Finite Group Databasebeta

← #29646054

Subgroups — #29646054

Order 15616.

976 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840384C211yesmaximal, characteristic
2#21454859C211yesmaximal, characteristic, complemented
3#21454842C211yesmaximal, characteristic, complemented
4#21444181C214yesmaximal, complemented
5#21444166C214yesmaximal, complemented
6#21444143C211yesmaximal, characteristic, complemented
7#21444131C211yesmaximal, characteristic, complemented
8#21444131C211yesmaximal, characteristic, complemented
9C61×#1461C211yesmaximal, characteristic, complemented
10#16840384—22no—
11#16840384—22no—
12#16840384—22no—
13#16840384—22no—
14C2×#14519776C411yescharacteristic
15C2×#14519771C412yes—
16C2×#14519766C411yescharacteristic
17C2×#14519765C412yes—
18C2×#14519762C2212yes—
19C2×#14519755C2212yes—
20C2×#14519753C411yescharacteristic
21C2×#14519752C411yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×#14519734C2211yescharacteristic
24#16839841—22no—
25#16839839—22no—
26C8×D244—22no—
27#16839827—22no—
28#16839815C2211yescharacteristic
29#16839815C2211yescharacteristic
30#16839724C2214yes—
31C22⋊C8×D61C2214yes—
32#16839488C2214yes—
33#16839487C2212yes—
34#16839487C2212yes—
35#16839485C2212yes—
36#16839485C2212yes—
37C2×C4×Q8×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#431C2211yescharacteristic
40C61×#324C2214yes—
41C22×Dic122C2×C412yes—
42C22×D244C2×C411yescharacteristic
43C22×D244C2×C412yes—
44C22×C4×D61C2×C411yescharacteristic
45C22×C4×D61C2312yes—
46C2×#11758189—24no—
47C2×C8×D61—24no—
48#14519776—22no—
49#14519776—22no—
50#14519771—24no—
51#14519771—24no—
52#14519766—22no—
53#14519766—22no—
54#14519765—24no—
55#14519765—24no—
56#14519762—24no—
57#14519762—24no—
58#14519755—24no—
59#14519755—24no—
60#14519753—22no—
61#14519753—22no—
62#14519752—22no—
63#14519752—22no—
64C4×D244D412yes—
65C4×D244D412yes—
66C4×D244D412yes—
67C4×D244D412yes—
68#14519734—21no—
69#14519734—21no—
70#14519734—21no—
71#14519734—21no—
72C2×C4⋊Dic61C2×C411yescharacteristic
73C2×#11758196C2×C411yescharacteristic
74C2×#11758196C2×C412yes—
75C2×#11758196C2×C412yes—
76C2×#11758196C2311yescharacteristic
77C2×#11758198C2×C411yescharacteristic
78C2×#11758198C2×C412yes—
79C2×#11758198C2×C412yes—
80C2×#11758198C2×C412yes—
81C2×#11758198C2×C412yes—
82C2×#11758198C2×C412yes—
83C2×#11758198C2311yescharacteristic
84C2×C4×Dic61C2×C411yescharacteristic
85C4⋊#92418—21no—
86C4⋊#92418—21no—
87C22⋊#92418C2312yes—
88C22⋊#92418C2312yes—
89#14519689—24no—
90#14519689—24no—
91C4×#92418—21no—
92C4×#92418—21no—
93C22×Q8×C61C2×C411yescharacteristic
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61—21no—
97C4×Q8×C61—21no—
98C2×C4⋊C4×C61C2×C411yescharacteristic
99C2×C4⋊C4×C61C2312yes—
100C2×C42×C61C2×C412yes—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—21no—
103C4⋊C8×C61—21no—
104C22⋊C8×C61C2312yes—
105C22⋊C8×C61C2312yes—
106C4×C8×C61—21no—
107C4×C8×C61—21no—
108C23×D61C22×C411yescharacteristic
109C23×D61C22×C412yes—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C22×C4×C61C22×C412yes—
115C22×C4×C61C22×C412yes—
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C22×Dic61C22×C411yescharacteristic
120C22×Dic61C22×C411yescharacteristic
121C22×Dic61C22×C411yescharacteristic
122C22×Dic61C22×C412yes—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—24no—
126C2×D244—82no—
127C2×D244C22⋊C412yes—
128C2×D244C22⋊C412yes—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—82no—
134C2×C4×D61C2×D414yes—
135C2×C4×D61C2×D414yes—
136C2×Dic122—24no—
137C2×Dic122—82no—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C42×C61—22no—
145C42×C61—22no—
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150#11758198—24no—
151#11758198—24no—
152#11758198—24no—
153#11758198—21no—
154#11758198—21no—
155#11758198—21no—
156#11758198—21no—
157#11758198—24no—
158#11758198C22⋊C414yes—
159#11758198C22⋊C414yes—
160C4⋊Dic61C22⋊C412yes—
161C4⋊Dic61C22⋊C412yes—
162#11758196—21no—
163#11758196—21no—
164#11758196—21no—
165#11758196—21no—
166#11758196—24no—
167#11758196—24no—
168C4×Dic61—21no—
169C4×Dic61—21no—
170C2×#92418—22no—
171C2×#92418—22no—
172#11758189—84no—
173C8×D61—84no—
174C23×C61C23×C411yescharacteristic
175C22×D61—22no—
176C22×D61—22no—
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—21no—
180C22×D61—42no—
181C22×D61—42no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61C2×C22⋊C414yes—
185Q8×C61—42no—
186Q8×C61—42no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61C2×C22⋊C411yescharacteristic
195C2×C4×C61C2×C22⋊C411yescharacteristic
196C2×C4×C61C2×C22⋊C411yescharacteristic
197C2×C4×C61C2×C22⋊C411yescharacteristic
198C2×C4×C61#13112yes—
199C2×C4×C61#13112yes—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61C2×C22⋊C414yes—
211D244—24no—
212D244—24no—
213D244—84no—
214C4×D61—44no—
215C4×D61—44no—
216C4×D61—42no—
217C4×D61—42no—
218Dic122—84no—
219C8×C61—42no—
220C8×C61—42no—
221#92418—42no—
222#92418—42no—
223#17388—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61C22×C22⋊C411yescharacteristic
227C22×C61C2×#13111yescharacteristic
228C22×C61C2×#13111yescharacteristic
229C2×D61—24no—
230C2×D61—24no—
231C2×D61—24no—
232C2×D61—41no—
233C2×D61—41no—
234C2×D61—41no—
235C2×D61—82no—
236C2×D61—82no—
237C2×D61—81no—
238C2×D61—81no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—42no—
244C4×C61—42no—
245C4×C61#40812yes—
246C4×C61#40812yes—
247Dic61—24no—
248Dic61—41no—
249Dic61—41no—
250Dic61—41no—
251C2×#542—611no—
252#2975—611no—
253#2972—611no—
254#1472—614no—
255#1468—614no—
256#1461—611no—
257#1461D6111yescharacteristic, complemented
258#1458—611no—
259#1458—611no—
260C2×C61—21no—
261C2×C61—21no—
262C2×C61C2×#40811yescharacteristic
263C2×C61#288311yescharacteristic
264C2×C61#288911yescharacteristic
265D61—44no—
266D61—81no—
267#542—1222no—
268#542—1222no—
269#542—1222no—
270#542—1222no—
271C2×C4⋊Q8—611no—
272C2×#126—612no—
273C2×#124—611no—
274C2×#124—612no—
275C2×#123—612no—
276C2×C22⋊Q8—611no—
277C2×C22⋊Q8—612no—
278C2×#121—611no—
279C2×C4×Q8C2×D6111yescharacteristic
280C2×C4×D4—611no—
281C2×#117—611no—
282#436—1222no—
283#436—1222no—
284C8×D4—1222no—
285C8×D4—1222no—
286#431—611no—
287#431—611no—
288#431C2×D6111yescharacteristic
289#431C2×D6111yescharacteristic
290#407—614no—
291C2×C22⋊C8—614no—
292#324—614no—
293#324—612no—
294#324—612no—
295#324C2×D6114yes—
296#323—612no—
297#323—612no—
298C61#1738811yescharacteristic, complemented, Sylow
299C22×Q8—612no—
300C22×Q8C4×D6111yescharacteristic
301C22×D4—611no—
302C22×D4—612no—
303C23×C4—612no—
304C23×C4—611no—
305C2×M16—1224no—
306C22×C8—1224no—
307C4⋊Q8—1222no—
308C4⋊Q8—1222no—
309#126—1224no—
310#126—1224no—
311#124—1224no—
312#124—1224no—
313#124—1222no—
314#124—1222no—
315#123—1224no—
316#123—1224no—
317C22⋊Q8—1224no—
318C22⋊Q8—1224no—
319C22⋊Q8—1222no—
320C22⋊Q8—1222no—
321#121—1222no—
322#121—1222no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×D4—612no—
328C4×D4—612no—
329C4×D4—612no—
330C4×D4—612no—
331#117—1221no—
332#117—1221no—
333#117—1221no—
334#117—1221no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4C4×D6111yescharacteristic
341C2×C4⋊C4C22×D6112yes—
342C2×C22⋊C4—612no—
343C2×C22⋊C4—611no—
344C2×C22⋊C4—612no—
345C2×C22⋊C4—612no—
346C2×C22⋊C4—612no—
347C2×C22⋊C4—612no—
348C2×C22⋊C4—611no—
349C2×C42—611no—
350C2×C42C4×D6112yes—
351C2×C42C22×D6111yescharacteristic
352C4⋊C8—21no—
353C4⋊C8—21no—
354C4⋊C8—1221no—
355C4⋊C8—1221no—
356C22⋊C8—612no—
357C22⋊C8—612no—
358C22⋊C8—1224no—
359C22⋊C8—1224no—
360C22⋊C8C22×D6112yes—
361C22⋊C8C22×D6112yes—
362C4×C8—21no—
363C4×C8—21no—
364C4×C8—1221no—
365C4×C8—1221no—
366C24—612no—
367C24—611no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—1224no—
373C2×Q8—4882no—
374C2×D4—612no—
375C2×D4—612no—
376C2×D4—1224no—
377C2×D4—1222no—
378C2×D4—1222no—
379C2×D4—4882no—
380C22×C4—612no—
381C22×C4—614no—
382C22×C4—614no—
383C22×C4—611no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4—4882no—
391C22×C4C2×C4×D6112yes—
392C22×C4C2×C4×D6112yes—
393C22×C4C2×C4×D6111yescharacteristic
394C22×C4C2×C4×D6111yescharacteristic
395C22×C4C23×D6111yescharacteristic
396M16—4884no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—4884no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—612no—
407C4⋊C4—612no—
408C4⋊C4—1224no—
409C4⋊C4—1224no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—1221no—
413C4⋊C4—1221no—
414C22⋊C4—614no—
415C22⋊C4—614no—
416C22⋊C4—1224no—
417C22⋊C4—1224no—
418C22⋊C4—1224no—
419C22⋊C4—1224no—
420C22⋊C4—1221no—
421C22⋊C4—1221no—
422C22⋊C4—1221no—
423C22⋊C4—1221no—
424C42—22no—
425C42—22no—
426C42—1221no—
427C42—1221no—
428C42D4×D6111yescharacteristic
429C42D4×D6111yescharacteristic
430C42D4×D6111yescharacteristic
431C42D4×D6111yescharacteristic
432C23—614no—
433C23—1222no—
434C23—1222no—
435C23—1221no—
436C23—1221no—
437C23—1221no—
438C23—2442no—
439C23—2442no—
440C23—2441no—
441C23—2441no—
442C23C22×C4×D6111yescharacteristic
443Q8—42no—
444Q8—42no—
445Q8—4884no—
446D4—1224no—
447D4—1224no—
448D4—4884no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—614no—
457C2×C4—1221no—
458C2×C4—1221no—
459C2×C4—1221no—
460C2×C4—1221no—
461C2×C4—1221no—
462C2×C4—2444no—
463C2×C4—2444no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4C22⋊C4×D6111yescharacteristic
470C2×C4C22⋊C4×D6111yescharacteristic
471C2×C4C22⋊C4×D6111yescharacteristic
472C2×C4C22⋊C4×D6111yescharacteristic
473C2×C4#1451978312yes—
474C2×C4#1451978512yes—
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4C2×D4×D6111yescharacteristic
477C8—42no—
478C8—42no—
479C8—2442no—
480C8—2442no—
481C22—21no—
482C22—21no—
483C22—1224no—
484C22—1224no—
485C22—1224no—
486C22—4881no—
487C22—4881no—
488C22—4882no—
489C22—4882no—
490C22—2441no—
491C22—2441no—
492C22—2441no—
493C22C2×C22⋊C4×D6111yescharacteristic
494C22C2×#1451978311yescharacteristic
495C22C2×#1451978511yescharacteristic
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—42no—
501C4—42no—
502C4—1224no—
503C4—2441no—
504C4—2441no—
505C4—2441no—
506C4#1683972712yes—
507C4#1683972712yes—
508C2—21no—
509C2—21no—
510C2—4881no—
511C2—2444no—
512C2C2×#1683972711yescharacteristic
513C2#2145427811yescharacteristic
514C2#2145429911yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1461C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519752C4
cosocleC22×C4×C61C24
hypercenter#1461D61
hypocenterC61#17388
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #21454859C2semidirect
3. #21454842C2semidirect
4. #21444181C2semidirect
5. #21444166C2semidirect
6. #21444143C2semidirect
7. #21444131C2semidirect
8. #21444131C2semidirect
9. C61×#1461C2semidirect
257. #1461D61semidirect
298. C61#17388semidirect

Maximal subgroups

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

subgroupindex
C2×#168403842
#214548592
#214548422
#214441812
#214441662
#214441432
#214441312
C61×#14612
#1738861