Finite Group Databasebeta

← #29653990

Subgroups — #29653990

Order 15616.

1192 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459152C211yesmaximal, characteristic
2#21456744C212yesmaximal, complemented
3#21444787C214yesmaximal, complemented
4#21444770C214yesmaximal, 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#16840136—22no—
19#16840112—22no—
20#16839964—24no—
21#16839901C2218yescomplemented
22#16839815C2212yescomplemented
23#16839495C2218yescomplemented
24#16839488C2212yescomplemented
25#16839486C2212yes—
26C2×C4⋊Q8×C61C2211yescharacteristic, complemented
27C2×#117×C61C2211yescharacteristic
28C61×#400—21no—
29C61×#399—21no—
30C61×#395C2212yescomplemented
31C61×#393C2212yes—
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—
45C2×#11758201—28no—
46#14519777—21no—
47#14519777D412yes—
48#14519774—21no—
49#14519774D412yes—
50#14519765—22no—
51#14519765—22no—
52#14519761—24no—
53#14519761—24no—
54#14519757—24no—
55#14519757—24no—
56#14519752—24no—
57#14519752—24no—
58#14519751—22no—
59#14519751—22no—
60#14519751—22no—
61#14519751C2314yes—
62#14519747—24no—
63#14519747—24no—
64C4×D244—22no—
65C4×D244—22no—
66C4×D244—22no—
67C4×D244D414yes—
68C4⋊#92418—22no—
69#14519701—28no—
70C22⋊#92418C2314yes—
71C4×#92418—22no—
72C22×Q8×C61C2312yes—
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—21no—
81C2×C4⋊C4×C61—22no—
82C2×C4⋊C4×C61D411yescharacteristic
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—24no—
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×C61SD1614yescomplemented
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—42no—
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#11758201—88no—
148#11758198—24no—
149#11758198—24no—
150#11758198—24no—
151#11758198—24no—
152C4⋊Dic61—22no—
153C4⋊Dic61—22no—
154C4⋊Dic61—22no—
155C4⋊Dic61C2×D414yes—
156C2×#92418—24no—
157C23×C61—21no—
158C23×C61—21no—
159C23×C61—41no—
160C23×C61—41no—
161C23×C61C2×SD1612yes—
162C23×C61C22×D411yescharacteristic
163C22×D61—24no—
164C22×D61—44no—
165C22×D61—82no—
166C22×D61—82no—
167Q8×C61—48no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—24no—
171C2×C4×C61—24no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61#12012yes—
185C2×C4×C61#12011yescharacteristic
186C2×C4×C61#12011yescharacteristic
187C2×C4×C61C2×SD1612yes—
188C2×C4×C61C2×SD1612yes—
189C2×C4×C61AGL(1,ℤ/8)12yes—
190C2×C4×C61C22×D412yes—
191C22⋊D61—48no—
192C22⋊D61—44no—
193C22⋊D61—44no—
194C2×Dic61—24no—
195C2×Dic61—44no—
196D244—44no—
197D244—44no—
198D244—44no—
199D244—82no—
200D244—82no—
201D244—82no—
202C4×D61—48no—
203C4×D61—48no—
204Dic122—44no—
205Dic122—44no—
206#92418—44no—
207#18301—611nomaximal, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—22no—
211C22×C61—42no—
212C22×C61—41no—
213C22×C61—41no—
214C22×C61—81no—
215C22×C61C2×#12011yescharacteristic
216C22×C61C22×SD1611yescharacteristic
217C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
218C2×D61—44no—
219C2×D61—44no—
220C2×D61—44no—
221C2×D61—88no—
222C4×C61—22no—
223C4×C61—22no—
224C4×C61—22no—
225C4×C61—41no—
226C4×C61—41no—
227C4×C61—82no—
228C4×C61—82no—
229C4×C61#45012yes—
230C4×C61#45012yes—
231Dic61—44no—
232#3449—611no—
233#3206—612no—
234#2628D6111yescharacteristic, complemented
235#1604—614no—
236#1601—614no—
237#1517—612no—
238#1458—611no—
239C2×C61—21no—
240C2×C61—21no—
241C2×C61—42no—
242C2×C61C2×#45011yescharacteristic
243C2×C61#302211yescharacteristic
244C2×C61#302711yescharacteristic
245D61—84no—
246C22×#41—1221no—
247#537—612no—
248#537—1221no—
249#536—1222no—
250#535—1222no—
251#534—612no—
252#534—1221no—
253#532—612no—
254#532—1221no—
255C2×C4⋊Q8C2×D6111yescharacteristic, complemented
256#525—1224no—
257C2×#117C2×D6111yescharacteristic
258#492—1222no—
259#487—1222no—
260#461—1224no—
261#448—618no—
262#431—612no—
263#400—21no—
264#399—21no—
265#395C2×D6112yescomplemented
266#393C2×D6112yes—
267#389—21no—
268#382C2×D6111yescharacteristic, complemented
269#380—21no—
270#327—618no—
271#324—612no—
272#323—612no—
273C61#1830111yescharacteristic, complemented, Sylow
274C2×#41—1222no—
275C2×#41—1222no—
276C2×#41—1222no—
277C2×#41—1222no—
278C2×#41—2444no—
279C2×#41—2444no—
280C22×Q8—1221no—
281C22×Q8C22×D6112yes—
282C22×D4—1221no—
283C22×D4—1222no—
284C23×C4—1222no—
285C23×C4C22⋊D6111yescharacteristic
286C2×SD16—1228no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—612no—
290C4⋊Q8—1221no—
291#127—612no—
292#127—1221no—
293#124—1222no—
294#124—1222no—
295#123—1224no—
296#123—1224no—
297C22⋊Q8—1224no—
298C22⋊Q8—1224no—
299#121—614no—
300#121—1224no—
301#121—1224no—
302#121—1222no—
303#121—1222no—
304#121—1222no—
305#120—1224no—
306#120—1224no—
307C4×D4—614no—
308C4×D4—1222no—
309C4×D4—1222no—
310C4×D4—1222no—
311#117—22no—
312#117—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4C22⋊D6111yescharacteristic
318C2×C4⋊C4C22×D6111yescharacteristic
319C2×C22⋊C4—22no—
320C2×C22⋊C4C22⋊D6111yescharacteristic
321C2×C22⋊C4C22×D6111yescharacteristic
322C2×C42C22⋊D6111yescharacteristic
323C2×C42C22×D6111yescharacteristic
324C4⋊C8—1222no—
325#103—1228no—
326C22⋊C8—614no—
327C4×C8—1222no—
328#95—22no—
329#95—21no—
330#95—22no—
331#95C22×D6112yes—
332C24—1222no—
333C24C2×C22⋊D6111yescharacteristic
334#41—2444no—
335#41—2448no—
336#41—2448no—
337#41—2444no—
338#41—2444no—
339#41—2444no—
340C2×Q8—24no—
341C2×Q8—82no—
342C2×Q8—1222no—
343C2×Q8—1222no—
344C2×Q8—2444no—
345C2×D4—614no—
346C2×D4—1222no—
347C2×D4—1222no—
348C2×D4—1222no—
349C2×D4—1224no—
350C2×D4—1224no—
351C2×D4—1222no—
352C2×D4—1222no—
353C2×D4—2444no—
354C2×D4—2444no—
355C2×D4—2444no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—1224no—
362C22×C4—1222no—
363C22×C4—1224no—
364C22×C4—2448no—
365C22×C4#1175820114yescomplemented
366C22×C4C2×C22⋊D6111yescharacteristic
367C22×C4C2×C22⋊D6111yescharacteristic
368C22×C4C2×C22⋊D6111yescharacteristic
369C22×C4C2×C22⋊D6111yescharacteristic
370C22×C4C2×C22⋊D6111yescharacteristic
371C22×C4C23×D6111yescharacteristic
372SD16—4888no—
373C2×C8—1224no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—82no—
381C4⋊C4—614no—
382C4⋊C4—1222no—
383C4⋊C4—1222no—
384C4⋊C4—1222no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—82no—
388C22⋊C4—1224no—
389C22⋊C4—1224no—
390C22⋊C4—1224no—
391C22⋊C4—1224no—
392C42—21no—
393C42—21no—
394C42D4×D6112yes—
395C42D4×D6112yes—
396C23—21no—
397C23—21no—
398C23—41no—
399C23—41no—
400C23—1224no—
401C23—4882no—
402C23—4882no—
403C23—2444no—
404C23C2×#1175820112yes—
405C23C22×C22⋊D6111yescharacteristic
406Q8—48no—
407Q8—2444no—
408Q8—2444no—
409D4—4882no—
410D4—4882no—
411D4—4882no—
412D4—2444no—
413D4—2444no—
414D4—2448no—
415D4—2444no—
416D4—2444no—
417D4—2444no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—1224no—
435C2×C4—2448no—
436C2×C4—2444no—
437C2×C4—2448no—
438C2×C4#1451974512yes—
439C2×C4#1451974511yescharacteristic
440C2×C4#1451974511yescharacteristic
441C2×C4C2×#1175820112yes—
442C2×C4C2×#1175820112yes—
443C2×C4#1451980212yes—
444C2×C4C2×D4×D6112yes—
445C8—2444no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—42no—
450C22—41no—
451C22—41no—
452C22—81no—
453C22—4888no—
454C22—2444no—
455C22—2444no—
456C22—2444no—
457C22C2×#1451974511yescharacteristic
458C22C22×#1175820111yescharacteristic
459C22C2×#1451980211yescharacteristic
460C4—22no—
461C4—22no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—82no—
466C4—82no—
467C4—2444no—
468C4#1683991212yes—
469C4#1683991212yes—
470C2—21no—
471C2—21no—
472C2—42no—
473C2—4884no—
474C2C2×#1683991211yescharacteristic
475C2#2145519711yescharacteristic
476C2#2145522011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2628C2
socleC22×C61C2×#120
prime radical#21459152C2
hypercenter#2628D61
hypocenterC61#18301
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. #21456744C2semidirect
3. #21444787C2semidirect
4. #21444770C2semidirect
5. #21444378C2semidirect
6. #21444133C2semidirect
7. C61×#2628C2semidirect
21. #16839901C22semidirect
22. #16839815C22semidirect
23. #16839495C22semidirect
24. #16839488C22semidirect
26. C2×C4⋊Q8×C61C22semidirect
30. C61×#395C22semidirect
33. C61×#382C22semidirect
102. C22×C4×C61SD16semidirect
234. #2628D61semidirect
255. C2×C4⋊Q8C2×D61semidirect
265. #395C2×D61semidirect
268. #382C2×D61semidirect
273. C61#18301semidirect
365. C22×C4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214591522
#214567442
#214447872
#214447702
#214443782
#214441332
C61×#26282
#1830161