Finite Group Databasebeta

← #29714072

Subgroups — #29714072

Order 15616.

1184 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840469C211yesmaximal, characteristic, complemented
2C2×#16840042C211yesmaximal, characteristic, complemented
3C2×#16839862C211yesmaximal, characteristic, complemented
4#21447022C212yesmaximal, complemented
5#21446822C212yesmaximal, complemented
6#21446791C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446137C212yesmaximal, complemented
9#21446006C212yesmaximal, complemented
10#21445831C211yesmaximal, characteristic, complemented
11C61×#2535C211yesmaximal, characteristic
12C22×#11758200C2211yescharacteristic
13#16840469—24no—
14#16840469—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519768—21no—
17C2×#14519756C2212yes—
18C2×#14519754—22no—
19C2×C4×Dic122C2211yescharacteristic
20C2×D4×Dic61—21no—
21C22×C4⋊Dic61C2212yes—
22#16840042—24no—
23#16839862—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519710C2212yes—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519700C2212yes—
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2212yes—
31C2×C22⋊#92418C2212yes—
32C2×C22⋊#92418C2212yes—
33C2×#14519688C2211yescharacteristic
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37C2×#121×C61C2212yes—
38C2×C4×D4×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C61×#397—21no—
41C61×#392—21no—
42C61×#386—22no—
43C61×#382C2211yescharacteristic
44C61×#380C2212yes—
45C22×Dic122—21no—
46C22×Dic122C2311yescharacteristic
47C23×Dic61—22no—
48C2×#11758200—22no—
49C2×#11758200—22no—
50C2×#11758200—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2312yes—
53C22×#92418C2311yescharacteristic
54#14519777—22no—
55#14519777D414yescomplemented
56#14519768—28no—
57#14519756—24no—
58#14519756—24no—
59#14519754—216no—
60C4×Dic122—22no—
61C4×Dic122D414yescomplemented
62D4×Dic61—28no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—24no—
66C2×C4⋊Dic61—24no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×#11758196—21no—
70C2×#11758196—22no—
71C2×C22⋊Dic61—22no—
72C2×C4×Dic61—21no—
73#14519710—24no—
74#14519710—22no—
75C4⋊#92418—22no—
76#14519700—22no—
77#14519700—42no—
78#14519698—24no—
79#14519698—22no—
80C22⋊#92418—24no—
81C22⋊#92418—42no—
82#14519688—22no—
83C22×D4×C61—22no—
84C22×D4×C61D411yescharacteristic
85C22×D4×C61C2311yescharacteristic
86C23×C4×C61—21no—
87C23×C4×C61C2312yes—
88#121×C61—24no—
89#121×C61—42no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94C2×C4⋊C4×C61D411yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C22⋊C4×C61—22no—
98C2×C22⋊C4×C61—21no—
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61D412yes—
101C2×C42×C61C2311yescharacteristic
102#95×C61—22no—
103#95×C61—21no—
104#95×C61—21no—
105C24×C61—21no—
106C24×C61C2×D412yes—
107C2×D4×C61—24no—
108C2×D4×C61—22no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—44no—
115C2×D4×C61—44no—
116C2×D4×C61—82no—
117C2×D4×C61#4112yes—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61—42no—
127C22×C4×C61—82no—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2411yescharacteristic
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61—21no—
135C22×Dic61—28no—
136C22×Dic61—82no—
137C22×Dic61—82no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—24no—
144C2×Dic122—82no—
145C2×Dic122C2×D412yes—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—44no—
153C22⋊C4×C61—42no—
154C22⋊C4×C61—82no—
155C42×C61C2×D412yes—
156C42×C61C2×D412yes—
157C22⋊Dic61—28no—
158#11758200—28no—
159C4⋊Dic61—28no—
160C4⋊Dic61—24no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61C2×D412yes—
164C4⋊Dic61C2×D414yes—
165#11758196—28no—
166#11758196—24no—
167C4×Dic61—24no—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—21no—
174C2×#92418—22no—
175C2×#92418—21no—
176C2×#92418#4112yes—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61#13714yes—
189C23×C61C22×D411yescharacteristic
190D4×C61—28no—
191D4×C61—22no—
192D4×C61—22no—
193D4×C61—41no—
194D4×C61—41no—
195D4×C61—88no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—84no—
211C2×C4×C61#12012yes—
212C2×C4×C61#12012yes—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61H(5,𝔽2)11yescharacteristic
218C2×C4×C612−1+411yescharacteristic
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—24no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—88no—
228Dic122—22no—
229Dic122—22no—
230Dic122—41no—
231Dic122—41no—
232Dic122—84no—
233#92418—22no—
234#92418—41no—
235#92418—42no—
236#24130—611nomaximal, Sylow
237C22×C61—24no—
238C22×C61—24no—
239C22×C61—24no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—44no—
246C22×C61—82no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61C2×#12011yescharacteristic
250C22×C61#54611yescharacteristic
251C22×C61#54711yescharacteristic
252C22×C61C2×#13712yes—
253C22×C61#57612yes—
254C4×C61—22no—
255C4×C61—22no—
256C4×C61—42no—
257C4×C61—81no—
258C4×C61—82no—
259C4×C61#52511yescharacteristic
260C4×C61#53711yescharacteristic
261C4×C61#54811yescharacteristic
262C4×C61#55011yescharacteristic
263Dic61—42no—
264Dic61—44no—
265Dic61—42no—
266C2×#554—611no—
267C2×#474—611no—
268C2×#440—611no—
269#2535D6111yescharacteristic
270#2036—612no—
271#2000—612no—
272#1993—611no—
273#1955—611no—
274#1874—612no—
275#1852—612no—
276#1805—611no—
277C2×C61—44no—
278C2×C61—82no—
279C2×C61#240711yescharacteristic
280C2×C61#300411yescharacteristic
281C2×C61#300811yescharacteristic
282C2×C61#331412yes—
283C2×C61#334612yes—
284C22×SD16—611no—
285#554—1224no—
286#554—1224no—
287C2×C4⋊Q8—611no—
288C2×#125—1221no—
289C2×C22⋊Q8—612no—
290C2×C22⋊Q8—1222no—
291C2×#121C2×D6112yes—
292C2×C4×Q8—611no—
293C2×C4×D4—1221no—
294C2×C4×D4C2×D6111yescharacteristic
295C2×C4×D4C2×D6111yescharacteristic
296C22×C4⋊C4—612no—
297#474—1224no—
298#440—1224no—
299C2×#107—611no—
300C2×#107—612no—
301C2×C4⋊C8—611no—
302C2×#103—611no—
303C2×#103—612no—
304C2×#102—611no—
305C2×#102—612no—
306C2×C22⋊C8—612no—
307C2×C22⋊C8—612no—
308C2×#97—611no—
309#397—21no—
310#392—21no—
311#386—22no—
312#382C2×D6111yescharacteristic
313#380C2×D6112yes—
314#340—612no—
315#340—612no—
316#340—612no—
317C61#2413011yescharacteristic, complemented, Sylow
318C22×Q8—611no—
319C22×Q8—1221no—
320C22×D4—22no—
321C22×D4C22⋊D6111yescharacteristic
322C22×D4C22×D6111yescharacteristic
323C23×C4—21no—
324C23×C4—1222no—
325C23×C4C22×D6112yes—
326C2×SD16—1222no—
327C2×SD16—1222no—
328C2×SD16—1222no—
329C22×C8—611no—
330C22×C8—611no—
331C22×C8—612no—
332C4⋊Q8—614no—
333C4⋊Q8—1222no—
334#125—1228no—
335C22⋊Q8—12216no—
336C22⋊Q8—1224no—
337C22⋊Q8—1224no—
338#121—24no—
339#121—42no—
340C4×Q8—614no—
341C4×Q8—1222no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—22no—
346C4×D4—1228no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—1224no—
350C2×C4⋊C4—1224no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4C22⋊D6111yescharacteristic
356C2×C4⋊C4C22×D6111yescharacteristic
357C2×C4⋊C4C22×D6112yes—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4C22⋊D6112yes—
363C2×C42—1221no—
364C2×C42C22×D6111yescharacteristic
365#107—1222no—
366#107—1224no—
367C4⋊C8—1222no—
368#103—1222no—
369#103—2442no—
370#102—1222no—
371#102—1224no—
372C22⋊C8—1224no—
373C22⋊C8—2442no—
374#97—1222no—
375#95—22no—
376#95—21no—
377#95—21no—
378C24—21no—
379C24C2×C22⋊D6112yes—
380C2×Q8—612no—
381C2×Q8—1221no—
382C2×Q8—1224no—
383C2×Q8—1221no—
384C2×Q8—1221no—
385C2×Q8—1221no—
386C2×Q8—1221no—
387C2×Q8—4882no—
388C2×D4—24no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—82no—
398C2×D4#1175822412yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—42no—
408C22×C4—82no—
409C22×C4—1228no—
410C22×C4—1221no—
411C22×C4—1221no—
412C22×C4—1222no—
413C22×C4—4882no—
414C22×C4—4882no—
415C22×C4C2×C22⋊D6112yes—
416C22×C4C2×C22⋊D6111yescharacteristic
417C22×C4C2×C22⋊D6111yescharacteristic
418C22×C4C23×D6111yescharacteristic
419SD16—1228no—
420C2×C8—612no—
421C2×C8—1221no—
422C2×C8—1221no—
423C2×C8—1221no—
424C2×C8—1222no—
425C2×C8—1222no—
426C2×C8—1221no—
427C2×C8—1222no—
428C2×C8—1221no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—612no—
435C4⋊C4—614no—
436C4⋊C4—1228no—
437C4⋊C4—1228no—
438C4⋊C4—1221no—
439C4⋊C4—1222no—
440C4⋊C4—1224no—
441C4⋊C4—1224no—
442C22⋊C4—24no—
443C22⋊C4—44no—
444C22⋊C4—42no—
445C22⋊C4—82no—
446C22⋊C4—1228no—
447C42—1224no—
448C42D4×D6112yes—
449C42D4×D6112yes—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—41no—
461C23#1451980714yes—
462C23C22×C22⋊D6111yescharacteristic
463Q8—1222no—
464Q8—1222no—
465Q8—4884no—
466Q8—2441no—
467Q8—2441no—
468D4—28no—
469D4—22no—
470D4—22no—
471D4—41no—
472D4—41no—
473D4—88no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—84no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1224no—
492C2×C4—4888no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4#1451974512yes—
499C2×C4#1451974512yes—
500C2×C4C2×D4×D6111yescharacteristic
501C2×C4C2×D4×D6111yescharacteristic
502C2×C4#41×D6111yescharacteristic
503C2×C4C2×#1175822411yescharacteristic
504C2×C4#1451982211yescharacteristic
505C2×C4#1451982411yescharacteristic
506C8—1222no—
507C8—2441no—
508C8—2442no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—44no—
518C22—82no—
519C22—81no—
520C22—81no—
521C22C2×#1451974511yescharacteristic
522C22#1684040611yescharacteristic
523C22#1684041111yescharacteristic
524C22C2×#1451980712yes—
525C22#1684056512yes—
526C4—22no—
527C4—22no—
528C4—42no—
529C4—82no—
530C4—81no—
531C4—2442no—
532C4—2442no—
533C4—2444no—
534C4#1684027211yescharacteristic
535C4#1684034111yescharacteristic
536C4#1684041811yescharacteristic
537C4#1684043011yescharacteristic
538C2—44no—
539C2—82no—
540C2#2144916011yescharacteristic
541C2#2145505211yescharacteristic
542C2#2145508511yescharacteristic
543C2#2145790012yes—
544C2#2145830412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2535C2
socleC23×C61C22×D4
hypercenter#2535D61
hypocenterC61#24130
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. C2×#16840469C2semidirect
2. C2×#16840042C2semidirect
3. C2×#16839862C2semidirect
4. #21447022C2semidirect
5. #21446822C2semidirect
6. #21446791C2semidirect
7. #21446618C2semidirect
8. #21446137C2semidirect
9. #21446006C2semidirect
10. #21445831C2semidirect
55. #14519777D4semidirect
61. C4×Dic122D4semidirect
317. C61#24130semidirect

Maximal subgroups

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

subgroupindex
C2×#168404692
C2×#168400422
C2×#168398622
#214470222
#214468222
#214467912
#214466182
#214461372
#214460062
#214458312
C61×#25352
#2413061