Finite Group Databasebeta

← #29652421

Subgroups — #29652421

Order 15616.

1398 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1398 conjugacy classes of subgroups — too many to draw. Only its 167 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×#16840324C211yesmaximal, characteristic
2#21455759C212yesmaximal, complemented
3#21444838C214yesmaximal, complemented
4#21444804C214yesmaximal, complemented
5#21444347C212yesmaximal, complemented
6#21444146C211yesmaximal, characteristic, complemented
7C61×#2482C211yesmaximal, characteristic, complemented
8C23×C22⋊D61—21no—
9#16840324—24no—
10#16840324—22no—
11#16840324C2214yes—
12C2×#14519761—22no—
13C2×#14519751—21no—
14C2×#14519751C2212yes—
15C2×#14519747—22no—
16#16840086—24no—
17#16840080C2214yescomplemented
18#16840075C2214yescomplemented
19#16839805—22no—
20#16839766—22no—
21#16839726C2212yescomplemented
22#16839495C2218yescomplemented
23#16839486C2214yescomplemented
24C2×#117×C61C2211yescharacteristic, complemented
25C22×C22⋊C4×C61C2211yescharacteristic
26C61×#399C2212yescomplemented
27C61×#397C2212yescomplemented
28C61×#388—22no—
29C61×#385—21no—
30C61×#381C2211yescharacteristic, complemented
31C61×#379—21no—
32C24×D61—21no—
33C22×D244—21no—
34C22×D244D412yes—
35C22×C22⋊D61—22no—
36C22×C22⋊D61—24no—
37C22×C22⋊D61—22no—
38C22×C22⋊D61—24no—
39C22×C22⋊D61—48no—
40C23×Dic61—21no—
41C2×#11758194—22no—
42#14519761—28no—
43#14519761—28no—
44#14519751—24no—
45#14519751—24no—
46#14519751—22no—
47#14519751—22no—
48#14519751D414yes—
49#14519751C2314yes—
50#14519747—28no—
51#14519747—28no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61D412yes—
54C2×#11758198—22no—
55C2×#11758198—22no—
56#14519710—24no—
57#14519708—24no—
58#14519697—24no—
59#14519697—24no—
60C22⋊#92418—22no—
61C22⋊#92418C2314yescomplemented
62C25×C61D411yescharacteristic
63C23×C4×C61D411yescharacteristic
64C23×C4×C61C2311yescharacteristic
65#117×C61—22no—
66#117×C61—22no—
67C2×C4⋊C4×C61—22no—
68C2×C4⋊C4×C61C2311yescharacteristic, complemented
69C2×C4⋊C4×C61C2311yescharacteristic, complemented
70C2×C22⋊C4×C61—22no—
71C2×C22⋊C4×C61—22no—
72C2×C22⋊C4×C61—21no—
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61—22no—
76C2×C22⋊C4×C61D412yes—
77C2×C22⋊C4×C61C2312yes—
78C2×C42×C61—21no—
79#95×C61—21no—
80#95×C61—22no—
81#95×C61C2312yescomplemented
82C24×C61—21no—
83C24×C61—21no—
84C24×C61—24no—
85C24×C61—42no—
86C24×C61—81no—
87C24×C61C2×D411yescharacteristic
88C24×C61C2×D412yes—
89C23×D61—22no—
90C23×D61—24no—
91C23×D61—44no—
92C23×D61—82no—
93C23×D61—82no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—22no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—42no—
100C22×C4×C61—81no—
101C22×C4×C61C2×D412yes—
102C22×C4×C61C2×D412yes—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D412yes—
105C22×C4×C61C2411yescharacteristic
106C2×C22⋊D61—28no—
107C2×C22⋊D61—24no—
108C2×C22⋊D61—24no—
109C2×C22⋊D61—48no—
110C2×C22⋊D61—416no—
111C2×C22⋊D61—48no—
112C2×C22⋊D61—48no—
113C2×C22⋊D61—48no—
114C22×Dic61—24no—
115C22×Dic61—22no—
116C22×Dic61—44no—
117C2×D244—24no—
118C2×D244—22no—
119C2×D244—44no—
120C2×D244—82no—
121C2×D244C2×D414yes—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—82no—
125C22⋊C4×C61—24no—
126C22⋊C4×C61—24no—
127C22⋊C4×C61—42no—
128C22⋊C4×C61—41no—
129C22⋊C4×C61—41no—
130C22⋊C4×C61—44no—
131C22⋊C4×C61—82no—
132C42×C61—42no—
133#11758198—28no—
134#11758198—24no—
135#11758198—24no—
136C4⋊Dic61—24no—
137C4⋊Dic61—22no—
138C4⋊Dic61C2×D414yes—
139#11758194—44no—
140C2×#92418—24no—
141C23×C61—24no—
142C23×C61—24no—
143C23×C61—22no—
144C23×C61—22no—
145C23×C61—22no—
146C23×C61—22no—
147C23×C61—41no—
148C23×C61—42no—
149C23×C61—42no—
150C23×C61—42no—
151C23×C61—42no—
152C23×C61—81no—
153C23×C61—81no—
154C23×C61—88no—
155C23×C61#12011yescharacteristic
156C23×C61#12011yescharacteristic
157C23×C61AGL(1,ℤ/8)14yes—
158C23×C61C22×D411yescharacteristic
159C22×D61—24no—
160C22×D61—44no—
161C22×D61—44no—
162C22×D61—48no—
163C22×D61—44no—
164C22×D61—84no—
165C22×D61—84no—
166C22×D61—816no—
167C2×C4×C61—24no—
168C2×C4×C61—22no—
169C2×C4×C61—42no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—81no—
182C2×C4×C61#12012yes—
183C2×C4×C61C22×D412yes—
184C22⋊D61—416no—
185C22⋊D61—48no—
186C22⋊D61—48no—
187C2×Dic61—24no—
188C2×Dic61—44no—
189C2×Dic61—48no—
190D244—44no—
191D244—82no—
192D244—84no—
193#92418—44no—
194#18137—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—24no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—44no—
204C22×C61—44no—
205C22×C61—48no—
206C22×C61—42no—
207C22×C61—42no—
208C22×C61—82no—
209C22×C61—84no—
210C22×C61#45812yes—
211C22×C61C2×#12011yescharacteristic
212C22×C61C2×AGL(1,ℤ/8)12yes—
213C2×D61—44no—
214C2×D61—44no—
215C2×D61—44no—
216C2×D61—88no—
217C2×D61—816no—
218C4×C61—22no—
219C4×C61—22no—
220C4×C61—42no—
221C4×C61—82no—
222C4×C61—81no—
223C4×C61—81no—
224Dic61—44no—
225C2×#534—611no—
226#3094—612no—
227#2482D6111yescharacteristic, complemented
228#1613—614no—
229#1607—614no—
230#1510—612no—
231#1462—611no—
232C2×C61—22no—
233C2×C61—22no—
234C2×C61—44no—
235C2×C61—41no—
236C2×C61#300311yescharacteristic
237C2×C61#301911yescharacteristic
238C2×C61C2×#45811yescharacteristic
239D61—84no—
240C23×D4—1221no—
241#534—614no—
242#534—1224no—
243#534—1222no—
244C2×#123—1222no—
245C2×#121—612no—
246C2×#121—1221no—
247C2×#120—1222no—
248C2×#117C2×D6111yescharacteristic, complemented
249C22×C22⋊C4C2×D6111yescharacteristic
250#482—1224no—
251#481—614no—
252#480—614no—
253#428—1222no—
254#418—1222no—
255#407—612no—
256#399C2×D6112yescomplemented
257#397C2×D6112yescomplemented
258#388—22no—
259#385—21no—
260#381C2×D6111yescharacteristic, complemented
261#379—21no—
262#327—618no—
263#323—614no—
264C61#1813711yescharacteristic, complemented, Sylow
265C25—1221no—
266C25C22⋊D6111yescharacteristic
267C22×D4—612no—
268C22×D4—1221no—
269C22×D4—1222no—
270C22×D4—1224no—
271C22×D4—1222no—
272C22×D4—1224no—
273C22×D4—2448no—
274C23×C4—1221no—
275C23×C4C22⋊D6111yescharacteristic
276C23×C4C22×D6111yescharacteristic
277C2×M16—1222no—
278#123—1228no—
279#123—1228no—
280#121—614no—
281#121—614no—
282#121—1224no—
283#121—1222no—
284#121—1222no—
285#121—1224no—
286#120—1228no—
287#120—1228no—
288#117—22no—
289#117—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—612no—
292C2×C4⋊C4—1221no—
293C2×C4⋊C4C22×D6111yescharacteristic, complemented
294C2×C4⋊C4C22×D6111yescharacteristic, complemented
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—1222no—
302C2×C22⋊C4—1222no—
303C2×C22⋊C4C22⋊D6112yes—
304C2×C22⋊C4C22×D6112yes—
305C2×C42—21no—
306#107—1224no—
307#106—1224no—
308#102—1224no—
309#102—1224no—
310C22⋊C8—614no—
311C22⋊C8—1222no—
312#95—21no—
313#95—22no—
314#95C22×D6112yescomplemented
315C24—21no—
316C24—21no—
317C24—24no—
318C24—42no—
319C24—81no—
320C24—1224no—
321C24—1222no—
322C24—4882no—
323C24—4882no—
324C24—2444no—
325C24C2×C22⋊D6111yescharacteristic
326C24C2×C22⋊D6112yes—
327C2×D4—614no—
328C2×D4—1224no—
329C2×D4—1228no—
330C2×D4—1224no—
331C2×D4—1224no—
332C2×D4—1222no—
333C2×D4—4882no—
334C2×D4—2444no—
335C2×D4—2448no—
336C2×D4—24416no—
337C2×D4—2448no—
338C2×D4—2448no—
339C2×D4—2448no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—42no—
346C22×C4—81no—
347C22×C4—1224no—
348C22×C4—1222no—
349C22×C4—2444no—
350C22×C4D4×D6112yes—
351C22×C4D4×D6112yes—
352C22×C4C2×C22⋊D6112yes—
353C22×C4C2×C22⋊D6111yescharacteristic
354C22×C4C23×D6111yescharacteristic
355M16—2444no—
356C2×C8—1224no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—82no—
360C4⋊C4—614no—
361C4⋊C4—1224no—
362C4⋊C4—1222no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—42no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—44no—
369C22⋊C4—82no—
370C22⋊C4—1228no—
371C22⋊C4—1224no—
372C22⋊C4—1224no—
373C42—42no—
374C23—24no—
375C23—24no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—41no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—81no—
386C23—81no—
387C23—88no—
388C23—1224no—
389C23—48816no—
390C23—4884no—
391C23—4884no—
392C23—2444no—
393C23—2444no—
394C23—2448no—
395C23—2444no—
396C23#1451974511yescharacteristic
397C23#1451974511yescharacteristic
398C23#1451980214yes—
399C23C22×C22⋊D6111yescharacteristic
400D4—4882no—
401D4—4884no—
402D4—2444no—
403D4—24416no—
404D4—2448no—
405D4—2448no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—81no—
421C2×C4—1224no—
422C2×C4—2444no—
423C2×C4—2448no—
424C2×C4#1451974512yes—
425C2×C4C2×D4×D6112yes—
426C8—2444no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—24no—
434C22—21no—
435C22—44no—
436C22—44no—
437C22—48no—
438C22—42no—
439C22—42no—
440C22—82no—
441C22—84no—
442C22—4888no—
443C22—48816no—
444C22—2444no—
445C22—2444no—
446C22—2444no—
447C22#1683995112yes—
448C22C2×#1451974511yescharacteristic
449C22C2×#1451980212yes—
450C4—22no—
451C4—22no—
452C4—42no—
453C4—82no—
454C4—81no—
455C4—81no—
456C4—2444no—
457C2—22no—
458C2—22no—
459C2—41no—
460C2—44no—
461C2—4884no—
462C2#2145504511yescharacteristic
463C2#2145517511yescharacteristic
464C2C2×#1683995111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2482C2
socleC22×C61C2×#120
prime radicalC2×#16840324C2
hypercenter#2482D61
hypocenterC61#18137
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
2. #21455759C2semidirect
3. #21444838C2semidirect
4. #21444804C2semidirect
5. #21444347C2semidirect
6. #21444146C2semidirect
7. C61×#2482C2semidirect
17. #16840080C22semidirect
18. #16840075C22semidirect
21. #16839726C22semidirect
22. #16839495C22semidirect
23. #16839486C22semidirect
24. C2×#117×C61C22semidirect
26. C61×#399C22semidirect
27. C61×#397C22semidirect
30. C61×#381C22semidirect
61. C22⋊#92418C23semidirect
68. C2×C4⋊C4×C61C23semidirect
69. C2×C4⋊C4×C61C23semidirect
81. #95×C61C23semidirect
227. #2482D61semidirect
248. C2×#117C2×D61semidirect
256. #399C2×D61semidirect
257. #397C2×D61semidirect
260. #381C2×D61semidirect
264. C61#18137semidirect
293. C2×C4⋊C4C22×D61semidirect
294. C2×C4⋊C4C22×D61semidirect
314. #95C22×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403242
#214557592
#214448382
#214448042
#214443472
#214441462
C61×#24822
#1813761