Finite Group Databasebeta

← #29689710

Subgroups — #29689710

Order 15616.

1138 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840298C211yesmaximal, characteristic
2C2×#16839812C211yesmaximal, characteristic, complemented
3#21448222C211yesmaximal, characteristic, complemented
4#21446782C212yesmaximal, complemented
5#21446627C212yesmaximal, complemented
6#21446141C211yesmaximal, characteristic
7#21446139C211yesmaximal, characteristic
8#21446132C211yesmaximal, characteristic, complemented
9#21445822C212yesmaximal, complemented
10#21445818C212yesmaximal, complemented
11C61×#1873C211yesmaximal, characteristic, complemented
12#16840298—24no—
13#16840298C418yescomplemented
14C2×#14519772—21no—
15C2×#14519768C2211yescharacteristic
16C2×#14519766C2211yescharacteristic
17C2×#14519761—21no—
18C2×#14519751C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21C2×#14519737—21no—
22C2×#14519736C2211yescharacteristic
23C2×#14519735—21no—
24C22×C4×Dic61C2211yescharacteristic
25#16839812—24no—
26C2×#14519706C2212yes—
27C2×#14519701C2212yes—
28C2×#14519699C2212yes—
29C2×#14519697C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31C2×C8×Dic61C2212yes—
32#16839631C2212yes—
33#16839627C2212yes—
34#16839620—21no—
35#16839620—21no—
36C4×#11758196—21no—
37C4×#11758196—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C22⋊Q8×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#340C2212yes—
46C22×D244C2311yescharacteristic
47C22×C22⋊D61—21no—
48C23×Dic61D411yescharacteristic
49C23×Dic61D411yescharacteristic
50C22×C4×D61—21no—
51C22×#92418C2312yes—
52#14519772—28no—
53#14519768—22no—
54#14519768C2×C414yes—
55#14519766—22no—
56#14519766C2×C414yes—
57#14519761—28no—
58#14519751—22no—
59#14519751C2×C414yes—
60Q8×Dic61—22no—
61Q8×Dic61C2×C414yes—
62#14519739—22no—
63#14519739C2×C414yes—
64#14519737—28no—
65#14519736—42no—
66#14519735—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—22no—
73C2×#11758196—22no—
74C2×C22⋊Dic61—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61C2×C414yes—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D414yes—
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519706—24no—
88#14519701—42no—
89#14519699—42no—
90#14519697—42no—
91C22⋊#92418—22no—
92C8×Dic61—24no—
93#14519684—22no—
94#14519684—22no—
95C22×Q8×C61C2311yescharacteristic
96C23×C4×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61C2×C414yes—
100#117×C61—42no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—21no—
107#103×C61—42no—
108C22⋊C8×C61—22no—
109C24×C61C2×D411yescharacteristic
110C23×D61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61C22×C412yes—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61C22×C412yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—24no—
125C2×C22⋊D61—44no—
126C22×Dic61—24no—
127C22×Dic61—21no—
128C22×Dic61—22no—
129C22×Dic61—42no—
130C22×Dic61C22⋊C414yes—
131C22×Dic61C22⋊C414yes—
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—21no—
139C2×D244C22×C412yes—
140C2×C4×D61—24no—
141C2×C4×D61—44no—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61C22×C412yes—
150C22⋊C4×C61—24no—
151C22⋊C4×C61—42no—
152C42×C61—42no—
153C22⋊Dic61—42no—
154#11758198—24no—
155#11758198—24no—
156#11758198—24no—
157#11758198—24no—
158C4⋊Dic61—21no—
159C4⋊Dic61—24no—
160C4⋊Dic61—24no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61C22×C412yes—
164#11758196—24no—
165#11758196—44no—
166#11758196—44no—
167C4×Dic61—24no—
168C4×Dic61—42no—
169C4×Dic61C22⋊C414yes—
170C4×Dic61C22⋊C414yes—
171C4×Dic61C22×C412yes—
172C4×Dic61C22×C412yes—
173C4×Dic61C2×D412yes—
174C4×Dic61C2×D412yes—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61C2×C22⋊C412yes—
183C23×C61C22×D411yescharacteristic
184C22×D61—22no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—42no—
188C22×D61—42no—
189Q8×C61—24no—
190Q8×C61—42no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—24no—
194C2×C4×C61—24no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61C2×C22⋊C412yes—
204C2×C4×C61C23×C411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61H(5,𝔽2)12yes—
207C2×C4×C612−1+412yes—
208C22⋊D61—48no—
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—28no—
213C2×Dic61—22no—
214C2×Dic61—44no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61C2×C22⋊C412yes—
220C2×Dic61C2×C22⋊C412yes—
221C2×Dic61C2×C22⋊C412yes—
222C2×Dic61C2×C22⋊C412yes—
223C2×Dic61#13514yes—
224D244—24no—
225D244—42no—
226C4×D61—48no—
227C8×C61—42no—
228#92418—42no—
229#21862—611nomaximal, Sylow
230C22×C61—22no—
231C22×C61—22no—
232C22×C61—22no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61C22×C22⋊C411yescharacteristic
236C22×C61#53511yescharacteristic
237C22×C61#53611yescharacteristic
238C22×C61C2×#13512yes—
239C22×C61#57512yes—
240C2×D61—42no—
241C2×D61—42no—
242C2×D61—42no—
243C2×D61—82no—
244C2×D61—82no—
245C4×C61—22no—
246C4×C61—42no—
247C4×C61—42no—
248C4×C61—82no—
249C4×C61#52012yes—
250C4×C61#53712yes—
251Dic61—22no—
252Dic61—22no—
253Dic61—22no—
254Dic61—42no—
255Dic61—82no—
256Dic61#41614yes—
257C2×#529—611no—
258C2×#431—611no—
259#2285—611no—
260#1991—612no—
261#1958—612no—
262#1875—611no—
263#1875—611no—
264#1873—611no—
265#1873D6111yescharacteristic, complemented
266#1803—612no—
267#1802—612no—
268C2×C61—42no—
269C2×C61#231111yescharacteristic
270C2×C61C2×#41611yescharacteristic
271C2×C61#289611yescharacteristic
272C2×C61#321212yes—
273C2×C61#323312yes—
274D61—82no—
275#529—618no—
276#529—1224no—
277C2×#126—1221no—
278C2×#125—611no—
279C2×#124—611no—
280C2×#123—1221no—
281C2×C22⋊Q8C2×D6111yescharacteristic
282C2×#121—611no—
283C2×C4×Q8—611no—
284C2×C4×D4—611no—
285C2×C4×D4—1221no—
286C2×#117—611no—
287C2×#117—1221no—
288C2×#117C2×D6111yescharacteristic
289C22×C42—611no—
290#431—1224no—
291C2×C4⋊C8—612no—
292C2×#103—612no—
293C2×#103C2×D6112yes—
294C2×#102—612no—
295C2×#102—612no—
296C2×C22⋊C8—611no—
297C2×C22⋊C8C2×D6111yescharacteristic
298C2×C4×C8—612no—
299#384—612no—
300#382—612no—
301#380—1221no—
302#380—1221no—
303C4×C4⋊C4—1221no—
304C4×C4⋊C4—1221no—
305#340—612no—
306#340—612no—
307#340—612no—
308#340C2×D6112yes—
309C61#2186211yescharacteristic, complemented, Sylow
310C22×Q8C22×D6111yescharacteristic
311C22×D4—611no—
312C22×D4—1221no—
313C23×C4—611no—
314C23×C4—611no—
315C23×C4—1221no—
316C23×C4C22×D6111yescharacteristic
317C22×C8—612no—
318C22×C8C22×D6112yes—
319#126—1228no—
320#125—614no—
321#125—1222no—
322#124—614no—
323#124—1222no—
324#123—1228no—
325C22⋊Q8—22no—
326C22⋊Q8C4×D6114yes—
327#121—614no—
328#121—1222no—
329C4×Q8—614no—
330C4×Q8—1222no—
331C4×D4—614no—
332C4×D4—1222no—
333C4×D4—1228no—
334#117—42no—
335#117—1228no—
336#117—2442no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—612no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—1222no—
345C2×C4⋊C4C22×D6112yes—
346C2×C4⋊C4C22×D6111yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1221no—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C42—21no—
355C2×C42—614no—
356C2×C42—611no—
357C2×C42—611no—
358C2×C42—614no—
359C2×C42—611no—
360C2×C42—611no—
361C2×C42—1221no—
362C2×C42—1221no—
363C4⋊C8—1224no—
364#103—42no—
365#103—2442no—
366#102—2442no—
367#102—2442no—
368C22⋊C8—22no—
369C22⋊C8—1222no—
370C4×C8—1224no—
371#95—1222no—
372#95—1222no—
373C24—1221no—
374C24D4×D6111yescharacteristic
375C2×Q8—21no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8C2×C4×D6112yes—
379C2×D4—612no—
380C2×D4—1224no—
381C2×D4—1221no—
382C2×D4—1222no—
383C2×D4—1222no—
384C2×D4—2444no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—611no—
391C22×C4—611no—
392C22×C4—611no—
393C22×C4—611no—
394C22×C4—614no—
395C22×C4—614no—
396C22×C4—1224no—
397C22×C4—1224no—
398C22×C4—1221no—
399C22×C4—1222no—
400C22×C4—2444no—
401C22×C4—2442no—
402C22×C4C2×C4×D6112yes—
403C22×C4D4×D6112yes—
404C22×C4D4×D6111yescharacteristic
405C22×C4C23×D6111yescharacteristic
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—1222no—
410C2×C8—1222no—
411C2×C8—1222no—
412C4⋊C4—21no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—612no—
417C4⋊C4—1224no—
418C4⋊C4—1224no—
419C4⋊C4—1224no—
420C4⋊C4—1221no—
421C4⋊C4—1222no—
422C4⋊C4—1222no—
423C4⋊C4—2444no—
424C4⋊C4—2444no—
425C4⋊C4C2×C4×D6112yes—
426C22⋊C4—24no—
427C22⋊C4—42no—
428C22⋊C4—1224no—
429C22⋊C4—1224no—
430C22⋊C4—1224no—
431C22⋊C4—1224no—
432C22⋊C4—2442no—
433C42—42no—
434C42—612no—
435C42—612no—
436C42—612no—
437C42—612no—
438C42—614no—
439C42—614no—
440C42—1224no—
441C42—2442no—
442C23—22no—
443C23—22no—
444C23—21no—
445C23—21no—
446C23—1222no—
447C23—2442no—
448C23—2442no—
449C23—2441no—
450C23—2441no—
451C23C22⋊C4×D6112yes—
452C23C2×D4×D6111yescharacteristic
453Q8—24no—
454Q8—42no—
455D4—1224no—
456D4—2448no—
457D4—2442no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—612no—
471C2×C4—614no—
472C2×C4—612no—
473C2×C4—612no—
474C2×C4—612no—
475C2×C4—1228no—
476C2×C4—1222no—
477C2×C4—1222no—
478C2×C4—1222no—
479C2×C4—1222no—
480C2×C4—2448no—
481C2×C4—2444no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4C22⋊C4×D6112yes—
487C2×C4C22×C4×D6111yescharacteristic
488C2×C4C2×D4×D6111yescharacteristic
489C2×C4#1451982212yes—
490C2×C4#1451982412yes—
491C8—42no—
492C8—2442no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—42no—
497C22—42no—
498C22—4882no—
499C22—4882no—
500C22—2442no—
501C22—2442no—
502C22—2442no—
503C22C2×C22⋊C4×D6111yescharacteristic
504C22#1684032711yescharacteristic
505C22#1684033411yescharacteristic
506C22#135×D6112yes—
507C22#1684056112yes—
508C4—22no—
509C4—42no—
510C4—42no—
511C4—82no—
512C4—614no—
513C4—1222no—
514C4—1222no—
515C4—1222no—
516C4—4882no—
517C4—2442no—
518C4#1684024112yes—
519C4#1684033912yes—
520C2—42no—
521C2—4882no—
522C2#2144840611yescharacteristic
523C2#416×D6111yescharacteristic
524C2#2145433711yescharacteristic
525C2#2145679812yes—
526C2#2145700112yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1873C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
cosocleC22×C4×C61C24
hypercenter#1873D61
hypocenterC61#21862
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
2. C2×#16839812C2semidirect
3. #21448222C2semidirect
4. #21446782C2semidirect
5. #21446627C2semidirect
8. #21446132C2semidirect
9. #21445822C2semidirect
10. #21445818C2semidirect
11. C61×#1873C2semidirect
13. #16840298C4semidirect
265. #1873D61semidirect
309. C61#21862semidirect

Maximal subgroups

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

subgroupindex
C2×#168402982
C2×#168398122
#214482222
#214467822
#214466272
#214461412
#214461392
#214461322
#214458222
#214458182
C61×#18732
#2186261