Finite Group Databasebeta

← #30601819

Subgroups — #30601819

Order 15616.

764 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 112 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#21461674C211yesmaximal, characteristic, complemented
2C2×#16840614C212yesmaximal, complemented
3C2×#16840613C212yesmaximal, complemented
4#21460995C211yesmaximal, characteristic, complemented
5#21454163C211yesmaximal, characteristic
6C2×C4⋊#92420—21no—
7C2×#14519834C2212yes—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4—21no—
10C2×(C2×#6923)⋊C4C2212yes—
11#16840628—21no—
12#16840614—24no—
13#16840613—24no—
14C2×#14519773C411yescharacteristic, complemented
15C2×#14519752C2211yescharacteristic
16C2×#14519750—21no—
17C2×#14519750—21no—
18C2×#14519748C2211yescharacteristic
19C2×C42×D61C2211yescharacteristic
20#16839664C411yescharacteristic, complemented
21#16839662—21no—
22#16839662—21no—
23#16839631C411yescharacteristic, complemented
24C61×#390C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27C4⋊#92420—42no—
28#14519834—22no—
29#14519834—22no—
30(C2×#6923)⋊C4—22no—
31(C2×#6923)⋊C4—42no—
32(C2×#6923)⋊C4—42no—
33(C2×#6923)⋊C4D414yescomplemented
34C22×D244—21no—
35C22×D244C2×C411yescharacteristic
36C22×C22⋊D61—21no—
37C22×C22⋊D61—21no—
38C22×C22⋊D61—21no—
39C22×C4×D61D411yescharacteristic
40C22×C4×D61D411yescharacteristic
41C22×C4×D61C2311yescharacteristic
42#14519773—42no—
43#14519752—22no—
44#14519752—22no—
45#14519750—44no—
46#14519750—42no—
47#14519750—42no—
48#14519748—22no—
49#14519748—22no—
50C2×C4⋊Dic61C2×C411yescharacteristic
51C2×#11758196—21no—
52C2×#11758196—21no—
53C2×C22⋊Dic61—21no—
54C2×#11758198—21no—
55C2×#11758198—21no—
56C2×#11758198—21no—
57C2×C4×Dic61C2×C411yescharacteristic
58C2×C4×Dic61D411yescharacteristic
59C2×C4×Dic61D411yescharacteristic
60C42×D61—42no—
61C22×D4×C61C2×C411yescharacteristic
62C2×C4⋊C4×C61C2×C411yescharacteristic
63C2×C22⋊C4×C61—21no—
64C2×C22⋊C4×C61—21no—
65C2×C42×C61C2×C411yescharacteristic
66C24×C61—21no—
67C23×D61—21no—
68C23×D61C2×D411yescharacteristic
69C22×#6923—24no—
70C22×#6923—42no—
71C22×#6923—42no—
72C22×#6923—42no—
73C22×#6923—42no—
74C22×#6923—42no—
75C22×#6923—42no—
76C2×D4×C61—22no—
77C2×D4×C61—41no—
78C2×D4×C61—41no—
79C22×C4×C61—21no—
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22×C411yescharacteristic
83C2×C22⋊D61—42no—
84C2×C22⋊D61—42no—
85C2×C22⋊D61—42no—
86C2×C22⋊D61—42no—
87C2×C22⋊D61—42no—
88C2×C22⋊D61—41no—
89C2×C22⋊D61—41no—
90C2×C22⋊D61—42no—
91C2×C22⋊D61—42no—
92C2×C22⋊D61—42no—
93C22×Dic61—21no—
94C22×Dic61C22⋊C411yescharacteristic
95C22×Dic61C22⋊C411yescharacteristic
96C22×Dic61C2×D411yescharacteristic
97C2×D244—22no—
98C2×D244—41no—
99C2×D244—41no—
100C2×D244—42no—
101C2×D244—41no—
102C2×D244—41no—
103C2×D244—42no—
104C2×C4×D61—22no—
105C2×C4×D61—22no—
106C2×C4×D61—41no—
107C2×C4×D61—41no—
108C2×C4×D61—41no—
109C2×C4×D61—41no—
110C2×C4×D61—41no—
111C2×C4×D61D812yes—
112C2×C4×D61SD1612yes—
113C2×C4×D61C2×D412yes—
114C2×C4×D61#4112yes—
115C2×#92420—42no—
116C2×#92420—42no—
117C2×#92420—42no—
118C4⋊C4×C61—22no—
119C22⋊C4×C61—42no—
120C22⋊C4×C61—42no—
121C42×C61—41no—
122C22⋊Dic61—41no—
123C22⋊Dic61—41no—
124#11758198—42no—
125#11758198—42no—
126#11758198—41no—
127#11758198—41no—
128C4⋊Dic61—22no—
129#11758196—42no—
130#11758196—42no—
131C4×Dic61—22no—
132C4×Dic61—22no—
133C4×Dic61—41no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61C2×C22⋊C411yescharacteristic
142C22×D61—21no—
143C22×D61—21no—
144C22×D61—21no—
145C22×D61—21no—
146C22×D61—41no—
147C22×D61—41no—
148C22×D61—41no—
149C22×D61—41no—
150C22×D61—41no—
151C22×D61—41no—
152C22×D61—41no—
153C22×D61#12112yes—
154C22×D61C2×D811yescharacteristic
155C22×D61C2×SD1611yescharacteristic
156C22×D61AGL(1,ℤ/8)12yes—
157C2×#6923—44no—
158C2×#6923—44no—
159C2×#6923—44no—
160C2×#6923—82no—
161C2×#6923—82no—
162C2×#6923—82no—
163C2×#6923—82no—
164D4×C61—82no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61#10212yes—
176C2×C4×C61C4×D412yes—
177C22⋊D61—84no—
178C22⋊D61—84no—
179C22⋊D61—82no—
180C22⋊D61—82no—
181C2×Dic61—21no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61#9912yes—
191C2×Dic61#10212yes—
192C2×Dic61#12411yescharacteristic
193C2×Dic61#12711yescharacteristic
194D244—82no—
195D244—82no—
196D244—82no—
197C4×D61—42no—
198C4×D61—42no—
199C4×D61—42no—
200C4×D61—42no—
201C4×D61—44no—
202#92420—82no—
203#9715—611nomaximal, Sylow
204C22×C61—81no—
205C22×C61—81no—
206C22×C61—81no—
207C22×C61—81no—
208C22×C61—81no—
209C22×C61—81no—
210C22×C61—81no—
211C22×C61#39011yescharacteristic
212C22×C61C2×#9911yescharacteristic
213C22×C61#41011yescharacteristic
214C22×C61C2×#10211yescharacteristic
215C22×C61#41811yescharacteristic
216C22×C61#42111yescharacteristic
217C22×C61#42111yescharacteristic
218C2×D61—22no—
219C2×D61—22no—
220C2×D61—22no—
221C2×D61—22no—
222C2×D61—21no—
223C2×D61—21no—
224C2×D61—21no—
225C2×D61—21no—
226C2×D61—81no—
227C2×D61—81no—
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C2×D61—81no—
232C2×D61—81no—
233C2×D61#45912yes—
234C2×D61#46012yes—
235#6923—84no—
236C4×C61—22no—
237C4×C61—22no—
238C4×C61—41no—
239C4×C61—81no—
240Dic61—21no—
241Dic61—21no—
242Dic61—22no—
243Dic61—22no—
244Dic61—81no—
245#2856—611no—
246#1797—611no—
247C2×#327—612no—
248C2×#327—612no—
249#1434—611no—
250C2×C61—161no—
251C2×C61#151011yescharacteristic
252C2×C61#151411yescharacteristic
253C2×C61#151711yescharacteristic
254C2×C61#151911yescharacteristic
255C2×C61#196311yescharacteristic
256C2×C61#196411yescharacteristic
257C2×C61#198211yescharacteristic
258D61—22no—
259D61—22no—
260D61—161no—
261C2×#127—611no—
262C2×#121—611no—
263C2×#121—611no—
264C2×#121—1221no—
265C2×#121—1221no—
266C22×C42—611no—
267C2×C4⋊C8—1221no—
268C2×C22⋊C8—612no—
269#390—611no—
270#390—1221no—
271#390—1221no—
272#390#692311yescharacteristic, complemented
273#384—611no—
274C2×#95—612no—
275C2×#95—1221no—
276C2×#95—1221no—
277#336—1221no—
278#327—1224no—
279#327—1224no—
280C61#971511yescharacteristic, complemented, Sylow
281C22×D4—611no—
282C22×D4—1221no—
283C22×D4—1221no—
284C22×D4—1221no—
285C22×D4—1221no—
286C22×D4C2×#692311yescharacteristic
287C23×C4—611no—
288C23×C4—611no—
289C23×C4—611no—
290C23×C4—1222no—
291C22×C8—1222no—
292#127—2442no—
293#121—1222no—
294#121—1222no—
295#121—1222no—
296#121—1222no—
297#121—2442no—
298#121—2442no—
299#121—2444no—
300C2×C4⋊C4—611no—
301C2×C4⋊C4—1221no—
302C2×C4⋊C4—1221no—
303C2×C4⋊C4C2×#692311yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—1221no—
307C2×C22⋊C4—1221no—
308C2×C22⋊C4—1221no—
309C2×C22⋊C4—1221no—
310C2×C42—611no—
311C2×C42—611no—
312C2×C42—611no—
313C2×C42—2442no—
314C2×C42C2×#692311yescharacteristic
315C4⋊C8—2442no—
316C22⋊C8—1222no—
317C22⋊C8—1222no—
318#95—614no—
319#95—1222no—
320#95—2442no—
321#95—2442no—
322C24—21no—
323C24—611no—
324C24—1221no—
325C2×D4—22no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—1222no—
329C2×D4—2441no—
330C2×D4—2441no—
331C2×D4—2442no—
332C2×D4—2441no—
333C2×D4—2441no—
334C2×D4—2442no—
335C2×D4—2442no—
336C2×D4—2442no—
337C2×D4—2442no—
338C2×D4—2442no—
339C2×D4—2442no—
340C2×D4—2442no—
341C2×D4—2442no—
342C2×D4—2441no—
343C2×D4—2441no—
344C2×D4—2442no—
345C22×C4—21no—
346C22×C4—611no—
347C22×C4—611no—
348C22×C4—611no—
349C22×C4—612no—
350C22×C4—612no—
351C22×C4—612no—
352C22×C4—612no—
353C22×C4—1222no—
354C22×C4—1222no—
355C22×C4—1224no—
356C22×C4—1221no—
357C22×C4—2441no—
358C22×C4—2441no—
359C22×C4—2441no—
360C22×C4—2441no—
361C22×C4—2441no—
362C22×C4—2442no—
363C22×C4—2442no—
364C22×C4—2442no—
365C22×C4—2442no—
366C22×C4—2442no—
367C22×C4—2442no—
368C22×C4C22⋊#692311yescharacteristic
369C22×C4C22⋊#692311yescharacteristic
370C22×C4C22×#692311yescharacteristic
371C2×C8—2442no—
372C2×C8—2442no—
373C2×C8—2442no—
374C4⋊C4—22no—
375C4⋊C4—1222no—
376C4⋊C4—2442no—
377C4⋊C4—2442no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—2442no—
381C22⋊C4—2441no—
382C22⋊C4—2441no—
383C22⋊C4—2441no—
384C22⋊C4—2441no—
385C22⋊C4—2442no—
386C42—41no—
387C42—1222no—
388C42—1222no—
389C42—2441no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—612no—
398C23—611no—
399C23—611no—
400C23—612no—
401C23—1221no—
402C23—1221no—
403C23—1221no—
404C23—1221no—
405C23—2441no—
406C23—2441no—
407C23—2441no—
408C23—2441no—
409C23—2441no—
410C23—2441no—
411C23—2441no—
412C23C2×C22⋊#692311yescharacteristic
413D4—82no—
414D4—4882no—
415D4—4882no—
416D4—4882no—
417D4—4884no—
418D4—4884no—
419D4—4882no—
420D4—4882no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—611no—
432C2×C4—611no—
433C2×C4—612no—
434C2×C4—612no—
435C2×C4—1221no—
436C2×C4—1221no—
437C2×C4—1221no—
438C2×C4—1221no—
439C2×C4—1221no—
440C2×C4—1221no—
441C2×C4—4882no—
442C2×C4—4882no—
443C2×C4—4882no—
444C2×C4—4882no—
445C2×C4—2442no—
446C2×C4—2442no—
447C2×C4—2442no—
448C2×C4—2442no—
449C2×C4—2444no—
450C2×C4—2444no—
451C2×C4—2444no—
452C2×C4—2444no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4#1451983912yes—
457C2×C4D4×#692312yes—
458C8—4882no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—612no—
467C22—612no—
468C22—1221no—
469C22—1221no—
470C22—1221no—
471C22—1221no—
472C22—1222no—
473C22—1222no—
474C22—1222no—
475C22—1222no—
476C22—4881no—
477C22—4881no—
478C22—4881no—
479C22—4881no—
480C22—4881no—
481C22—4881no—
482C22—4881no—
483C22#1684071411yescharacteristic
484C22D61⋊#9911yescharacteristic
485C22#1684074111yescharacteristic
486C22C2×#1451983911yescharacteristic
487C22#1684075411yescharacteristic
488C22#1684075911yescharacteristic
489C22#1684075911yescharacteristic
490C4—22no—
491C4—22no—
492C4—41no—
493C4—81no—
494C4—1222no—
495C4—1221no—
496C4—1221no—
497C4—1222no—
498C4—4884no—
499C4—4881no—
500C2—161no—
501C2—1222no—
502C2—1222no—
503C2—9761no—
504C2#2146122511yescharacteristic
505C2#2146123811yescharacteristic
506C2#2146124911yescharacteristic
507C2#2146125411yescharacteristic
508C2#2146195611yescharacteristic
509C2#2146196011yescharacteristic
510C2#2146198611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radical#21454163C2
cosocleC22×C4×D61C23
hypercenter#390#6923
hypocenterC61#9715
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21461674C2semidirect
2. C2×#16840614C2semidirect
3. C2×#16840613C2semidirect
4. #21460995C2semidirect
14. C2×#14519773C4semidirect
20. #16839664C4semidirect
23. #16839631C4semidirect
24. C61×#390C4semidirect
33. (C2×#6923)⋊C4D4semidirect
272. #390#6923semidirect
280. C61#9715semidirect

Maximal subgroups

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

subgroupindex
#214616742
C2×#168406142
C2×#168406132
#214609952
#214541632
#971561