Finite Group Databasebeta

← #29656319

Subgroups — #29656319

Order 15616.

982 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840477C211yesmaximal, characteristic
2#21456201C211yesmaximal, characteristic, complemented
3#21455914C211yesmaximal, characteristic, complemented
4#21445046C214yesmaximal, complemented
5#21445045C214yesmaximal, complemented
6#21444544C211yesmaximal, characteristic, complemented
7#21444262C211yesmaximal, characteristic, complemented
8#21444261C211yesmaximal, characteristic, complemented
9C61×#3110C211yesmaximal, characteristic, complemented
10#16840477—22no—
11#16840477—22no—
12#16840477—22no—
13#16840477C2214yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519763—22no—
16C2×#14519751—21no—
17C2×#14519748—21no—
18C2×C4×D244C2211yescharacteristic
19C2×#14519736—21no—
20C2×#14519735—21no—
21#16840132—22no—
22#16840114C2212yescomplemented
23#16840114C2212yescomplemented
24#16840095—21no—
25#16840039C2212yescomplemented
26#16839964C2212yescomplemented
27#16839912—21no—
28#16839860—22no—
29#16839819C2211yescharacteristic
30#16839770—22no—
31C4×#11758194C2211yescharacteristic
32#16839510C2214yescomplemented
33D4⋊#92418C2214yescomplemented
34#16839489C2214yescomplemented
35C61×#533C2211yescharacteristic
36C2×C4×D4×C61C2211yescharacteristic
37C61×#484—21no—
38C61×#474C2212yescomplemented
39C61×#461C2212yescomplemented
40C61×#450—21no—
41C61×#440—22no—
42C61×#432C2211yescharacteristic
43C22×Dic122D412yes—
44C22×D244D411yescharacteristic
45C22×C22⋊D61—22no—
46C22×C4×D61—21no—
47C2×#11758201—22no—
48C2×#11758194—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777D412yes—
52#14519777C2312yes—
53#14519763—28no—
54#14519763—28no—
55#14519751—24no—
56#14519751—24no—
57#14519748—24no—
58#14519748—24no—
59C4×D244—21no—
60C4×D244—21no—
61C4×D244D412yes—
62C4×D244C2312yes—
63#14519736—24no—
64#14519736—24no—
65#14519735—24no—
66#14519735—24no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×C22⋊Dic61—22no—
71C2×#11758198—22no—
72C2×#11758198—21no—
73C2×C4×Dic61—22no—
74#14519710—22no—
75C4⋊#92418C2312yes—
76#14519701—22no—
77#14519700—24no—
78#14519700—24no—
79#14519697—22no—
80C22⋊#92418—21no—
81#14519688—21no—
82#14519688—21no—
83C4×#92418C2312yes—
84C22×D4×C61D411yescharacteristic
85C23×C4×C61—21no—
86C2×SD16×C61—22no—
87C4⋊Q8×C61C2311yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89#125×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61D412yes—
97C4×D4×C61C2312yes—
98#117×C61—21no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C22⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#107×C61—22no—
103C4⋊C8×C61C2312yes—
104#103×C61—22no—
105#102×C61—22no—
106C22⋊C8×C61—21no—
107#97×C61—21no—
108C24×C61—21no—
109C23×D61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—42no—
114C2×D4×C61C2×D412yes—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—44no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C2×C22⋊D61—28no—
123C2×C22⋊D61—84no—
124C22×Dic61—21no—
125C22×Dic61—22no—
126C2×D244—21no—
127C2×D244—42no—
128C2×D244C2×D412yes—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—44no—
132C2×Dic122—22no—
133C2×Dic122—44no—
134C2×Dic122C2×D414yes—
135SD16×C61—44no—
136C2×C8×C61—22no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61C2×D412yes—
145C22⋊C4×C61—22no—
146C22⋊C4×C61—22no—
147C22⋊C4×C61—41no—
148C42×C61—41no—
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61C2411yescharacteristic
153C22⋊Dic61—28no—
154#11758201—44no—
155#11758198—28no—
156#11758198—22no—
157#11758198—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61C2×D412yes—
164C4×Dic61—28no—
165#11758194—42no—
166C2×#92418—22no—
167C2×#92418—22no—
168C23×C61—22no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61#12011yescharacteristic
173C22×D61—22no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177Q8×C61—42no—
178Q8×C61—42no—
179D4×C61—22no—
180D4×C61—22no—
181D4×C61—42no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—44no—
187C2×C4×C61—42no—
188C2×C4×C61—44no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61#12011yescharacteristic
192C2×C4×C61#12011yescharacteristic
193C2×C4×C61#12011yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C22⋊D61—88no—
198C2×Dic61—22no—
199C2×Dic61—24no—
200C2×Dic61—42no—
201C2×Dic61—44no—
202D244—22no—
203D244—22no—
204D244—42no—
205C4×D61—44no—
206C4×D61—44no—
207Dic122—44no—
208Dic122—82no—
209C8×C61—42no—
210#92418—42no—
211#92418—42no—
212#18552—611nomaximal, Sylow
213C22×C61—21no—
214C22×C61—21no—
215C22×C61—42no—
216C22×C61—42no—
217C22×C61—42no—
218C22×C61—84no—
219C22×C61#45512yes—
220C22×C61C2×#12011yescharacteristic
221C2×D61—42no—
222C2×D61—42no—
223C2×D61—42no—
224C2×D61—84no—
225C4×C61—21no—
226C4×C61—21no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—42no—
230C4×C61—81no—
231C4×C61—81no—
232C4×C61#57512yes—
233C4×C61#57512yes—
234Dic61—42no—
235Dic61—44no—
236C2×#555—611no—
237#3139—611no—
238#3110—611no—
239#3110D6111yescharacteristic, complemented
240#1646—614no—
241#1646—614no—
242#1558—611no—
243#1490—611no—
244#1490—611no—
245C2×C61—22no—
246C2×C61—82no—
247C2×C61#300511yescharacteristic
248C2×C61#300511yescharacteristic
249C2×C61C2×#45511yescharacteristic
250D61—82no—
251#555—614no—
252#555—1222no—
253#555—1222no—
254#555—1222no—
255#533C2×D6111yescharacteristic
256C2×C4⋊Q8—611no—
257C2×#124—1222no—
258C2×#121—1221no—
259C2×#121—1221no—
260C2×C4×D4—611no—
261C2×C4×D4C2×D6111yescharacteristic
262C2×#117—1221no—
263C2×#117—1221no—
264#491—1222no—
265#487—612no—
266#487—612no—
267#484—21no—
268#484—1221no—
269#474—612no—
270#474C2×D6112yescomplemented
271#461—612no—
272#461C2×D6112yescomplemented
273#450—21no—
274#450—1221no—
275#440—22no—
276#440—1222no—
277#432—611no—
278#432C2×D6111yescharacteristic
279#419—1222no—
280C4×M16—611no—
281#333—614no—
282D4⋊C8—614no—
283D4⋊C8—614no—
284C61#1855211yescharacteristic, complemented, Sylow
285C22×Q8—612no—
286C22×D4—611no—
287C22×D4—1222no—
288C22×D4C22⋊D6111yescharacteristic
289C23×C4—21no—
290C23×C4—1221no—
291C2×SD16—22no—
292C2×SD16—1222no—
293C2×M16—1221no—
294C4⋊Q8—612no—
295C4⋊Q8—612no—
296C4⋊Q8—1221no—
297C4⋊Q8—1221no—
298C4⋊Q8C22×D6111yescharacteristic
299C4⋊Q8C22×D6111yescharacteristic
300#125—21no—
301#124—1228no—
302#124—1228no—
303C22⋊Q8—21no—
304C22⋊Q8—21no—
305#121—1224no—
306#121—1224no—
307#121—1224no—
308#121—1224no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—612no—
314C4×D4—612no—
315C4×D4—1221no—
316C4×D4—1221no—
317C4×D4C22⋊D6112yes—
318C4×D4C22×D6112yes—
319#117—21no—
320#117—1224no—
321#117—1224no—
322#117—1224no—
323#117—1224no—
324C2×C4⋊C4—611no—
325C2×C4⋊C4—1221no—
326C2×C4⋊C4—1221no—
327C2×C4⋊C4C22⋊D6111yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—1222no—
330C2×C22⋊C4—1222no—
331C2×C22⋊C4—1221no—
332C2×C42—1222no—
333C2×C42C22×D6111yescharacteristic
334#107—22no—
335#107—1222no—
336C4⋊C8—612no—
337C4⋊C8C22×D6112yes—
338#103—22no—
339#103—1224no—
340#103—1224no—
341#103—1222no—
342#102—22no—
343#102—1222no—
344C22⋊C8—21no—
345C22⋊C8—1221no—
346#97—21no—
347#97—1221no—
348#97—1221no—
349C4×C8—612no—
350C24—21no—
351C24—1221no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—614no—
355C2×Q8—1222no—
356C2×Q8—2444no—
357C2×D4—21no—
358C2×D4—42no—
359C2×D4—612no—
360C2×D4—1228no—
361C2×D4—1221no—
362C2×D4—4884no—
363C2×D4—2442no—
364C2×D4C2×C22⋊D6112yes—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—44no—
369C22×C4—1221no—
370C22×C4—1222no—
371C22×C4—1222no—
372C22×C4—1222no—
373C22×C4—2444no—
374C22×C4D4×D6111yescharacteristic
375C22×C4D4×D6111yescharacteristic
376C22×C4C2×C22⋊D6111yescharacteristic
377SD16—44no—
378SD16—2444no—
379M16—2442no—
380C2×C8—22no—
381C2×C8—1222no—
382C2×C8—1222no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—612no—
391C4⋊C4—1222no—
392C4⋊C4—1222no—
393C4⋊C4—1221no—
394C4⋊C4—1222no—
395C4⋊C4—1222no—
396C4⋊C4C2×C22⋊D6112yes—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—41no—
400C22⋊C4—1228no—
401C22⋊C4—1228no—
402C22⋊C4—1222no—
403C22⋊C4—1222no—
404C42—41no—
405C42—1228no—
406C42D4×D6111yescharacteristic
407C42D4×D6111yescharacteristic
408C42C2×C22⋊D6111yescharacteristic
409C42C23×D6111yescharacteristic
410C23—22no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—1222no—
415C23—2442no—
416C23—2442no—
417C23—2442no—
418C23#1451974511yescharacteristic
419Q8—42no—
420Q8—42no—
421Q8—4882no—
422Q8—2444no—
423D4—22no—
424D4—22no—
425D4—42no—
426D4—1222no—
427D4—1222no—
428D4—4888no—
429D4—2442no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—1222no—
440C2×C4—1224no—
441C2×C4—2444no—
442C2×C4—2442no—
443C2×C4—2444no—
444C2×C4—2444no—
445C2×C4#1451974511yescharacteristic
446C2×C4#1451974511yescharacteristic
447C2×C4#1451974511yescharacteristic
448C2×C4C2×D4×D6111yescharacteristic
449C2×C4C2×D4×D6111yescharacteristic
450C2×C4C22×C22⋊D6111yescharacteristic
451C8—42no—
452C8—2442no—
453C8—2442no—
454C22—21no—
455C22—21no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—84no—
460C22—4884no—
461C22—2442no—
462C22—2442no—
463C22—2442no—
464C22#1683993812yes—
465C22C2×#1451974511yescharacteristic
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—42no—
471C4—81no—
472C4—81no—
473C4—2444no—
474C4—2442no—
475C4#1684055812yes—
476C4#1684056012yes—
477C2—22no—
478C2—82no—
479C2—4882no—
480C2#2145505811yescharacteristic
481C2#2145505911yescharacteristic
482C2C2×#1683993811yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3110C2
socleC22×C61C2×#120
prime radicalC2×#16840477C2
hypercenter#3110D61
hypocenterC61#18552
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
2. #21456201C2semidirect
3. #21455914C2semidirect
4. #21445046C2semidirect
5. #21445045C2semidirect
6. #21444544C2semidirect
7. #21444262C2semidirect
8. #21444261C2semidirect
9. C61×#3110C2semidirect
22. #16840114C22semidirect
23. #16840114C22semidirect
25. #16840039C22semidirect
26. #16839964C22semidirect
32. #16839510C22semidirect
33. D4⋊#92418C22semidirect
34. #16839489C22semidirect
38. C61×#474C22semidirect
39. C61×#461C22semidirect
239. #3110D61semidirect
270. #474C2×D61semidirect
272. #461C2×D61semidirect
284. C61#18552semidirect

Maximal subgroups

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

subgroupindex
C2×#168404772
#214562012
#214559142
#214450462
#214450452
#214445442
#214442622
#214442612
C61×#31102
#1855261