Finite Group Databasebeta

← #30595374

Subgroups — #30595374

Order 15616.

884 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 126 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#21461676C211yesmaximal, characteristic, complemented
2C4×#14519834C211yesmaximal, characteristic, complemented
3#21460636C212yesmaximal, complemented
4#21460635C212yesmaximal, complemented
5#390×D61C211yesmaximal, characteristic
6C2×#14519844—21no—
7C2×#14519834—21no—
8C2×#14519834—21no—
9C2×C4×#92420—21no—
10#16840628C2212yescomplemented
11#16840628C2212yescomplemented
12C22×D4×D61C411yescharacteristic
13C2×C4⋊C4×D61C411yescharacteristic
14C2×C22⋊C4×D61—21no—
15C2×C22⋊C4×D61—21no—
16C2×C42×D61C2211yescharacteristic
17#16839665C2211yescharacteristic
18#16839664C411yescharacteristic
19#16839663C2211yescharacteristic
20#16839662—21no—
21#16839662—21no—
22C61×#390C411yescharacteristic
23C22×#92420—22no—
24C22×#92420—22no—
25#14519844—42no—
26#14519834—24no—
27#14519834—42no—
28C4×#92420—42no—
29C24×D61—21no—
30C22×D244C2×C411yescharacteristic
31C22×C22⋊D61—21no—
32C2×D4×D61—22no—
33C2×D4×D61—42no—
34C2×D4×D61—42no—
35C2×D4×D61C814yescomplemented
36C22×C4×D61—21no—
37C22×C4×D61C2×C411yescharacteristic
38C22×C4×D61D411yescharacteristic
39C22×C4×D61D411yescharacteristic
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×#11758196—21no—
42C4⋊C4×D61—42no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×#11758198—21no—
49C22⋊C4×D61—44no—
50C22⋊C4×D61—42no—
51C22⋊C4×D61—42no—
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61C2311yescharacteristic
55C42×D61—42no—
56C22×D4×C61C2×C411yescharacteristic
57C2×C4⋊C4×C61C2×C411yescharacteristic
58C2×C22⋊C4×C61—21no—
59C2×C22⋊C4×C61—21no—
60C2×C42×C61C2×C411yescharacteristic
61C24×C61—21no—
62C23×D61—21no—
63C23×D61—24no—
64C23×D61—42no—
65C23×D61—42no—
66C23×D61—42no—
67C23×D61—42no—
68C23×D61—42no—
69C23×D61—42no—
70C23×D61C22⋊C411yescharacteristic
71C2×D4×C61—21no—
72C2×D4×C61—41no—
73C2×D4×C61—41no—
74C2×D4×C61C2×C812yes—
75C22×C4×C61—21no—
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22×C411yescharacteristic
79C2×C22⋊D61—24no—
80C2×C22⋊D61—42no—
81C2×C22⋊D61—42no—
82C22×Dic61—21no—
83C22×Dic61C22⋊C411yescharacteristic
84C22×Dic61C2×D411yescharacteristic
85C22×Dic61C2×D411yescharacteristic
86D4×D61—48no—
87D4×D61—48no—
88C2×D244—21no—
89C2×D244—41no—
90C2×D244—41no—
91C2×D244C2×C812yes—
92C2×C4×D61—22no—
93C2×C4×D61—22no—
94C2×C4×D61—22no—
95C2×C4×D61—22no—
96C2×C4×D61—41no—
97C2×C4×D61—41no—
98C2×C4×D61—41no—
99C2×C4×D61—42no—
100C2×C4×D61—42no—
101C2×C4×D61—42no—
102C2×C4×D61C2×C812yes—
103C2×C4×D61M1612yes—
104C2×C4×D61D812yes—
105C2×C4×D61SD1612yes—
106C2×#92420—24no—
107C2×#92420—42no—
108C2×#92420—42no—
109C2×#92420—42no—
110C2×#92420—42no—
111C2×#92420—42no—
112C4⋊C4×C61—41no—
113C22⋊C4×C61—42no—
114C22⋊C4×C61—42no—
115C42×C61—41no—
116C22⋊Dic61—41no—
117C22⋊Dic61—41no—
118C22⋊Dic61—42no—
119#11758198—42no—
120#11758198—42no—
121#11758198—41no—
122#11758198—41no—
123#11758198—42no—
124C4⋊Dic61—41no—
125#11758196—42no—
126C4×Dic61—22no—
127C4×Dic61—22no—
128C4×Dic61—41no—
129C23×C61—22no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61C2×C22⋊C411yescharacteristic
137C22×D61—21no—
138C22×D61—21no—
139C22×D61—21no—
140C22×D61—21no—
141C22×D61—22no—
142C22×D61—48no—
143C22×D61—48no—
144C22×D61—41no—
145C22×D61—41no—
146C22×D61—41no—
147C22×D61—41no—
148C22×D61—41no—
149C22×D61—41no—
150C22×D61—48no—
151C22×D61—84no—
152C22×D61—84no—
153C22×D61—84no—
154C22×D61—84no—
155C22×D61C22⋊C812yes—
156C22×D61#10212yes—
157C22×D61#10412yes—
158D4×C61—42no—
159D4×C61—42no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61#10212yes—
171C2×C4×C61C22×C811yescharacteristic
172C2×C4×C61C2×M1611yescharacteristic
173C22⋊D61—48no—
174C2×Dic61—21no—
175C2×Dic61—21no—
176C2×Dic61—21no—
177C2×Dic61—21no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61C22⋊C812yes—
182C2×Dic61#9912yes—
183C2×Dic61#10012yes—
184C2×Dic61C2×D811yescharacteristic
185C2×Dic61C2×SD1611yescharacteristic
186C2×Dic61AGL(1,ℤ/8)12yes—
187D244—42no—
188D244—42no—
189C4×D61—24no—
190C4×D61—24no—
191C4×D61—42no—
192C4×D61—42no—
193C4×D61—44no—
194C4×D61—84no—
195#92420—44no—
196#92420—82no—
197#7572—611nomaximal, Sylow
198C22×C61—42no—
199C22×C61—42no—
200C22×C61—42no—
201C22×C61—81no—
202C22×C61—81no—
203C22×C61—81no—
204C22×C61—81no—
205C22×C61C2×C22⋊C811yescharacteristic
206C22×C61C2×#9911yescharacteristic
207C22×C61C2×#10011yescharacteristic
208C22×C61C2×#10211yescharacteristic
209C22×C61#41811yescharacteristic
210C22×C61C2×#10411yescharacteristic
211C22×C61#42111yescharacteristic
212C2×D61—22no—
213C2×D61—22no—
214C2×D61—22no—
215C2×D61—22no—
216C2×D61—22no—
217C2×D61—22no—
218C2×D61—42no—
219C2×D61—42no—
220C2×D61—42no—
221C2×D61—816no—
222C2×D61—81no—
223C2×D61—81no—
224C2×D61—81no—
225C2×D61—81no—
226C2×D61D4⋊C814yes—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—22no—
230C4×C61—41no—
231C4×C61—81no—
232Dic61—22no—
233Dic61—21no—
234Dic61—21no—
235Dic61—81no—
236Dic61#32314yes—
237C2×#390—611no—
238#1797—611no—
239C4×C22⋊C8—611no—
240#1270—612no—
241#1270—612no—
242C2×C61—82no—
243C2×C61C2×#32311yescharacteristic
244C2×C61C2×D4⋊C811yescharacteristic
245C2×C61#147811yescharacteristic
246C2×C61#150711yescharacteristic
247C2×C61#151911yescharacteristic
248C2×C61#152711yescharacteristic
249C2×C61#154511yescharacteristic
250D61—24no—
251D61—82no—
252C23×D4—611no—
253C22×C4⋊C4—611no—
254C22×C22⋊C4—1221no—
255C22×C22⋊C4—1221no—
256C22×C42—611no—
257C2×C4⋊C8—1221no—
258C2×C22⋊C8—1221no—
259C2×C22⋊C8—1221no—
260C2×C4×C8—1221no—
261#390—611no—
262#390—611no—
263#390—611no—
264#390—1221no—
265#390—1221no—
266#390#692311yescharacteristic
267#336—612no—
268#336—612no—
269C61#757211yescharacteristic, complemented, Sylow
270C25—1221no—
271C22×D4—614no—
272C22×D4—611no—
273C22×D4—1222no—
274C22×D4—1221no—
275C22×D4—2442no—
276C22×D4—2442no—
277C22×D4C2×#692311yescharacteristic
278C23×C4—611no—
279C23×C4—611no—
280C23×C4—611no—
281C23×C4—1221no—
282C22×C8—1222no—
283C22×C8—1222no—
284C2×C4⋊C4—611no—
285C2×C4⋊C4—1221no—
286C2×C4⋊C4—2442no—
287C2×C4⋊C4C2×#692311yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—1221no—
291C2×C22⋊C4—1221no—
292C2×C22⋊C4—1221no—
293C2×C22⋊C4—1221no—
294C2×C22⋊C4—1221no—
295C2×C22⋊C4—1221no—
296C2×C22⋊C4—2442no—
297C2×C22⋊C4—2442no—
298C2×C22⋊C4—2444no—
299C2×C42—611no—
300C2×C42—611no—
301C2×C42—611no—
302C2×C42—2442no—
303C2×C42C2×#692311yescharacteristic
304C4⋊C8—2442no—
305C22⋊C8—1224no—
306C22⋊C8—2442no—
307C4×C8—2442no—
308C24—21no—
309C24—611no—
310C24—1224no—
311C24—1221no—
312C24—2442no—
313C24—2442no—
314C24—2442no—
315C24—2442no—
316C24—2442no—
317C24—2442no—
318C2×D4—21no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—612no—
322C2×D4—1224no—
323C2×D4—1221no—
324C2×D4—2441no—
325C2×D4—2441no—
326C2×D4—2442no—
327C2×D4—2442no—
328C2×D4—2448no—
329C2×D4—2448no—
330C2×D4D61⋊C812yes—
331C22×C4—21no—
332C22×C4—611no—
333C22×C4—611no—
334C22×C4—611no—
335C22×C4—612no—
336C22×C4—612no—
337C22×C4—612no—
338C22×C4—612no—
339C22×C4—1222no—
340C22×C4—1222no—
341C22×C4—1222no—
342C22×C4—1222no—
343C22×C4—1221no—
344C22×C4—2441no—
345C22×C4—2441no—
346C22×C4—2441no—
347C22×C4—2442no—
348C22×C4—2442no—
349C22×C4—2442no—
350C22×C4C22⋊#692311yescharacteristic
351C22×C4C22⋊#692311yescharacteristic
352C22×C4C22×#692311yescharacteristic
353C2×C8—1224no—
354C2×C8—2442no—
355C2×C8—2442no—
356C2×C8—2442no—
357C2×C8—2442no—
358C2×C8—2442no—
359C4⋊C4—41no—
360C4⋊C4—2441no—
361C4⋊C4—2442no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—2442no—
365C22⋊C4—2441no—
366C22⋊C4—2441no—
367C22⋊C4—2442no—
368C22⋊C4—2441no—
369C22⋊C4—2441no—
370C22⋊C4—2442no—
371C22⋊C4—2442no—
372C42—41no—
373C42—1222no—
374C42—1222no—
375C42—2441no—
376C23—22no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—612no—
384C23—612no—
385C23—612no—
386C23—1221no—
387C23—1221no—
388C23—1221no—
389C23—1221no—
390C23—1222no—
391C23—4884no—
392C23—4884no—
393C23—4884no—
394C23—4884no—
395C23—2448no—
396C23—2448no—
397C23—2448no—
398C23—2441no—
399C23—2441no—
400C23—2441no—
401C23—2441no—
402C23—2441no—
403C23—2441no—
404C23C2×C22⋊#692311yescharacteristic
405D4—42no—
406D4—42no—
407D4—2442no—
408D4—2442no—
409D4—2448no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—612no—
421C2×C4—612no—
422C2×C4—612no—
423C2×C4—611no—
424C2×C4—611no—
425C2×C4—612no—
426C2×C4—1224no—
427C2×C4—1224no—
428C2×C4—1221no—
429C2×C4—1221no—
430C2×C4—1221no—
431C2×C4—1221no—
432C2×C4—4884no—
433C2×C4—2442no—
434C2×C4—2442no—
435C2×C4—2444no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4#1451983912yes—
440C2×C4C2×D61⋊C811yescharacteristic
441C2×C4D61⋊M1611yescharacteristic
442C8—4882no—
443C8—2444no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—614no—
452C22—1222no—
453C22—1222no—
454C22—1222no—
455C22—1222no—
456C22—1222no—
457C22—1222no—
458C22—48816no—
459C22—4881no—
460C22—4881no—
461C22—4881no—
462C22—4881no—
463C22—2442no—
464C22—2442no—
465C22—2442no—
466C22D61⋊(C22⋊C8)11yescharacteristic
467C22D61⋊#9911yescharacteristic
468C22D61⋊#10011yescharacteristic
469C22C2×#1451983911yescharacteristic
470C22#1684075411yescharacteristic
471C22D61⋊#10411yescharacteristic
472C22#1684075911yescharacteristic
473C4—22no—
474C4—21no—
475C4—21no—
476C4—41no—
477C4—81no—
478C4—614no—
479C4—1221no—
480C4—1221no—
481C4—1222no—
482C4—4881no—
483C2—82no—
484C2—1224no—
485C2—4882no—
486C2D61⋊#32311yescharacteristic
487C2D61⋊(D4⋊C8)11yescharacteristic
488C2#2146111911yescharacteristic
489C2#2146121411yescharacteristic
490C2#2146125411yescharacteristic
491C2#2146127911yescharacteristic
492C2#2146132711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D4×D61C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#7572
LCSG ▷ C2×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. #21461676C2semidirect
2. C4×#14519834C2semidirect
3. #21460636C2semidirect
4. #21460635C2semidirect
10. #16840628C22semidirect
11. #16840628C22semidirect
35. C2×D4×D61C8semidirect
269. C61#7572semidirect

Maximal subgroups

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

subgroupindex
#214616762
C4×#145198342
#214606362
#214606352
#390×D612
#757261