Finite Group Databasebeta

← #29654051

Subgroups — #29654051

Order 15616.

1188 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459152C211yesmaximal, characteristic
2#21456940C212yesmaximal, complemented
3#21444821C214yesmaximal, complemented
4#21444804C214yesmaximal, complemented
5#21444378C212yesmaximal, complemented
6#21444133C211yesmaximal, characteristic, complemented
7C61×#2621C211yesmaximal, 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#16840165—22no—
19#16840110—22no—
20#16840075C2218yescomplemented
21#16840039—24no—
22#16839815C2212yescomplemented
23#16839495C2218yescomplemented
24#16839488C2212yescomplemented
25#16839486C2212yes—
26C2×C4⋊Q8×C61C2211yescharacteristic, complemented
27C2×C22⋊Q8×C61—21no—
28C2×#117×C61C2211yescharacteristic
29C61×#398—21no—
30C61×#398C2212yescomplemented
31C61×#397—21no—
32C61×#397C2212yes—
33C61×#391—21no—
34C61×#382C2211yescharacteristic, complemented
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#14519710—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—
76C22⋊Q8×C61—44no—
77#117×C61—22no—
78#117×C61—22no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61—22no—
82C2×C4⋊C4×C61D411yescharacteristic
83C2×C4⋊C4×C61C2312yes—
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C22⋊C4×C61—22no—
86C2×C22⋊C4×C61D411yescharacteristic
87C2×C22⋊C4×C61C2311yescharacteristic
88C2×C42×C61D411yescharacteristic
89C2×C42×C61C2311yescharacteristic
90#95×C61—21no—
91#95×C61—22no—
92#95×C61—22no—
93#95×C61C2312yes—
94C24×C61C2×D411yescharacteristic
95C23×D61—22no—
96C2×Q8×C61—41no—
97C2×Q8×C61—41no—
98C2×Q8×C61—42no—
99C2×Q8×C61—42no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61D814yes—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2411yescharacteristic
112C2×C22⋊D61—24no—
113C2×C22⋊D61—24no—
114C2×C22⋊D61—44no—
115C2×C22⋊D61—44no—
116C22×Dic61—22no—
117#11758222—44no—
118#11758222—48no—
119#11758222—48no—
120#11758222—44no—
121#11758222—44no—
122#11758222—44no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—44no—
129C2×D244C2×D414yes—
130C2×C4×D61—24no—
131C2×C4×D61—24no—
132C2×C4×D61—48no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122—44no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—24no—
139C4⋊C4×C61—42no—
140C4⋊C4×C61—42no—
141C4⋊C4×C61—44no—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—44no—
145C42×C61—21no—
146C42×C61—21no—
147C42×C61C2×D412yes—
148C42×C61C2×D412yes—
149#11758198—24no—
150#11758198—24no—
151#11758198—24no—
152#11758198—24no—
153C4⋊Dic61—22no—
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156C4⋊Dic61C2×D414yes—
157C2×#92418—24no—
158C23×C61—21no—
159C23×C61—21no—
160C23×C61—41no—
161C23×C61—41no—
162C23×C61C2×D812yes—
163C23×C61C22×D411yescharacteristic
164C22×D61—24no—
165C22×D61—44no—
166C22×D61—82no—
167C22×D61—82no—
168Q8×C61—82no—
169Q8×C61—82no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—24no—
173C2×C4×C61—24no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61#12012yes—
187C2×C4×C61#12011yescharacteristic
188C2×C4×C61#12011yescharacteristic
189C2×C4×C61C2×D812yes—
190C2×C4×C61C2×D812yes—
191C2×C4×C61AGL(1,ℤ/8)12yes—
192C2×C4×C61C22×D412yes—
193C22⋊D61—48no—
194C22⋊D61—44no—
195C22⋊D61—44no—
196C2×Dic61—24no—
197C2×Dic61—44no—
198D244—44no—
199D244—44no—
200D244—44no—
201D244—82no—
202D244—82no—
203D244—82no—
204C4×D61—48no—
205C4×D61—48no—
206Dic122—44no—
207Dic122—44no—
208#92418—44no—
209#18308—611nomaximal, Sylow
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—22no—
213C22×C61—42no—
214C22×C61—41no—
215C22×C61—41no—
216C22×C61—81no—
217C22×C61C2×#12011yescharacteristic
218C22×C61C22×D811yescharacteristic
219C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
220C2×D61—44no—
221C2×D61—44no—
222C2×D61—44no—
223C2×D61—88no—
224C4×C61—22no—
225C4×C61—22no—
226C4×C61—22no—
227C4×C61—41no—
228C4×C61—41no—
229C4×C61—82no—
230C4×C61—82no—
231C4×C61#44712yes—
232C4×C61#44712yes—
233Dic61—44no—
234#3449—611no—
235#3226—612no—
236#2621D6111yescharacteristic, complemented
237#1610—614no—
238#1607—614no—
239#1517—612no—
240#1458—611no—
241C2×C61—21no—
242C2×C61—21no—
243C2×C61—42no—
244C2×C61C2×#44711yescharacteristic
245C2×C61#302211yescharacteristic
246C2×C61#302811yescharacteristic
247D61—84no—
248C22×#41—1221no—
249#537—612no—
250#537—1221no—
251#536—1222no—
252#535—1222no—
253#534—612no—
254#534—1221no—
255#532—612no—
256#532—1221no—
257C2×C4⋊Q8C2×D6111yescharacteristic, complemented
258#525—1224no—
259C2×C22⋊Q8—21no—
260C2×#117C2×D6111yescharacteristic
261#500—1222no—
262#486—1222no—
263#480—618no—
264#474—1224no—
265#431—612no—
266#398—21no—
267#398C2×D6112yescomplemented
268#397—21no—
269#397C2×D6112yes—
270#391—21no—
271#382C2×D6111yescharacteristic, complemented
272#327—618no—
273#324—612no—
274#323—612no—
275C61#1830811yescharacteristic, complemented, Sylow
276C2×#41—1222no—
277C2×#41—1222no—
278C2×#41—1222no—
279C2×#41—1222no—
280C2×#41—2444no—
281C2×#41—2444no—
282C22×Q8—21no—
283C22×Q8—1221no—
284C22×D4—1221no—
285C22×D4—1222no—
286C23×C4—1222no—
287C23×C4C22⋊D6111yescharacteristic
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—612no—
291C4⋊Q8—1221no—
292#127—612no—
293#127—1221no—
294#124—1222no—
295#124—1222no—
296#123—1224no—
297#123—1224no—
298C22⋊Q8—44no—
299C22⋊Q8—1224no—
300C22⋊Q8—1224no—
301#121—614no—
302#121—1224no—
303#121—1224no—
304#121—1222no—
305#121—1222no—
306#121—1222no—
307#120—1224no—
308#120—1224no—
309C4×D4—614no—
310C4×D4—1222no—
311C4×D4—1222no—
312C4×D4—1222no—
313#117—22no—
314#117—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4C22⋊D6111yescharacteristic
319C2×C4⋊C4C22×D6112yes—
320C2×C4⋊C4C22×D6111yescharacteristic
321C2×C22⋊C4—22no—
322C2×C22⋊C4C22⋊D6111yescharacteristic
323C2×C22⋊C4C22×D6111yescharacteristic
324C2×C42C22⋊D6111yescharacteristic
325C2×C42C22×D6111yescharacteristic
326#107—1228no—
327C4⋊C8—1222no—
328#102—1228no—
329C22⋊C8—614no—
330C4×C8—1222no—
331#95—21no—
332#95—22no—
333#95—22no—
334#95C22×D6112yes—
335C24—1222no—
336C24C2×C22⋊D6111yescharacteristic
337#41—2444no—
338#41—2448no—
339#41—2448no—
340#41—2444no—
341#41—2444no—
342#41—2444no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—1222no—
348C2×Q8—1222no—
349C2×Q8—2444no—
350C2×D4—614no—
351C2×D4—1222no—
352C2×D4—1222no—
353C2×D4—1222no—
354C2×D4—1224no—
355C2×D4—1224no—
356C2×D4—1222no—
357C2×D4—1222no—
358C2×D4—2444no—
359C2×D4—2444no—
360C2×D4—2444no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—1224no—
367C22×C4—1222no—
368C22×C4—1224no—
369C22×C4—2448no—
370C22×C4#1175819914yes—
371C22×C4C2×C22⋊D6111yescharacteristic
372C22×C4C2×C22⋊D6111yescharacteristic
373C22×C4C2×C22⋊D6111yescharacteristic
374C22×C4C2×C22⋊D6111yescharacteristic
375C22×C4C2×C22⋊D6111yescharacteristic
376C22×C4C23×D6111yescharacteristic
377C2×C8—1224no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—614no—
385C4⋊C4—1222no—
386C4⋊C4—1222no—
387C4⋊C4—1222no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—44no—
391C22⋊C4—1224no—
392C22⋊C4—1224no—
393C22⋊C4—1224no—
394C22⋊C4—1224no—
395C42—21no—
396C42—21no—
397C42D4×D6112yes—
398C42D4×D6112yes—
399C23—21no—
400C23—21no—
401C23—41no—
402C23—41no—
403C23—1224no—
404C23—4882no—
405C23—4882no—
406C23—2444no—
407C23C2×#1175819912yes—
408C23C22×C22⋊D6111yescharacteristic
409Q8—82no—
410Q8—82no—
411Q8—2444no—
412Q8—2444no—
413D4—4882no—
414D4—4882no—
415D4—4882no—
416D4—2444no—
417D4—2444no—
418D4—2448no—
419D4—2444no—
420D4—2444no—
421D4—2444no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—1224no—
439C2×C4—2448no—
440C2×C4—2444no—
441C2×C4—2448no—
442C2×C4#1451974512yes—
443C2×C4#1451974511yescharacteristic
444C2×C4#1451974511yescharacteristic
445C2×C4C2×#1175819912yes—
446C2×C4C2×#1175819912yes—
447C2×C4#1451980212yes—
448C2×C4C2×D4×D6112yes—
449C8—2444no—
450C22—21no—
451C22—21no—
452C22—22no—
453C22—42no—
454C22—41no—
455C22—41no—
456C22—81no—
457C22—4888no—
458C22—2444no—
459C22—2444no—
460C22—2444no—
461C22C2×#1451974511yescharacteristic
462C22C22×#1175819911yescharacteristic
463C22C2×#1451980211yescharacteristic
464C4—22no—
465C4—22no—
466C4—22no—
467C4—41no—
468C4—41no—
469C4—82no—
470C4—82no—
471C4—2444no—
472C4#1683989512yes—
473C4#1683989512yes—
474C2—21no—
475C2—21no—
476C2—42no—
477C2—4884no—
478C2C2×#1683989511yescharacteristic
479C2#2145519711yescharacteristic
480C2#2145522411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2621C2
socleC22×C61C2×#120
prime radical#21459152C2
hypercenter#2621D61
hypocenterC61#18308
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2621

Complements

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

NH ≅ G/Ntype
2. #21456940C2semidirect
3. #21444821C2semidirect
4. #21444804C2semidirect
5. #21444378C2semidirect
6. #21444133C2semidirect
7. C61×#2621C2semidirect
20. #16840075C22semidirect
22. #16839815C22semidirect
23. #16839495C22semidirect
24. #16839488C22semidirect
26. C2×C4⋊Q8×C61C22semidirect
30. C61×#398C22semidirect
34. C61×#382C22semidirect
236. #2621D61semidirect
257. C2×C4⋊Q8C2×D61semidirect
267. #398C2×D61semidirect
271. #382C2×D61semidirect
275. C61#18308semidirect

Maximal subgroups

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

subgroupindex
#214591522
#214569402
#214448212
#214448042
#214443782
#214441332
C61×#26212
#1830861