Finite Group Databasebeta

← #29646068

Subgroups — #29646068

Order 15616.

940 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840377C211yesmaximal, characteristic
2#21454902C211yesmaximal, characteristic, complemented
3#21454899C211yesmaximal, characteristic, complemented
4#21444185C214yesmaximal, complemented
5#21444169C214yesmaximal, complemented
6#21444140C211yesmaximal, characteristic, complemented
7#21444131C211yesmaximal, characteristic, complemented
8#21444131C211yesmaximal, characteristic, complemented
9C61×#1460C211yesmaximal, characteristic, complemented
10#16840377—22no—
11#16840377—22no—
12#16840377—22no—
13#16840377—22no—
14C2×#14519776C411yescharacteristic
15C2×#14519770C2212yes—
16C2×#14519764C411yescharacteristic
17C2×#14519763C2212yes—
18C2×#14519760C412yes—
19C2×#14519757C411yescharacteristic
20C2×#14519756C412yes—
21C2×#14519748C411yescharacteristic
22C2×C4×Dic122C2211yescharacteristic
23C2×#14519734C2211yescharacteristic
24#16839892—22no—
25#16839889—22no—
26C8×Dic122—22no—
27#16839886—22no—
28#16839816C2214yes—
29#16839815C2211yescharacteristic
30#16839815C2211yescharacteristic
31#16839812C2214yes—
32#16839487C2212yes—
33#16839487C2212yes—
34#16839486C2214yes—
35#16839485C2212yes—
36#16839485C2212yes—
37C2×C4×D4×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#431C2211yescharacteristic
40C61×#323C2214yes—
41C22×Dic122C2×C411yescharacteristic
42C22×Dic122C2×C412yes—
43C22×C22⋊D61C2×C412yes—
44C22×C4×D61C2×C411yescharacteristic
45#14519776—22no—
46#14519776—22no—
47#14519770—24no—
48#14519770—24no—
49#14519764—22no—
50#14519764—22no—
51#14519763—24no—
52#14519763—24no—
53#14519760—24no—
54#14519760—24no—
55#14519757—22no—
56#14519757—22no—
57#14519756—24no—
58#14519756—24no—
59#14519748—22no—
60#14519748—22no—
61C4×Dic122D412yes—
62C4×Dic122D412yes—
63C4×Dic122D412yes—
64C4×Dic122D412yes—
65#14519734—21no—
66#14519734—21no—
67#14519734—21no—
68#14519734—21no—
69C2×C4⋊Dic61C2×C412yes—
70C2×C4⋊Dic61C2×C411yescharacteristic
71C2×#11758196C2×C411yescharacteristic
72C2×#11758196C2×C412yes—
73C2×#11758196C2×C412yes—
74C2×#11758196C2311yescharacteristic
75C2×C22⋊Dic61C2×C412yes—
76C2×C22⋊Dic61C2×C412yes—
77C2×C22⋊Dic61C2×C412yes—
78C2×#11758198C2×C411yescharacteristic
79C2×#11758198C2×C412yes—
80C2×#11758198C2311yescharacteristic
81C2×C4×Dic61C2×C411yescharacteristic
82C2×C4×Dic61C2312yes—
83C4⋊#92418—21no—
84C4⋊#92418—21no—
85#14519706—24no—
86#14519706—24no—
87C22⋊#92418C2312yes—
88C22⋊#92418C2312yes—
89#14519687—24no—
90C4×#92418—21no—
91C4×#92418—21no—
92C8×Dic61—24no—
93C22×D4×C61C2×C411yescharacteristic
94C23×C4×C61C2×C412yes—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C2×C4⋊C4×C61C2×C411yescharacteristic
100C2×C22⋊C4×C61C2312yes—
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—
108C24×C61C22×C412yes—
109C23×D61C22×C411yescharacteristic
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—82no—
117C22×C4×C61C22×C412yes—
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C2×C22⋊D61—24no—
122C2×C22⋊D61—82no—
123C22×Dic61C22×C411yescharacteristic
124C22×Dic61C22×C412yes—
125C22×Dic61C22×C411yescharacteristic
126C22×Dic61C22×C411yescharacteristic
127C22×Dic61C22×C412yes—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×Dic122—22no—
133C2×Dic122—22no—
134C2×Dic122—24no—
135C2×Dic122—82no—
136C2×Dic122C22⋊C412yes—
137C2×Dic122C22⋊C412yes—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C42×C61C2×D411yescharacteristic
145C42×C61C2×D411yescharacteristic
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C22⋊Dic61—24no—
149C22⋊Dic61—24no—
150C22⋊Dic61—24no—
151#11758198—21no—
152#11758198—21no—
153#11758198—21no—
154#11758198—21no—
155#11758198—24no—
156C4⋊Dic61—24no—
157C4⋊Dic61C22⋊C412yes—
158C4⋊Dic61C22⋊C412yes—
159#11758196—21no—
160#11758196—21no—
161#11758196—21no—
162#11758196—21no—
163#11758196—24no—
164#11758196C22⋊C414yes—
165#11758196C22⋊C414yes—
166C4×Dic61—21no—
167C4×Dic61—21no—
168C4×Dic61C2×D414yes—
169C4×Dic61C2×D414yes—
170C2×#92418—22no—
171C2×#92418—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61C23×C411yescharacteristic
178C22×D61—21no—
179C22×D61—21no—
180C22×D61—21no—
181C22×D61—41no—
182C22×D61—41no—
183D4×C61—42no—
184D4×C61—42no—
185C2×C4×C61—22no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—82no—
190C2×C4×C61—82no—
191C2×C4×C61C2×C22⋊C411yescharacteristic
192C2×C4×C61C2×C22⋊C411yescharacteristic
193C2×C4×C61C2×C22⋊C411yescharacteristic
194C2×C4×C61C2×C22⋊C411yescharacteristic
195C2×C4×C61#13112yes—
196C2×C4×C61#13112yes—
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C22⋊D61—84no—
200C2×Dic61—22no—
201C2×Dic61—22no—
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—
211C2×Dic61C2×C22⋊C414yes—
212C4×D61—42no—
213C4×D61—42no—
214Dic122—24no—
215Dic122—24no—
216Dic122—84no—
217C8×C61—42no—
218C8×C61—42no—
219#92418—42no—
220#92418—42no—
221#17390—611nomaximal, Sylow
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—42no—
227C22×C61—82no—
228C22×C61—82no—
229C22×C61C22×C22⋊C411yescharacteristic
230C22×C61C2×#13111yescharacteristic
231C22×C61C2×#13111yescharacteristic
232C2×D61—41no—
233C2×D61—41no—
234C2×D61—41no—
235C2×D61—81no—
236C2×D61—81no—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—42no—
242C4×C61#40812yes—
243C4×C61#40812yes—
244Dic61—24no—
245Dic61—24no—
246Dic61—41no—
247Dic61—41no—
248Dic61—41no—
249C2×#542—611no—
250#2983—611no—
251#2982—611no—
252#1473—614no—
253#1469—614no—
254#1460—611no—
255#1460D6111yescharacteristic, complemented
256#1458—611no—
257#1458—611no—
258C2×C61—21no—
259C2×C61—21no—
260C2×C61—82no—
261C2×C61C2×#40811yescharacteristic
262C2×C61#288411yescharacteristic
263C2×C61#288811yescharacteristic
264D61—81no—
265#542—1222no—
266#542—1222no—
267#542—1222no—
268#542—1222no—
269C2×C4⋊Q8—611no—
270C2×#126—612no—
271C2×#124—611no—
272C2×#124—612no—
273C2×#123—612no—
274C2×C22⋊Q8—612no—
275C2×C22⋊Q8—611no—
276C2×#121—611no—
277C2×C4×Q8—611no—
278C2×C4×D4C2×D6111yescharacteristic
279C2×#117—611no—
280#446—1222no—
281#446—1222no—
282C8×Q8—1222no—
283C8×Q8—1222no—
284#432—614no—
285#431—611no—
286#431—611no—
287#431—614no—
288#431C2×D6111yescharacteristic
289#431C2×D6111yescharacteristic
290#324—612no—
291#324—612no—
292#323—614no—
293#323—612no—
294#323—612no—
295#323C2×D6114yes—
296C61#1739011yescharacteristic, complemented, Sylow
297C22×Q8—611no—
298C22×Q8—612no—
299C22×D4—612no—
300C22×D4C4×D6111yescharacteristic
301C23×C4—611no—
302C23×C4C4×D6112yes—
303C4⋊Q8—1222no—
304C4⋊Q8—1222no—
305#126—1224no—
306#126—1224no—
307#124—1224no—
308#124—1224no—
309#124—1222no—
310#124—1222no—
311#123—1224no—
312#123—1224no—
313C22⋊Q8—1222no—
314C22⋊Q8—1222no—
315C22⋊Q8—1224no—
316C22⋊Q8—1224no—
317#121—1222no—
318#121—1222no—
319C4×Q8—612no—
320C4×Q8—612no—
321C4×Q8—612no—
322C4×Q8—612no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327#117—1221no—
328#117—1221no—
329#117—1221no—
330#117—1221no—
331C2×C4⋊C4—611no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4—612no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4C4×D6111yescharacteristic
338C2×C22⋊C4—612no—
339C2×C22⋊C4—612no—
340C2×C22⋊C4—612no—
341C2×C22⋊C4—611no—
342C2×C22⋊C4—612no—
343C2×C22⋊C4—611no—
344C2×C22⋊C4C22×D6112yes—
345C2×C42—612no—
346C2×C42—611no—
347C2×C42C22×D6111yescharacteristic
348C4⋊C8—21no—
349C4⋊C8—21no—
350C4⋊C8—1224no—
351C4⋊C8—1224no—
352C4⋊C8—1221no—
353C4⋊C8—1221no—
354C22⋊C8—612no—
355C22⋊C8—612no—
356C22⋊C8C22×D6112yes—
357C22⋊C8C22×D6112yes—
358#97—1224no—
359C4×C8—21no—
360C4×C8—21no—
361C4×C8—1224no—
362C4×C8—1221no—
363C4×C8—1221no—
364C24—611no—
365C24C2×C4×D6112yes—
366C2×Q8—612no—
367C2×Q8—612no—
368C2×Q8—1224no—
369C2×Q8—1222no—
370C2×Q8—1222no—
371C2×Q8—4882no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—1224no—
377C2×D4—4882no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—82no—
381C22×C4—612no—
382C22×C4—612no—
383C22×C4—611no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4C2×C4×D6112yes—
391C22×C4C2×C4×D6111yescharacteristic
392C22×C4C2×C4×D6111yescharacteristic
393C22×C4C23×D6111yescharacteristic
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—1222no—
397C2×C8—1222no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—614no—
401C4⋊C4—614no—
402C4⋊C4—612no—
403C4⋊C4—612no—
404C4⋊C4—1224no—
405C4⋊C4—1224no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—1224no—
413C22⋊C4—1224no—
414C22⋊C4—1224no—
415C22⋊C4—1224no—
416C22⋊C4—1221no—
417C22⋊C4—1221no—
418C22⋊C4—1221no—
419C22⋊C4—1221no—
420C42—614no—
421C42—614no—
422C42—1221no—
423C42—1221no—
424C42D4×D6111yescharacteristic
425C42D4×D6111yescharacteristic
426C42D4×D6111yescharacteristic
427C42D4×D6111yescharacteristic
428C23—22no—
429C23—22no—
430C23—22no—
431C23—42no—
432C23—42no—
433C23—1221no—
434C23—1221no—
435C23—1221no—
436C23—2441no—
437C23—2441no—
438C23C22×C4×D6111yescharacteristic
439Q8—1224no—
440Q8—1224no—
441Q8—4884no—
442D4—42no—
443D4—42no—
444D4—4884no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—614no—
452C2×C4—614no—
453C2×C4—1222no—
454C2×C4—1222no—
455C2×C4—1221no—
456C2×C4—1221no—
457C2×C4—1221no—
458C2×C4—1221no—
459C2×C4—1221no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2442no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4C22⋊C4×D6111yescharacteristic
466C2×C4C22⋊C4×D6111yescharacteristic
467C2×C4C22⋊C4×D6111yescharacteristic
468C2×C4C22⋊C4×D6111yescharacteristic
469C2×C4#1451978312yes—
470C2×C4#1451978512yes—
471C2×C4C2×D4×D6111yescharacteristic
472C2×C4C2×D4×D6111yescharacteristic
473C8—42no—
474C8—42no—
475C8—2442no—
476C8—2442no—
477C22—21no—
478C22—21no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—82no—
483C22—82no—
484C22—4881no—
485C22—4881no—
486C22—2441no—
487C22—2441no—
488C22—2441no—
489C22C2×C22⋊C4×D6111yescharacteristic
490C22C2×#1451978311yescharacteristic
491C22C2×#1451978511yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—42no—
497C4—1224no—
498C4—1224no—
499C4—2441no—
500C4—2441no—
501C4—2441no—
502C4#1683972712yes—
503C4#1683972712yes—
504C2—21no—
505C2—21no—
506C2—82no—
507C2—4881no—
508C2C2×#1683972711yescharacteristic
509C2#2145428111yescharacteristic
510C2#2145429511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1460C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519748C4
cosocleC22×C4×C61C24
hypercenter#1460D61
hypocenterC61#17390
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
2. #21454902C2semidirect
3. #21454899C2semidirect
4. #21444185C2semidirect
5. #21444169C2semidirect
6. #21444140C2semidirect
7. #21444131C2semidirect
8. #21444131C2semidirect
9. C61×#1460C2semidirect
255. #1460D61semidirect
296. C61#17390semidirect

Maximal subgroups

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

subgroupindex
C2×#168403772
#214549022
#214548992
#214441852
#214441692
#214441402
#214441312
C61×#14602
#1739061