Finite Group Databasebeta

← #29590470

Subgroups — #29590470

Order 15616.

778 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#21449138C211yesmaximal, characteristic
2C2×#16839529C211yesmaximal, characteristic, complemented
3#21443841C212yesmaximal, complemented
4#21443817C212yesmaximal, complemented
5C61×#1462C211yesmaximal, characteristic, complemented
6C22×#11758222C411yescharacteristic, complemented
7C2×#14519746—21no—
8C22×C22⋊Dic61—21no—
9#16839674C411yescharacteristic, complemented
10#16839674C2211yescharacteristic
11#16839671C411yescharacteristic, complemented
12#16839671C2211yescharacteristic
13#16839641—21no—
14#16839638—21no—
15#16839638—21no—
16#16839625C411yescharacteristic, complemented
17#16839529—21no—
18#16839529C2212yes—
19C22×C22⋊C4×C61C2211yescharacteristic
20C61×#407—21no—
21C61×#323—21no—
22C61×#323C2212yes—
23C2×#11758222—41no—
24C2×#11758222—41no—
25C2×#11758222—44no—
26C22×Dic122C2×C411yescharacteristic, complemented
27C22×D244C2×C411yescharacteristic, complemented
28C22×C22⋊D61—21no—
29C22×C22⋊D61—21no—
30C22×C22⋊D61—21no—
31C23×Dic61—41no—
32C22×C4×D61—21no—
33#14519746—44no—
34C2×C22⋊Dic61—24no—
35C2×C22⋊Dic61—22no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61—41no—
40C2×C22⋊Dic61—42no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684C2×C411yescharacteristic, complemented
47#14519684C2×C411yescharacteristic, complemented
48C25×C61D411yescharacteristic
49C23×C4×C61C2×C411yescharacteristic
50C23×C4×C61D411yescharacteristic
51C2×M16×C61—41no—
52C2×C22⋊C4×C61—41no—
53C2×C22⋊C4×C61—41no—
54C2×C22⋊C4×C61C2×C411yescharacteristic
55C2×C22⋊C4×C61D411yescharacteristic
56C2×C22⋊C4×C61D411yescharacteristic
57C2×C22⋊C4×C61C2311yescharacteristic
58C22⋊C8×C61—22no—
59C22⋊C8×C61—41no—
60C24×C61—21no—
61C24×C61—21no—
62C24×C61—22no—
63C24×C61—21no—
64C24×C61—21no—
65C24×C61—42no—
66C24×C61—81no—
67C24×C61C22⋊C411yescharacteristic
68C24×C61C22⋊C411yescharacteristic
69C24×C61C2×D411yescharacteristic
70C23×D61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—42no—
76C22×C4×C61—81no—
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22×C411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C2×C22⋊D61—24no—
82C2×C22⋊D61—42no—
83C2×C22⋊D61—41no—
84C2×C22⋊D61—41no—
85C2×C22⋊D61—42no—
86C2×C22⋊D61—42no—
87C2×C22⋊D61—42no—
88C2×C22⋊D61—41no—
89C2×C22⋊D61—41no—
90C2×C22⋊D61—81no—
91C2×C22⋊D61—81no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—44no—
96C22×Dic61—42no—
97C22×Dic61—82no—
98C22×Dic61—82no—
99#11758222—44no—
100#11758222—44no—
101#11758222—44no—
102#11758222—44no—
103C2×D244—41no—
104C2×D244—42no—
105C2×C4×D61—42no—
106C2×C4×D61—42no—
107C2×C4×D61—42no—
108C2×Dic122—41no—
109C2×Dic122—42no—
110M16×C61—82no—
111C2×C8×C61—42no—
112C22⋊C4×C61—41no—
113C22⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊Dic61—24no—
117C22⋊Dic61—42no—
118C22⋊Dic61—42no—
119C22⋊Dic61—44no—
120C22⋊Dic61—42no—
121C22⋊Dic61—42no—
122#11758198—81no—
123#11758198—81no—
124C23×C61—22no—
125C23×C61—22no—
126C23×C61—22no—
127C23×C61—22no—
128C23×C61—22no—
129C23×C61—21no—
130C23×C61—21no—
131C23×C61—21no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—21no—
135C23×C61—41no—
136C23×C61—42no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61—42no—
142C23×C61—81no—
143C23×C61—81no—
144C23×C61—84no—
145C23×C61—82no—
146C23×C61—82no—
147C23×C61#9911yescharacteristic
148C23×C61#9911yescharacteristic
149C23×C61#10412yes—
150C23×C61C2×C22⋊C411yescharacteristic
151C23×C61AGL(1,ℤ/8)11yescharacteristic
152C23×C61#13711yescharacteristic
153C22×D61—22no—
154C22×D61—41no—
155C22×D61—41no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61—41no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—81no—
172C22⋊D61—44no—
173C22⋊D61—44no—
174C22⋊D61—84no—
175C22⋊D61—82no—
176C22⋊D61—82no—
177C2×Dic61—22no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—44no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—88no—
188C2×Dic61—82no—
189C2×Dic61—82no—
190D244—42no—
191D244—42no—
192C4×D61—48no—
193Dic122—42no—
194Dic122—42no—
195C8×C61—82no—
196#11753—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—21no—
204C22×C61—21no—
205C22×C61—22no—
206C22×C61—21no—
207C22×C61—42no—
208C22×C61—42no—
209C22×C61—42no—
210C22×C61—41no—
211C22×C61—41no—
212C22×C61—44no—
213C22×C61—44no—
214C22×C61—41no—
215C22×C61—41no—
216C22×C61—41no—
217C22×C61—41no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61—82no—
221C22×C61—82no—
222C22×C61AΣL(1,𝔽16)12yes—
223C22×C61C2×#9911yescharacteristic
224C22×C61#41711yescharacteristic
225C22×C61C2×#10411yescharacteristic
226C2×D61—42no—
227C2×D61—42no—
228C2×D61—42no—
229C2×D61—81no—
230C2×D61—81no—
231C2×D61—81no—
232C2×D61—81no—
233C4×C61—41no—
234C4×C61—41no—
235C4×C61—41no—
236C4×C61—41no—
237Dic61—42no—
238Dic61—81no—
239Dic61—81no—
240Dic61—84no—
241#2404—611no—
242C2×#342—611no—
243#1462D6111yescharacteristic, complemented
244#1351—612no—
245#1343—612no—
246C2×C61—21no—
247C2×C61—21no—
248C2×C61—22no—
249C2×C61—41no—
250C2×C61—41no—
251C2×C61—42no—
252C2×C61—41no—
253C2×C61#150411yescharacteristic
254C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
255C2×C61#212711yescharacteristic
256D61—82no—
257C22×#41—611no—
258C2×#120—1221no—
259C22×C22⋊C4—1221no—
260C22×C22⋊C4C2×D6111yescharacteristic
261#407—21no—
262#393—611no—
263#393—611no—
264#392—611no—
265#392—611no—
266#386—1221no—
267#386—1221no—
268#386—1221no—
269#381—611no—
270#342—612no—
271#342—1221no—
272#323—21no—
273#323C2×D6112yes—
274C61#1175311yescharacteristic, complemented, Sylow
275C25D24411yescharacteristic
276C2×#41—2441no—
277C2×#41—2441no—
278C2×#41—2444no—
279C22×Q8—611no—
280C22×D4—611no—
281C22×D4—1221no—
282C22×D4—1221no—
283C22×D4—1221no—
284C23×C4—1221no—
285C23×C4—2441no—
286C23×C4C4×D6111yescharacteristic
287C23×C4C22⋊D6111yescharacteristic
288C2×M16—41no—
289#120—2444no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—1224no—
293C2×C22⋊C4—1222no—
294C2×C22⋊C4—1221no—
295C2×C22⋊C4—1221no—
296C2×C22⋊C4—1221no—
297C2×C22⋊C4—1221no—
298C2×C22⋊C4—1221no—
299C2×C22⋊C4—2441no—
300C2×C22⋊C4—2442no—
301C2×C22⋊C4C4×D6111yescharacteristic
302C2×C22⋊C4D24411yescharacteristic
303C2×C22⋊C4C22⋊D6111yescharacteristic
304C2×C22⋊C4C22×D6111yescharacteristic
305C22⋊C8—22no—
306C22⋊C8—41no—
307#95—611no—
308#95—611no—
309#95—1221no—
310#95—1221no—
311#95—1221no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—22no—
316C24—21no—
317C24—42no—
318C24—81no—
319C24—1221no—
320C24#1175819811yescharacteristic
321C24#1175819811yescharacteristic
322C24C2×D24411yescharacteristic
323#41—2444no—
324#41—2444no—
325#41—2444no—
326#41—2444no—
327C2×Q8—2441no—
328C2×Q8—2442no—
329C2×D4—1224no—
330C2×D4—4881no—
331C2×D4—4881no—
332C2×D4—2441no—
333C2×D4—2442no—
334C2×D4—2442no—
335C2×D4—2442no—
336C2×D4—2441no—
337C2×D4—2442no—
338C2×D4—2441no—
339C2×D4—2442no—
340C2×D4—2441no—
341C2×D4—2441no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—42no—
347C22×C4—81no—
348C22×C4—1221no—
349C22×C4—1221no—
350C22×C4—1221no—
351C22×C4—4882no—
352C22×C4—4882no—
353C22×C4—2442no—
354C22×C4—2442no—
355C22×C4—2442no—
356C22×C4—2444no—
357C22×C4—2442no—
358C22×C4#1175819811yescharacteristic
359C22×C4#1175819811yescharacteristic
360C22×C4C2×C4×D6111yescharacteristic
361C22×C4C2×C22⋊D6111yescharacteristic
362M16—82no—
363C2×C8—42no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—1224no—
369C22⋊C4—4881no—
370C22⋊C4—4881no—
371C22⋊C4—2442no—
372C22⋊C4—2442no—
373C22⋊C4—2444no—
374C22⋊C4—2442no—
375C22⋊C4—2442no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—22no—
385C23—21no—
386C23—21no—
387C23—41no—
388C23—42no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—42no—
394C23—81no—
395C23—81no—
396C23—82no—
397C23—84no—
398C23—82no—
399C23—1222no—
400C23—2441no—
401C23—2441no—
402C23—2441no—
403C23—2441no—
404C23—2441no—
405C23—2441no—
406C23#1451969111yescharacteristic
407C23#1451969111yescharacteristic
408C23#1451970312yes—
409C23C2×#1175819811yescharacteristic
410C23#1451980511yescharacteristic
411C23#1451981011yescharacteristic
412Q8—2442no—
413Q8—2442no—
414D4—4882no—
415D4—4884no—
416D4—4882no—
417D4—2442no—
418D4—2442no—
419D4—2444no—
420D4—2444no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—81no—
433C2×C4—1222no—
434C2×C4—4888no—
435C2×C4—4882no—
436C2×C4—4882no—
437C2×C4—2448no—
438C2×C4—2444no—
439C2×C4—2441no—
440C2×C4—2441no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C8—82no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—22no—
458C22—42no—
459C22—42no—
460C22—41no—
461C22—42no—
462C22—41no—
463C22—44no—
464C22—44no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—81no—
470C22—81no—
471C22—82no—
472C22—82no—
473C22—4881no—
474C22—4881no—
475C22—4881no—
476C22—4881no—
477C22—2442no—
478C22—2442no—
479C22—2442no—
480C22#1683955012yes—
481C22C2×#1451969111yescharacteristic
482C22#1683976411yescharacteristic
483C22C2×#1451970311yescharacteristic
484C4—41no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—4881no—
489C4—4881no—
490C4—4884no—
491C4—2442no—
492C2—21no—
493C2—21no—
494C2—22no—
495C2—41no—
496C2—41no—
497C2—41no—
498C2—42no—
499C2—4882no—
500C2#2144432411yescharacteristic
501C2C2×#1683955011yescharacteristic
502C2#2144748611yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1462C2
socleC22×C61C2×#99
prime radical#21449138C2
cosocleC2×C22⋊C4×C61C23
hypercenter#1462D61
hypocenterC61#11753
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
2. C2×#16839529C2semidirect
3. #21443841C2semidirect
4. #21443817C2semidirect
5. C61×#1462C2semidirect
6. C22×#11758222C4semidirect
9. #16839674C4semidirect
11. #16839671C4semidirect
16. #16839625C4semidirect
26. C22×Dic122C2×C4semidirect
27. C22×D244C2×C4semidirect
46. #14519684C2×C4semidirect
47. #14519684C2×C4semidirect
243. #1462D61semidirect
274. C61#11753semidirect

Maximal subgroups

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

subgroupindex
#214491382
C2×#168395292
#214438412
#214438172
C61×#14622
#1175361