Finite Group Databasebeta

← #29654058

Subgroups — #29654058

Order 15616.

1168 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459152C211yesmaximal, characteristic
2#21456947C212yesmaximal, complemented
3#21444855C214yesmaximal, complemented
4#21444838C214yesmaximal, complemented
5#21444378C212yesmaximal, complemented
6#21444133C211yesmaximal, characteristic, complemented
7C61×#2628C211yesmaximal, characteristic, complemented
8C22×#11758222—21no—
9#16840343—21no—
10#16840343C2212yes—
11#16840338—22no—
12#16840331—22no—
13#16840324—21no—
14#16840324C2212yes—
15#16840312—21no—
16#16840312C2212yes—
17#16840274—24no—
18#16840158—22no—
19#16840110—22no—
20#16840080C2218yescomplemented
21#16840049—24no—
22#16839815C2212yescomplemented
23#16839495C2218yescomplemented
24#16839488C2212yescomplemented
25#16839486C2212yes—
26C2×C4⋊Q8×C61C2211yescharacteristic, complemented
27C2×#117×C61C2211yescharacteristic
28C61×#400C2212yescomplemented
29C61×#399C2212yes—
30C61×#395—21no—
31C61×#393—21no—
32C61×#389—21no—
33C61×#382C2211yescharacteristic, complemented
34C61×#380—21no—
35C2×#11758222—22no—
36C2×#11758222—22no—
37C2×#11758222—22no—
38C2×#11758222—22no—
39C2×#11758222—44no—
40C2×#11758222—44no—
41C22×Dic122—21no—
42C22×D244—21no—
43C22×C22⋊D61—22no—
44C22×C4×D61—22no—
45#14519777—21no—
46#14519777D412yes—
47#14519774—21no—
48#14519774D412yes—
49#14519765—22no—
50#14519765—22no—
51#14519761—24no—
52#14519761—24no—
53#14519757—24no—
54#14519757—24no—
55#14519752—24no—
56#14519752—24no—
57#14519751—22no—
58#14519751—22no—
59#14519751—22no—
60#14519751C2314yes—
61#14519747—24no—
62#14519747—24no—
63C4×D244—22no—
64C4×D244—22no—
65C4×D244—22no—
66C4×D244D414yes—
67#14519708—28no—
68C4⋊#92418—22no—
69#14519697—28no—
70C22⋊#92418C2314yes—
71C4×#92418—22no—
72C22×Q8×C61—21no—
73C23×C4×C61D411yescharacteristic
74C4⋊Q8×C61—22no—
75C4⋊Q8×C61—22no—
76#117×C61—22no—
77#117×C61—22no—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61—22no—
81C2×C4⋊C4×C61D411yescharacteristic
82C2×C4⋊C4×C61C2312yes—
83C2×C4⋊C4×C61C2311yescharacteristic
84C2×C22⋊C4×C61—22no—
85C2×C22⋊C4×C61D411yescharacteristic
86C2×C22⋊C4×C61C2311yescharacteristic
87C2×C42×C61D411yescharacteristic
88C2×C42×C61C2311yescharacteristic
89#95×C61—22no—
90#95×C61—21no—
91#95×C61—22no—
92#95×C61C2312yes—
93C24×C61C2×D411yescharacteristic
94C23×D61—22no—
95C2×Q8×C61—42no—
96C2×Q8×C61—82no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61SD1614yes—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2411yescharacteristic
109C2×C22⋊D61—24no—
110C2×C22⋊D61—24no—
111C2×C22⋊D61—44no—
112C2×C22⋊D61—44no—
113C22×Dic61—22no—
114#11758222—44no—
115#11758222—48no—
116#11758222—48no—
117#11758222—44no—
118#11758222—44no—
119#11758222—44no—
120C2×D244—22no—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—44no—
126C2×D244C2×D414yes—
127C2×C4×D61—24no—
128C2×C4×D61—24no—
129C2×C4×D61—48no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×Dic122—44no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—24no—
136C4⋊C4×C61—41no—
137C4⋊C4×C61—41no—
138C4⋊C4×C61—42no—
139C4⋊C4×C61—82no—
140C22⋊C4×C61—22no—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—82no—
143C42×C61—21no—
144C42×C61—21no—
145C42×C61C2×D412yes—
146C42×C61C2×D412yes—
147#11758198—24no—
148#11758198—24no—
149#11758198—24no—
150#11758198—24no—
151C4⋊Dic61—22no—
152C4⋊Dic61—22no—
153C4⋊Dic61—22no—
154C4⋊Dic61C2×D414yes—
155C2×#92418—24no—
156C23×C61—21no—
157C23×C61—21no—
158C23×C61—41no—
159C23×C61—41no—
160C23×C61C2×SD1612yes—
161C23×C61C22×D411yescharacteristic
162C22×D61—24no—
163C22×D61—44no—
164C22×D61—82no—
165C22×D61—82no—
166Q8×C61—84no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—24no—
170C2×C4×C61—24no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61#12012yes—
184C2×C4×C61#12011yescharacteristic
185C2×C4×C61#12011yescharacteristic
186C2×C4×C61C2×SD1612yes—
187C2×C4×C61C2×SD1612yes—
188C2×C4×C61AGL(1,ℤ/8)12yes—
189C2×C4×C61C22×D412yes—
190C22⋊D61—48no—
191C22⋊D61—44no—
192C22⋊D61—44no—
193C2×Dic61—24no—
194C2×Dic61—44no—
195D244—44no—
196D244—44no—
197D244—44no—
198D244—82no—
199D244—82no—
200D244—82no—
201C4×D61—48no—
202C4×D61—48no—
203Dic122—44no—
204Dic122—44no—
205#92418—44no—
206#18309—611nomaximal, Sylow
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—22no—
210C22×C61—42no—
211C22×C61—41no—
212C22×C61—41no—
213C22×C61—81no—
214C22×C61C2×#12011yescharacteristic
215C22×C61C22×SD1611yescharacteristic
216C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
217C2×D61—44no—
218C2×D61—44no—
219C2×D61—44no—
220C2×D61—88no—
221C4×C61—22no—
222C4×C61—22no—
223C4×C61—22no—
224C4×C61—41no—
225C4×C61—41no—
226C4×C61—82no—
227C4×C61—82no—
228C4×C61#45012yes—
229C4×C61#45012yes—
230Dic61—44no—
231#3449—611no—
232#3227—612no—
233#2628D6111yescharacteristic, complemented
234#1616—614no—
235#1613—614no—
236#1517—612no—
237#1458—611no—
238C2×C61—21no—
239C2×C61—21no—
240C2×C61—42no—
241C2×C61C2×#45011yescharacteristic
242C2×C61#302111yescharacteristic
243C2×C61#302811yescharacteristic
244D61—84no—
245C22×#41—1221no—
246#537—612no—
247#537—1221no—
248#536—1222no—
249#535—1222no—
250#534—612no—
251#534—1221no—
252#532—612no—
253#532—1221no—
254C2×C4⋊Q8C2×D6111yescharacteristic, complemented
255#525—1224no—
256C2×#117C2×D6111yescharacteristic
257#498—1222no—
258#486—1222no—
259#481—618no—
260#476—1224no—
261#431—612no—
262#400C2×D6112yescomplemented
263#399C2×D6112yes—
264#395—21no—
265#393—21no—
266#389—21no—
267#382C2×D6111yescharacteristic, complemented
268#380—21no—
269#327—618no—
270#324—612no—
271#323—612no—
272C61#1830911yescharacteristic, complemented, Sylow
273C2×#41—1222no—
274C2×#41—1222no—
275C2×#41—1222no—
276C2×#41—1222no—
277C2×#41—2444no—
278C2×#41—2444no—
279C22×Q8—21no—
280C22×Q8—1221no—
281C22×D4—1221no—
282C22×D4—1222no—
283C23×C4—1222no—
284C23×C4C22⋊D6111yescharacteristic
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—612no—
288C4⋊Q8—1221no—
289#127—612no—
290#127—1221no—
291#124—1222no—
292#124—1222no—
293#123—1224no—
294#123—1224no—
295C22⋊Q8—1224no—
296C22⋊Q8—1224no—
297#121—614no—
298#121—1224no—
299#121—1224no—
300#121—1222no—
301#121—1222no—
302#121—1222no—
303#120—1224no—
304#120—1224no—
305C4×D4—614no—
306C4×D4—1222no—
307C4×D4—1222no—
308C4×D4—1222no—
309#117—22no—
310#117—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4C22⋊D6111yescharacteristic
315C2×C4⋊C4C22×D6112yes—
316C2×C4⋊C4C22×D6111yescharacteristic
317C2×C22⋊C4—22no—
318C2×C22⋊C4C22⋊D6111yescharacteristic
319C2×C22⋊C4C22×D6111yescharacteristic
320C2×C42C22⋊D6111yescharacteristic
321C2×C42C22×D6111yescharacteristic
322#106—1228no—
323C4⋊C8—1222no—
324#102—1228no—
325C22⋊C8—614no—
326C4×C8—1222no—
327#95—22no—
328#95—21no—
329#95—22no—
330#95C22×D6112yes—
331C24—1222no—
332C24C2×C22⋊D6111yescharacteristic
333#41—2444no—
334#41—2448no—
335#41—2448no—
336#41—2444no—
337#41—2444no—
338#41—2444no—
339C2×Q8—42no—
340C2×Q8—82no—
341C2×Q8—1222no—
342C2×Q8—1222no—
343C2×Q8—2444no—
344C2×D4—614no—
345C2×D4—1222no—
346C2×D4—1222no—
347C2×D4—1222no—
348C2×D4—1224no—
349C2×D4—1224no—
350C2×D4—1222no—
351C2×D4—1222no—
352C2×D4—2444no—
353C2×D4—2444no—
354C2×D4—2444no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—1224no—
361C22×C4—1222no—
362C22×C4—1224no—
363C22×C4—2448no—
364C22×C4#1175820114yes—
365C22×C4C2×C22⋊D6111yescharacteristic
366C22×C4C2×C22⋊D6111yescharacteristic
367C22×C4C2×C22⋊D6111yescharacteristic
368C22×C4C2×C22⋊D6111yescharacteristic
369C22×C4C2×C22⋊D6111yescharacteristic
370C22×C4C23×D6111yescharacteristic
371C2×C8—1224no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—82no—
379C4⋊C4—614no—
380C4⋊C4—1222no—
381C4⋊C4—1222no—
382C4⋊C4—1222no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—82no—
386C22⋊C4—1224no—
387C22⋊C4—1224no—
388C22⋊C4—1224no—
389C22⋊C4—1224no—
390C42—21no—
391C42—21no—
392C42D4×D6112yes—
393C42D4×D6112yes—
394C23—21no—
395C23—21no—
396C23—41no—
397C23—41no—
398C23—1224no—
399C23—4882no—
400C23—4882no—
401C23—2444no—
402C23C2×#1175820112yes—
403C23C22×C22⋊D6111yescharacteristic
404Q8—84no—
405Q8—2444no—
406Q8—2444no—
407D4—4882no—
408D4—4882no—
409D4—4882no—
410D4—2444no—
411D4—2444no—
412D4—2448no—
413D4—2444no—
414D4—2444no—
415D4—2444no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—1224no—
433C2×C4—2448no—
434C2×C4—2444no—
435C2×C4—2448no—
436C2×C4#1451974512yes—
437C2×C4#1451974511yescharacteristic
438C2×C4#1451974511yescharacteristic
439C2×C4C2×#1175820112yes—
440C2×C4C2×#1175820112yes—
441C2×C4#1451980212yes—
442C2×C4C2×D4×D6112yes—
443C8—2444no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—42no—
448C22—41no—
449C22—41no—
450C22—81no—
451C22—4888no—
452C22—2444no—
453C22—2444no—
454C22—2444no—
455C22C2×#1451974511yescharacteristic
456C22C22×#1175820111yescharacteristic
457C22C2×#1451980211yescharacteristic
458C4—22no—
459C4—22no—
460C4—22no—
461C4—41no—
462C4—41no—
463C4—82no—
464C4—82no—
465C4—2444no—
466C4#1683991212yes—
467C4#1683991212yes—
468C2—21no—
469C2—21no—
470C2—42no—
471C2—4884no—
472C2C2×#1683991211yescharacteristic
473C2#2145519011yescharacteristic
474C2#2145522411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2628C2
socleC22×C61C2×#120
prime radical#21459152C2
hypercenter#2628D61
hypocenterC61#18309
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2628

Complements

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

NH ≅ G/Ntype
2. #21456947C2semidirect
3. #21444855C2semidirect
4. #21444838C2semidirect
5. #21444378C2semidirect
6. #21444133C2semidirect
7. C61×#2628C2semidirect
20. #16840080C22semidirect
22. #16839815C22semidirect
23. #16839495C22semidirect
24. #16839488C22semidirect
26. C2×C4⋊Q8×C61C22semidirect
28. C61×#400C22semidirect
33. C61×#382C22semidirect
233. #2628D61semidirect
254. C2×C4⋊Q8C2×D61semidirect
262. #400C2×D61semidirect
267. #382C2×D61semidirect
272. C61#18309semidirect

Maximal subgroups

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

subgroupindex
#214591522
#214569472
#214448552
#214448382
#214443782
#214441332
C61×#26282
#1830961