Finite Group Databasebeta

← #29689726

Subgroups — #29689726

Order 15616.

1106 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840316C211yesmaximal, characteristic
2C2×#16839814C211yesmaximal, characteristic
3#21448225C211yesmaximal, characteristic
4#21446778C212yesmaximal, complemented
5#21446619C212yesmaximal, complemented
6#21446130C212yesmaximal
7#21446128C212yesmaximal
8#21445835C211yesmaximal, characteristic, complemented
9#21445834C211yesmaximal, characteristic, complemented
10#21445831C211yesmaximal, characteristic, complemented
11C61×#1805C211yesmaximal, characteristic, complemented
12#16840316—24no—
13#16840316C418yescomplemented
14C2×#14519776C2211yescharacteristic
15C2×#14519775C2211yescharacteristic
16C2×#14519767—21no—
17C2×#14519757C2211yescharacteristic
18C2×#14519755—21no—
19C2×#14519753C2211yescharacteristic
20C2×C4×Dic122—21no—
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519735C2212yes—
23C2×#14519734—21no—
24C2×C42×D61C2211yescharacteristic
25#16839814—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519702C2212yes—
28C2×#14519701C2212yes—
29C2×#14519700C2212yes—
30C2×#14519689C2212yes—
31C2×C8×Dic61C2212yes—
32#16839631C2212yes—
33#16839628C2211yescharacteristic
34#16839621—21no—
35#16839621—21no—
36#16839613—21no—
37#16839613—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C61×#382C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52#14519776—22no—
53#14519776C2×C414yes—
54#14519775—22no—
55#14519775C2×C414yes—
56#14519767—28no—
57#14519757—22no—
58#14519757C2×C414yes—
59#14519755—28no—
60#14519753—22no—
61#14519753C2×C414yes—
62C4×Dic122—22no—
63C4×Dic122—28no—
64C4×Dic122C2×C414yes—
65#14519735—42no—
66#14519735—42no—
67#14519734—28no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—22no—
77C2×#11758196—22no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—22no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61C2×C414yes—
87C42×D61D414yes—
88C4⋊#92418—22no—
89#14519702—42no—
90#14519701—42no—
91#14519700—42no—
92#14519689—24no—
93C8×Dic61—24no—
94#14519684—21no—
95#14519684—21no—
96C22×Q8×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98C4×Q8×C61—22no—
99C4×Q8×C61C2×C414yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2312yes—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C42×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—22no—
106#103×C61—42no—
107#95×C61—21no—
108#95×C61—21no—
109C23×D61C2×D411yescharacteristic
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61C22×C412yes—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61C2×D411yescharacteristic
126C22×Dic61C2×D411yescharacteristic
127C22×Dic61C2×D411yescharacteristic
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—24no—
133C2×C4×D61—24no—
134C2×C4×D61C22⋊C414yes—
135C2×C4×D61C22⋊C414yes—
136C2×C4×D61C22×C412yes—
137C2×C4×D61C2×D412yes—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—24no—
144C2×Dic122—44no—
145C2×Dic122C22×C412yes—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61C22×C412yes—
154C42×C61—24no—
155C42×C61C22×C412yes—
156C42×C61C2×D412yes—
157#11758198—24no—
158#11758198—24no—
159#11758198—44no—
160C4⋊Dic61—21no—
161C4⋊Dic61—24no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61C22×C412yes—
165#11758196—24no—
166#11758196—24no—
167#11758196—24no—
168#11758196—24no—
169#11758196—24no—
170#11758196—44no—
171#11758196—44no—
172C4×Dic61—24no—
173C4×Dic61—44no—
174C4×Dic61C22⋊C414yes—
175C4×Dic61C22⋊C414yes—
176C4×Dic61C22×C412yes—
177C4×Dic61C2×D412yes—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C23×C61C22×D411yescharacteristic
182C22×D61—21no—
183C22×D61—21no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61C2×C22⋊C412yes—
187Q8×C61—22no—
188Q8×C61—22no—
189Q8×C61—41no—
190Q8×C61—41no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61C2×C22⋊C412yes—
202C2×C4×C61C2×C22⋊C412yes—
203C2×C4×C61C23×C411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61H(5,𝔽2)11yescharacteristic
206C2×C4×C61H(5,𝔽2)11yescharacteristic
207C2×C4×C612−1+411yescharacteristic
208C2×C4×C612−1+411yescharacteristic
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—41no—
214C2×Dic61—41no—
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×Dic61#13514yes—
223C2×Dic61#13714yes—
224C4×D61—22no—
225C4×D61—22no—
226C4×D61—22no—
227C4×D61—22no—
228C4×D61—28no—
229C4×D61—28no—
230Dic122—22no—
231Dic122—22no—
232Dic122—41no—
233Dic122—41no—
234Dic122—48no—
235C8×C61—42no—
236#92418—42no—
237#21864—611nomaximal, Sylow
238C22×C61C22×C22⋊C411yescharacteristic
239C22×C61#53511yescharacteristic
240C22×C61#53611yescharacteristic
241C22×C61C2×#13512yes—
242C22×C61C2×#13712yes—
243C2×D61—22no—
244C2×D61—22no—
245C2×D61—22no—
246C2×D61—42no—
247C2×D61—42no—
248C4×C61—22no—
249C4×C61—22no—
250C4×C61—42no—
251C4×C61—42no—
252C4×C61—82no—
253C4×C61#52011yescharacteristic
254C4×C61#52011yescharacteristic
255C4×C61#53711yescharacteristic
256C4×C61#53711yescharacteristic
257Dic61—22no—
258Dic61—42no—
259Dic61—42no—
260Dic61—82no—
261Dic61#41614yes—
262Dic61#41914yes—
263C2×#533—611no—
264C2×#431—611no—
265#2285—611no—
266#1990—612no—
267#1956—612no—
268#1873—612no—
269#1873—612no—
270#1805—611no—
271#1805—611no—
272#1805—611no—
273#1805D6111yescharacteristic, complemented
274C2×C61#231111yescharacteristic
275C2×C61C2×#41611yescharacteristic
276C2×C61C2×#41911yescharacteristic
277C2×C61#321012yes—
278C2×C61#323112yes—
279D61—42no—
280#533—618no—
281#533—1224no—
282C2×C4⋊Q8—611no—
283C2×C4⋊Q8—611no—
284C2×#125—1221no—
285C2×C22⋊Q8—611no—
286C2×C22⋊Q8—611no—
287C2×C22⋊Q8—1221no—
288C2×C4×Q8—611no—
289C2×C4×Q8—1221no—
290C2×C4×Q8C2×D6111yescharacteristic
291C2×#117—612no—
292C2×#117—1221no—
293C22×C42—611no—
294#431—1224no—
295C2×C4⋊C8—611no—
296C2×C4⋊C8C2×D6111yescharacteristic
297C2×#103—612no—
298C2×#103—612no—
299C2×#103—612no—
300C2×#103C2×D6112yes—
301C2×C22⋊C8—612no—
302C2×C4×C8—612no—
303#384—612no—
304#382—611no—
305#382C2×D6111yescharacteristic
306#380—1221no—
307#380—1221no—
308C4×C4⋊C4—1221no—
309C4×C4⋊C4—1221no—
310#340—612no—
311#340—612no—
312#340—612no—
313#340C2×D6112yes—
314C61#2186411yescharacteristic, complemented, Sylow
315C22×Q8—611no—
316C22×Q8—1221no—
317C22×Q8C22×D6111yescharacteristic
318C23×C4—611no—
319C23×C4—611no—
320C23×C4—611no—
321C22×C8—612no—
322C22×C8C22×D6112yes—
323C4⋊Q8—614no—
324C4⋊Q8—614no—
325C4⋊Q8—1222no—
326C4⋊Q8—1222no—
327#125—1228no—
328C22⋊Q8—614no—
329C22⋊Q8—614no—
330C22⋊Q8—1222no—
331C22⋊Q8—1228no—
332C22⋊Q8—1222no—
333C4×Q8—22no—
334C4×Q8—614no—
335C4×Q8—1228no—
336C4×Q8—1222no—
337C4×Q8C4×D6114yes—
338#117—1228no—
339#117—2442no—
340#117—2442no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—611no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1222no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4C22×D6112yes—
353C2×C4⋊C4C22×D6111yescharacteristic
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1222no—
357C2×C42—21no—
358C2×C42—611no—
359C2×C42—611no—
360C2×C42—611no—
361C2×C42—614no—
362C2×C42—614no—
363C2×C42—1221no—
364C2×C42—1222no—
365C2×C42C22×D6111yescharacteristic
366C4⋊C8—22no—
367C4⋊C8—1222no—
368#103—42no—
369#103—2442no—
370#103—2442no—
371#103—2442no—
372C22⋊C8—1224no—
373C4×C8—1224no—
374#95—21no—
375#95—21no—
376#95—1221no—
377#95—1221no—
378C24—611no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—612no—
385C2×Q8—1224no—
386C2×Q8—1221no—
387C2×Q8—1221no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×Q8—2444no—
392C2×Q8C2×C4×D6112yes—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—612no—
400C22×C4—612no—
401C22×C4—614no—
402C22×C4—614no—
403C22×C4—1224no—
404C22×C4—1224no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1221no—
409C22×C4—1221no—
410C22×C4—1221no—
411C22×C4—1222no—
412C22×C4D4×D6111yescharacteristic
413C22×C4D4×D6111yescharacteristic
414C22×C4C23×D6111yescharacteristic
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—1222no—
421C4⋊C4—21no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—612no—
426C4⋊C4—1224no—
427C4⋊C4—1224no—
428C4⋊C4—1224no—
429C4⋊C4—1224no—
430C4⋊C4—1224no—
431C4⋊C4—1221no—
432C4⋊C4—1224no—
433C4⋊C4—1222no—
434C4⋊C4—1222no—
435C4⋊C4—2444no—
436C4⋊C4—2444no—
437C4⋊C4C2×C4×D6112yes—
438C22⋊C4—1224no—
439C22⋊C4—1224no—
440C22⋊C4—2444no—
441C42—24no—
442C42—612no—
443C42—612no—
444C42—614no—
445C42—614no—
446C42—1224no—
447C42—2444no—
448C42C2×C4×D6112yes—
449C42D4×D6112yes—
450C23—612no—
451C23—1222no—
452C23—1222no—
453C23—1221no—
454C23—1221no—
455C23C2×D4×D6111yescharacteristic
456Q8—22no—
457Q8—22no—
458Q8—41no—
459Q8—41no—
460Q8—1222no—
461Q8—1222no—
462Q8—2448no—
463Q8—2441no—
464Q8—2441no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—612no—
476C2×C4—612no—
477C2×C4—614no—
478C2×C4—614no—
479C2×C4—612no—
480C2×C4—1228no—
481C2×C4—1228no—
482C2×C4—1221no—
483C2×C4—1221no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2442no—
493C2×C4—2442no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4C22⋊C4×D6112yes—
497C2×C4C22⋊C4×D6112yes—
498C2×C4C22×C4×D6111yescharacteristic
499C2×C4C2×D4×D6111yescharacteristic
500C2×C4#1451982211yescharacteristic
501C2×C4#1451982311yescharacteristic
502C2×C4#1451982411yescharacteristic
503C2×C4#1451982511yescharacteristic
504C8—42no—
505C8—2442no—
506C22—1222no—
507C22—1222no—
508C22—1222no—
509C22—2442no—
510C22—2442no—
511C22C2×C22⋊C4×D6111yescharacteristic
512C22#1684032811yescharacteristic
513C22#1684033511yescharacteristic
514C22#135×D6112yes—
515C22#137×D6112yes—
516C4—22no—
517C4—22no—
518C4—42no—
519C4—42no—
520C4—82no—
521C4—614no—
522C4—614no—
523C4—1222no—
524C4—4882no—
525C4—2442no—
526C4—2442no—
527C4#1684024211yescharacteristic
528C4#1684024311yescharacteristic
529C4#1684034111yescharacteristic
530C4#1684034211yescharacteristic
531C2—2442no—
532C2#2144840811yescharacteristic
533C2#416×D6111yescharacteristic
534C2#419×D6111yescharacteristic
535C2#2145677412yes—
536C2#2145697712yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1805C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1805D61
hypocenterC61#21864
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
4. #21446778C2semidirect
5. #21446619C2semidirect
8. #21445835C2semidirect
9. #21445834C2semidirect
10. #21445831C2semidirect
11. C61×#1805C2semidirect
13. #16840316C4semidirect
273. #1805D61semidirect
314. C61#21864semidirect

Maximal subgroups

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

subgroupindex
C2×#168403162
C2×#168398142
#214482252
#214467782
#214466192
#214461302
#214461282
#214458352
#214458342
#214458312
C61×#18052
#2186461