611no—
263C2×C61—21no—
264C2×C61—22no—
265C2×C61—21no—
266C2×C61—161no—
267C2×C61#160112yes—
268C2×C61#160412yes—
269C2×C61#200711yescharacteristic
270C2×C61#201411yescharacteristic
271C2×C61#202111yescharacteristic
272D61—161no—
273C23×Q8—611no—
274C22×SD16—1221no—
275C22×SD16—1221no—
276C2×#121—611no—
277C2×#121—1222no—
278C2×#121C2×D6111yescharacteristic
279#448—1222no—
280#448—1222no—
281#448—1222no—
282#448—1222no—
283C2×#103—1221no—
284C2×#103—1221no—
285C2×C22⋊C8—611no—
286C2×C22⋊C8C2×D6111yescharacteristic
287#399—1221no—
288#399—1221no—
289#392—1221no—
290#392—1221no—
291C2×#95—611no—
292C2×#95—1222no—
293C2×#95C2×D6111yescharacteristic
294#327—22no—
295#327—1222no—
296C61#996411yescharacteristic, complemented, Sylow
297C22×Q8—612no—
298C22×Q8—1221no—
299C22×Q8—1221no—
300C22×Q8—1221no—
301C22×Q8—1221no—
302C22×Q8—1221no—
303C22×Q8—2444no—
304C22×D4—21no—
305C22×D4—611no—
306C22×D4—1222no—
307C22×D4—1221no—
308C22×D4D24411yescharacteristic
309C23×C4—21no—
310C23×C4—612no—
311C23×C4—1221no—
312C23×C4C22×D6111yescharacteristic
313C2×SD16—1224no—
314C2×SD16—1224no—
315C2×SD16—2444no—
316C2×SD16—2444no—
317C22×C8—21no—
318C22×C8—1221no—
319#121—21no—
320#121—21no—
321#121—21no—
322#121—21no—
323#121—1221no—
324#121—1221no—
325#121—1221no—
326#121—1221no—
327#121—2444no—
328#121—2444no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4D24411yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—1222no—
334C2×C22⋊C4—1222no—
335C2×C22⋊C4—1221no—
336#103—1224no—
337#103—1224no—
338C22⋊C8—21no—
339C22⋊C8—612no—
340C22⋊C8—1221no—
341C22⋊C8C22⋊D6112yescomplemented
342#95—21no—
343#95—21no—
344#95—1221no—
345#95—1221no—
346#95—2444no—
347C24—21no—
348C24—1221no—
349C24D4×D6111yescharacteristic
350C2×Q8—1224no—
351C2×Q8—1224no—
352C2×Q8—1222no—
353C2×Q8—1222no—
354C2×Q8—2442no—
355C2×Q8—2442no—
356C2×Q8—2444no—
357C2×Q8—2444no—
358C2×Q8—2444no—
359C2×Q8—2444no—
360C2×Q8—2441no—
361C2×Q8—2441no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—42no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—1221no—
370C2×D4—1221no—
371C2×D4—4882no—
372C2×D4—4882no—
373C2×D4—2442no—
374C2×D4—2444no—
375C2×D4—2441no—
376C2×D4—2441no—
377C2×D4—2442no—
378C2×D4—2442no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—614no—
387C22×C4—1222no—
388C22×C4—1224no—
389C22×C4—1222no—
390C22×C4—1222no—
391C22×C4—1221no—
392C22×C4—2442no—
393C22×C4—2442no—
394C22×C4—2442no—
395C22×C4—2441no—
396C22×C4—2441no—
397C22×C4C2×D24411yescharacteristic
398C22×C4#1175822211yescharacteristic
399C22×C4D4×D6111yescharacteristic
400C22×C4D4×D6111yescharacteristic
401C22×C4D4×D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403SD16—2448no—
404SD16—2448no—
405C2×C8—22no—
406C2×C8—42no—
407C2×C8—1222no—
408C2×C8—2442no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—1221no—
412C4⋊C4—1221no—
413C4⋊C4—2444no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—2444no—
417C22⋊C4—2444no—
418C22⋊C4—2441no—
419C22⋊C4—2441no—
420C23—21no—
421C23—21no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23—42no—
427C23—41no—
428C23—2441no—
429C23—2441no—
430C23—2442no—
431C23—2442no—
432C23—2441no—
433C23#1451974711yescharacteristic
434C23#1451975011yescharacteristic
435C23#1451977311yescharacteristic
436C23SD16×D6112yes—
437C23#1451980112yes—
438Q8—1224no—
439Q8—1224no—
440Q8—2442no—
441Q8—2442no—
442Q8—2444no—
443Q8—2444no—
444D4—42no—
445D4—81no—
446D4—82no—
447D4—81no—
448D4—81no—
449D4—4881no—
450D4—4882no—
451D4—4881no—
452D4—4881no—
453D4—4888no—
454D4—2442no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—1224no—
469C2×C4—1224no—
470C2×C4—4882no—
471C2×C4—2442no—
472C2×C4—2441no—
473C2×C4—2444no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2444no—
477C2×C4—2442no—
478C2×C4—2442no—
479C2×C4—2444no—
480C2×C4#1451974511yescharacteristic
481C2×C4#1451974711yescharacteristic
482C2×C4#1451975011yescharacteristic
483C2×C4#1451975111yescharacteristic
484C8—42no—
485C8—2442no—
486C22—21no—
487C22—21no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—21no—
493C22—21no—
494C22—81no—
495C22—82no—
496C22—82no—
497C22—81no—
498C22—81no—
499C22—4881no—
500C22—4882no—
501C22—4882no—
502C22—4881no—
503C22—4881no—
504C22#1683967111yescharacteristic
505C22#1683991412yes—
506C22#1683994111yescharacteristic
507C22#1683994611yescharacteristic
508C22#1683995012yes—
509C22#1683996112yes—
510C22#1684013311yescharacteristic
511C22#1684015011yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—82no—
517C4—81no—
518C4—4881no—
519C4—2444no—
520C4—2444no—
521C2—21no—
522C2—22no—
523C2—21no—
524C2—161no—
525C2—9761no—
526C2#2144477212yes—
527C2#2144478912yes—
528C2#2144686711yescharacteristic
529C2#2144690811yescharacteristic
530C2#2144694411yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#327C2
socleC23×C61#120
prime radical#21451923C2
hypercenterC2×#327D61
hypocenterC61#9964
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
1. C2×#16839903C2semidirect
2. C2×#16839900C2semidirect
4. C2×#16839495C2semidirect
5. #21444089C2semidirect
6. #21443661C2semidirect
7. C2×C61×#327C2semidirect
37. C22×Dic122D4semidirect
62. C22⋊#92418D4semidirect
78. C22⋊C8×C61D4semidirect
121. C22×Dic61SD16semidirect
260. C2×#327D61semidirect
296. C61#9964semidirect
341. C22⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399032
C2×#168399002
#214519232
C2×#168394952
#214440892
#214436612
C2×C61×#3272
#996461