Finite Group Databasebeta

← #29737008

Subgroups — #29737008

Order 15616.

1100 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840128C211yesmaximal, characteristic
2C2×#16839962C211yesmaximal, characteristic
3#21450818C211yesmaximal, characteristic
4#21446962C212yesmaximal, complemented
5#21446952C212yesmaximal, complemented
6#21446777C211yesmaximal, characteristic, complemented
7#21446760C211yesmaximal, characteristic, complemented
8#21446174C212yesmaximal, complemented
9#21446174C212yesmaximal, complemented
10#21445649C211yesmaximal, characteristic, complemented
11C2×C61×#557C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×#14519766—21no—
15#16840128—24no—
16#16839962—24no—
17C2×C4⋊#92418C2211yescharacteristic
18C2×#14519701C2212yes—
19C2×#14519697C2212yes—
20C2×#14519697C2212yes—
21C2×#14519697C2212yes—
22C2×#14519688C2211yescharacteristic
23#16839679—21no—
24#16839679—21no—
25#16839664C2212yes—
26#16839664C2212yes—
27#16839663—21no—
28C4×C4⋊Dic61C2211yescharacteristic
29#16839524C2212yes—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839517C2212yes—
33#16839517C2212yes—
34C61×#557—24no—
35C61×#557—24no—
36C2×C4⋊Q8×C61—21no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#125×C61—21no—
39C2×#125×C61—21no—
40C2×#125×C61C2211yescharacteristic
41C2×#125×C61C2211yescharacteristic
42C2×C4×Q8×C61—21no—
43C2×C4×Q8×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic
45C2×C4×Q8×C61C2211yescharacteristic
46C2×C4×Q8×C61C2211yescharacteristic
47C22×D244—21no—
48C22×D244—21no—
49C22×D244C2312yes—
50C2×#11758201—24no—
51C2×#11758201—24no—
52C2×#11758201—24no—
53C22×#92418C2312yes—
54C22×#92418C2312yes—
55#14519774—22no—
56#14519774—22no—
57#14519766—44no—
58C2×C4⋊Dic61C2312yes—
59C2×#11758198—22no—
60C2×#11758198—22no—
61C2×#11758198—22no—
62C2×#11758198—22no—
63C2×C4×Dic61—21no—
64C4⋊#92418—22no—
65#14519701—24no—
66#14519697—24no—
67#14519697—24no—
68#14519697—24no—
69#14519688—22no—
70#14519684—21no—
71C22×Q8×C61—21no—
72C22×Q8×C61D411yescharacteristic
73C22×Q8×C61C2311yescharacteristic
74C4⋊Q8×C61—28no—
75C4⋊Q8×C61—22no—
76C4⋊Q8×C61—22no—
77#125×C61—28no—
78#125×C61—28no—
79#125×C61—22no—
80#125×C61—22no—
81#125×C61—22no—
82#125×C61D414yes—
83C4×Q8×C61—28no—
84C4×Q8×C61—22no—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C4×Q8×C61—22no—
88C4×Q8×C61—22no—
89C4×Q8×C61—22no—
90C4×Q8×C61—22no—
91C4×Q8×C61D414yes—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C42×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111C2×C42×C61C2311yescharacteristic
112C23×D61—22no—
113C2×Q8×C61—24no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—21no—
117C2×Q8×C61—22no—
118C2×Q8×C61—41no—
119C2×Q8×C61—42no—
120C2×Q8×C61—42no—
121C2×Q8×C61C2×D412yes—
122C2×Q8×C61#4114yes—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—22no—
135C2×D244—22no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—22no—
141C2×D244—42no—
142C2×D244—42no—
143C2×D244—42no—
144C2×D244—42no—
145C2×D244—82no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61C2×D412yes—
161C4⋊C4×C61C2×D412yes—
162C4⋊C4×C61C2×D412yes—
163C42×C61—24no—
164C42×C61—24no—
165C42×C61—24no—
166C42×C61—21no—
167C42×C61—21no—
168C42×C61—21no—
169C42×C61—21no—
170C42×C61C2×D412yes—
171C42×C61C2×D412yes—
172#11758201—48no—
173#11758198—44no—
174#11758198—44no—
175#11758198—82no—
176#11758198—82no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4×Dic61—42no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C23×C61C22×D411yescharacteristic
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194Q8×C61—22no—
195Q8×C61—24no—
196Q8×C61—44no—
197Q8×C61—48no—
198Q8×C61#12414yescomplemented
199C2×C4×C61—22no—
200C2×C4×C61—21no—
201C2×C4×C61—24no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—21no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—42no—
221C2×C4×C61#12012yes—
222C2×C4×C61#12012yes—
223C2×C4×C61AGL(1,ℤ/8)12yes—
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C2×#4112yes—
227C2×C4×C612−1+412yes—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231D244—42no—
232D244—42no—
233D244—42no—
234D244—42no—
235D244—84no—
236D244—84no—
237#92418—42no—
238#92418—42no—
239#25944—611nomaximal, Sylow
240C22×C61C2×#12011yescharacteristic
241C22×C61#54812yes—
242C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
243C22×C61#57511yescharacteristic
244C22×C61#57712yes—
245C2×D61—82no—
246C2×D61—82no—
247C2×D61—82no—
248C2×D61—82no—
249C2×D61—82no—
250C2×D61—82no—
251C2×D61—82no—
252C4×C61—24no—
253C4×C61—22no—
254C4×C61—22no—
255C4×C61—22no—
256C4×C61—41no—
257C4×C61—42no—
258C4×C61—42no—
259C4×C61—41no—
260C4×C61—41no—
261C4×C61—42no—
262C4×C61C2×#12411yescharacteristic
263C4×C61#53611yescharacteristic
264C4×C61#54812yes—
265Dic61—82no—
266C2×#557D6111yescharacteristic, complemented
267C2×#490—611no—
268C2×#460—611no—
269#2548—611no—
270#2025—612no—
271#2023—612no—
272#1989—611no—
273#1985—611no—
274#1880—612no—
275#1880—612no—
276#1763—611no—
277C2×C61#243611yescharacteristic
278C2×C61#300611yescharacteristic
279C2×C61#301011yescharacteristic
280C2×C61#336812yes—
281C2×C61#337112yes—
282D61—162no—
283C22×SD16—612no—
284#557—24no—
285#557—24no—
286C2×C4⋊Q8—21no—
287C2×C4⋊Q8C2×D6111yescharacteristic
288C2×#127—611no—
289C2×#125—21no—
290C2×#125—21no—
291C2×#125C2×D6111yescharacteristic
292C2×#125C2×D6111yescharacteristic
293C2×#124—1221no—
294C2×C4×Q8—21no—
295C2×C4×Q8C2×D6111yescharacteristic
296C2×C4×Q8C2×D6111yescharacteristic
297C2×C4×Q8C2×D6111yescharacteristic
298C2×C4×Q8C2×D6111yescharacteristic
299#490—1224no—
300#460—1224no—
301C2×C4⋊C8—611no—
302C2×#103—612no—
303C2×#102—612no—
304C2×#102—612no—
305C2×#102—612no—
306C2×#97—611no—
307#394—1221no—
308#394—1221no—
309#390—612no—
310#390—612no—
311#390—1221no—
312C4×C4⋊C4—611no—
313#340—612no—
314#340—612no—
315#340—612no—
316#336—612no—
317#336—612no—
318C61#2594411yescharacteristic, complemented, Sylow
319C22×Q8—21no—
320C22×Q8C22⋊D6111yescharacteristic
321C22×Q8C22×D6111yescharacteristic
322C22×D4—612no—
323C22×D4—1221no—
324C22×D4—1221no—
325C2×SD16—1224no—
326C2×SD16—1224no—
327C2×SD16—1224no—
328C22×C8—612no—
329C22×C8—612no—
330C4⋊Q8—28no—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333#127—1222no—
334#127—1222no—
335#125—28no—
336#125—28no—
337#125—22no—
338#125—22no—
339#125—22no—
340#125C22⋊D6114yes—
341#124—2444no—
342C4×Q8—28no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—22no—
350C4×Q8C22⋊D6114yes—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—612no—
360C2×C4⋊C4C22⋊D6111yescharacteristic
361C2×C4⋊C4C22⋊D6111yescharacteristic
362C2×C4⋊C4C22⋊D6111yescharacteristic
363C2×C4⋊C4C22×D6111yescharacteristic
364C2×C4⋊C4C22×D6111yescharacteristic
365C2×C4⋊C4C22×D6111yescharacteristic
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1222no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—1221no—
374C2×C42C22×D6111yescharacteristic
375C2×C42C22×D6111yescharacteristic
376C2×C42C22×D6111yescharacteristic
377C4⋊C8—1222no—
378#103—1224no—
379#102—1224no—
380#102—1224no—
381#102—1224no—
382#97—1222no—
383#95—1221no—
384C24—1222no—
385C2×Q8—24no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—41no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8#1175822614yes—
394C2×Q8C2×C22⋊D6112yes—
395C2×D4—1222no—
396C2×D4—1222no—
397C2×D4—1222no—
398C2×D4—1222no—
399C2×D4—1222no—
400C2×D4—1222no—
401C2×D4—4882no—
402C2×D4—2442no—
403C2×D4—2442no—
404C2×D4—2442no—
405C2×D4—2442no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—1222no—
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6111yescharacteristic
413C22×C4C2×C22⋊D6111yescharacteristic
414C22×C4C2×C22⋊D6111yescharacteristic
415C22×C4C2×C22⋊D6111yescharacteristic
416C22×C4C2×C22⋊D6111yescharacteristic
417C22×C4C23×D6111yescharacteristic
418SD16—2448no—
419C2×C8—1222no—
420C2×C8—1222no—
421C2×C8—1222no—
422C2×C8—1222no—
423C2×C8—1222no—
424C2×C8—1222no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—1222no—
440C4⋊C4—1222no—
441C4⋊C4C2×C22⋊D6112yes—
442C4⋊C4C2×C22⋊D6112yes—
443C4⋊C4C2×C22⋊D6112yes—
444C22⋊C4—4882no—
445C22⋊C4—4882no—
446C22⋊C4—2444no—
447C22⋊C4—2444no—
448C42—24no—
449C42—24no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—24no—
455C42—2442no—
456C42C2×C22⋊D6112yes—
457C42C2×C22⋊D6112yes—
458C23—2442no—
459C23—2442no—
460C23—2442no—
461C23—2442no—
462C23—2442no—
463C23—2442no—
464C23—2442no—
465C23C22×C22⋊D6111yescharacteristic
466Q8—22no—
467Q8—24no—
468Q8—44no—
469Q8—48no—
470Q8#1451976614yescomplemented
471D4—4884no—
472D4—4884no—
473D4—2442no—
474D4—2442no—
475D4—2442no—
476D4—2442no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4#1451974612yes—
503C2×C4#1451974612yes—
504C2×C4#1451980212yes—
505C2×C4C22×C22⋊D6111yescharacteristic
506C2×C4C22×C22⋊D6111yescharacteristic
507C2×C4C2×#1175822612yes—
508C2×C4#1451982512yes—
509C8—2442no—
510C8—2442no—
511C22—4882no—
512C22—4882no—
513C22—4882no—
514C22—4882no—
515C22—4882no—
516C22—4882no—
517C22—4882no—
518C22C2×#1451974611yescharacteristic
519C22#1684042312yes—
520C22C2×#1451980211yescharacteristic
521C22#1684056011yescharacteristic
522C22#1684057012yes—
523C4—24no—
524C4—22no—
525C4—22no—
526C4—22no—
527C4—41no—
528C4—41no—
529C4—42no—
530C4—41no—
531C4—42no—
532C4—42no—
533C4—4882no—
534C4C2×#1451976611yescharacteristic
535C4#1684033611yescharacteristic
536C4#1684042312yes—
537C2—9762no—
538C2#2144947311yescharacteristic
539C2#2145507111yescharacteristic
540C2#2145510511yescharacteristic
541C2#2145854812yes—
542C2#2145858712yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#557C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenterC2×#557D61
hypocenterC61#25944
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #21446962C2semidirect
5. #21446952C2semidirect
6. #21446777C2semidirect
7. #21446760C2semidirect
8. #21446174C2semidirect
9. #21446174C2semidirect
10. #21445649C2semidirect
11. C2×C61×#557C2semidirect
198. Q8×C61#124semidirect
266. C2×#557D61semidirect
318. C61#25944semidirect
470. Q8#14519766semidirect

Maximal subgroups

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

subgroupindex
C2×#168401282
C2×#168399622
#214508182
#214469622
#214469522
#214467772
#214467602
#214461742
#214456492
C2×C61×#5572
#2594461