Finite Group Databasebeta

← #29733330

Subgroups — #29733330

Order 15616.

1162 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840482C211yesmaximal, characteristic, complemented
2C2×#16840121C211yesmaximal, characteristic, complemented
3C2×#16839981C211yesmaximal, characteristic, complemented
4#21446827C212yesmaximal, complemented
5#21446822C212yesmaximal, complemented
6#21446767C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446142C212yesmaximal, complemented
9#21446132C212yesmaximal, complemented
10#21445652C211yesmaximal, characteristic, complemented
11C61×#2540C211yesmaximal, characteristic
12C22×#11758202C2211yescharacteristic
13C22×#11758200C2211yescharacteristic
14#16840482—24no—
15#16840482—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519770—22no—
18C2×#14519764—21no—
19C2×#14519756C2212yes—
20C2×C4×Dic122C2211yescharacteristic
21C2×#14519742—21no—
22C2×#14519736C2212yes—
23#16840121—24no—
24#16839981—24no—
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519701C2212yes—
27C2×#14519701C2211yescharacteristic
28C2×#14519700C2212yes—
29C2×#14519698C2212yes—
30C2×#14519698C2211yescharacteristic
31C2×C22⋊#92418C2212yes—
32C2×C22⋊#92418C2212yes—
33C2×#14519688C2211yescharacteristic
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37C2×#124×C61C2211yescharacteristic
38C2×C22⋊Q8×C61C2212yes—
39C2×#121×C61C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C61×#399—21no—
42C61×#392—21no—
43C61×#390—22no—
44C4×C4⋊C4×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122C2311yescharacteristic
47C2×#11758202—22no—
48C2×#11758202—42no—
49C2×#11758200—22no—
50C2×#11758200—42no—
51C22×#92418C2312yes—
52C22×#92418C2311yescharacteristic
53C22×#92418C2311yescharacteristic
54#14519777—22no—
55#14519777D414yescomplemented
56#14519770—216no—
57#14519764—28no—
58#14519756—24no—
59#14519756—24no—
60C4×Dic122—22no—
61C4×Dic122D414yescomplemented
62#14519742—28no—
63#14519736—24no—
64#14519736—24no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×#11758196—22no—
69C2×#11758196—21no—
70C2×C22⋊Dic61—22no—
71C2×C22⋊Dic61—22no—
72C2×C4×Dic61—22no—
73C2×C4×Dic61—21no—
74C4⋊#92418—22no—
75#14519701—22no—
76#14519701—24no—
77#14519700—24no—
78#14519698—22no—
79#14519698—24no—
80C22⋊#92418—42no—
81C22⋊#92418—42no—
82#14519688—22no—
83C22×Q8×C61C2311yescharacteristic
84C22×D4×C61—22no—
85C22×D4×C61D411yescharacteristic
86C22×D4×C61C2311yescharacteristic
87C23×C4×C61C2312yes—
88#124×C61—22no—
89#124×C61—22no—
90C22⋊Q8×C61—24no—
91C22⋊Q8×C61—42no—
92#121×C61—24no—
93#121×C61—42no—
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61C2312yes—
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61—21no—
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61—21no—
103C2×C22⋊C4×C61D412yes—
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106#95×C61—21no—
107C24×C61—21no—
108C24×C61C2×D412yes—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—42no—
119C2×D4×C61—44no—
120C2×D4×C61—44no—
121C2×D4×C61—82no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—22no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—24no—
140C2×Dic122—21no—
141C2×Dic122—44no—
142C2×Dic122C2×D412yes—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—44no—
146C4⋊C4×C61#4114yes—
147C22⋊C4×C61—24no—
148C22⋊C4×C61—44no—
149C22⋊C4×C61—44no—
150C22⋊C4×C61—42no—
151C22⋊C4×C61—42no—
152C42×C61—81no—
153C42×C61C2×D412yes—
154C42×C61C2×D412yes—
155C22⋊Dic61—28no—
156C22⋊Dic61—28no—
157#11758202—44no—
158#11758200—44no—
159C4⋊Dic61—24no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61C2×D412yes—
163C4⋊Dic61C2×D414yes—
164#11758196—28no—
165#11758196—24no—
166C4×Dic61—28no—
167C4×Dic61—24no—
168C2×#92418—22no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—41no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61C22×D411yescharacteristic
188Q8×C61—41no—
189Q8×C61—41no—
190Q8×C61—42no—
191D4×C61—44no—
192D4×C61—41no—
193D4×C61—41no—
194D4×C61—42no—
195D4×C61—88no—
196C2×C4×C61—22no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—22no—
200C2×C4×C61—24no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61#12012yes—
212C2×C4×C61#12012yes—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4112yes—
216C2×C4×C61H(5,𝔽2)12yes—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—24no—
220C2×Dic61—42no—
221C2×Dic61—44no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224Dic122—22no—
225Dic122—22no—
226Dic122—48no—
227Dic122—42no—
228#92418—41no—
229#92418—41no—
230#92418—42no—
231#25580—611nomaximal, Sylow
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—81no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61—82no—
243C22×C61—82no—
244C22×C61C2×#12011yescharacteristic
245C22×C61#54612yes—
246C22×C61#57512yes—
247C22×C61#57711yescharacteristic
248C22×C61#57811yescharacteristic
249C4×C61—22no—
250C4×C61—22no—
251C4×C61—42no—
252C4×C61—44no—
253C4×C61—81no—
254C4×C61#53211yescharacteristic
255C4×C61#53511yescharacteristic
256C4×C61#55012yes—
257Dic61—42no—
258Dic61—44no—
259Dic61—42no—
260C2×#556—611no—
261C2×#489—611no—
262C2×#464—611no—
263#2540D6111yescharacteristic
264#2001—612no—
265#2000—612no—
266#1987—611no—
267#1955—611no—
268#1875—612no—
269#1873—612no—
270#1763—611no—
271C2×C61—82no—
272C2×C61—161no—
273C2×C61#243111yescharacteristic
274C2×C61#300511yescharacteristic
275C2×C61#301311yescharacteristic
276C2×C61#331112yes—
277C2×C61#331312yes—
278C22×Q16—611no—
279C22×SD16—611no—
280#556—1224no—
281#556—1224no—
282C2×C4⋊Q8—611no—
283C2×#126—1222no—
284C2×#124—1221no—
285C2×#124C2×D6111yescharacteristic
286C2×C22⋊Q8—612no—
287C2×C22⋊Q8C2×D6112yes—
288C2×#121C2×D6112yes—
289C2×C4×Q8—611no—
290C2×C4×Q8—1221no—
291C2×C4×D4C2×D6111yescharacteristic
292C2×#117—612no—
293#489—1224no—
294#464—1224no—
295C2×C4⋊C8—611no—
296C2×#103—612no—
297C2×#103—611no—
298C2×#103—612no—
299C2×#102—612no—
300C2×#102—611no—
301C2×C22⋊C8—612no—
302C2×C22⋊C8—612no—
303C2×#97—611no—
304#399—21no—
305#392—21no—
306#390—22no—
307C4×C4⋊C4C2×D6111yescharacteristic
308#340—612no—
309#340—612no—
310#340—612no—
311C61#2558011yescharacteristic, complemented, Sylow
312C22×Q8—611no—
313C22×Q8—1221no—
314C22×Q8C22×D6111yescharacteristic
315C22×D4—22no—
316C22×D4C22⋊D6111yescharacteristic
317C22×D4C22×D6111yescharacteristic
318C23×C4C22×D6112yes—
319C2×Q16—1222no—
320C2×Q16—2442no—
321C2×SD16—1222no—
322C2×SD16—2442no—
323C22×C8—611no—
324C22×C8—611no—
325C22×C8—612no—
326C4⋊Q8—614no—
327C4⋊Q8—1222no—
328#126—12216no—
329#124—22no—
330#124—22no—
331#124—1228no—
332C22⋊Q8—24no—
333C22⋊Q8—42no—
334C22⋊Q8—1224no—
335C22⋊Q8—1224no—
336#121—24no—
337#121—42no—
338C4×Q8—614no—
339C4×Q8—1228no—
340C4×Q8—1222no—
341C4×D4—22no—
342C4×D4—22no—
343#117—1224no—
344#117—1224no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4C22⋊D6111yescharacteristic
352C2×C4⋊C4C22×D6112yes—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—1222no—
358C2×C22⋊C4—1222no—
359C2×C22⋊C4C22⋊D6112yes—
360C2×C42—21no—
361C2×C42—1222no—
362C2×C42—1221no—
363C2×C42C22×D6111yescharacteristic
364C4⋊C8—1222no—
365#103—1222no—
366#103—1224no—
367#103—1224no—
368#102—1222no—
369#102—1224no—
370C22⋊C8—2442no—
371C22⋊C8—2442no—
372#97—1222no—
373#95—21no—
374C24—21no—
375C24C2×C22⋊D6112yes—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—612no—
381C2×Q8—1221no—
382C2×Q8—1224no—
383C2×Q8—1222no—
384C2×Q8—1222no—
385C2×Q8—2444no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—82no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—1221no—
404C22×C4—1222no—
405C22×C4—1221no—
406C22×C4C2×C22⋊D6112yes—
407C22×C4C2×C22⋊D6111yescharacteristic
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C23×D6111yescharacteristic
410Q16—2444no—
411SD16—2444no—
412C2×C8—1222no—
413C2×C8—1221no—
414C2×C8—1221no—
415C2×C8—1222no—
416C2×C8—1222no—
417C2×C8—1222no—
418C2×C8—1222no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—44no—
422C4⋊C4—612no—
423C4⋊C4—614no—
424C4⋊C4—1228no—
425C4⋊C4—1221no—
426C4⋊C4—1222no—
427C4⋊C4—1224no—
428C4⋊C4—1224no—
429C4⋊C4#1175822414yes—
430C22⋊C4—24no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—1228no—
436C22⋊C4—1228no—
437C42—81no—
438C42—1228no—
439C42—1224no—
440C42D4×D6112yes—
441C42D4×D6112yes—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—41no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23C22×C22⋊D6111yescharacteristic
455Q8—41no—
456Q8—41no—
457Q8—42no—
458Q8—1222no—
459Q8—1222no—
460Q8—2448no—
461Q8—2442no—
462D4—44no—
463D4—41no—
464D4—41no—
465D4—42no—
466D4—88no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—1222no—
483C2×C4—1224no—
484C2×C4—1222no—
485C2×C4—2444no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4—2442no—
489C2×C4#1451974512yes—
490C2×C4#1451974512yes—
491C2×C4C2×D4×D6111yescharacteristic
492C2×C4C2×D4×D6111yescharacteristic
493C2×C4C2×#1175822412yes—
494C2×C4#1451982212yes—
495C8—2441no—
496C8—2441no—
497C8—2442no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—81no—
506C22—82no—
507C22—81no—
508C22—81no—
509C22—82no—
510C22C2×#1451974511yescharacteristic
511C22#1684040612yes—
512C22#1684056012yes—
513C22#1684057311yescharacteristic
514C22#1684057711yescharacteristic
515C4—22no—
516C4—22no—
517C4—44no—
518C4—42no—
519C4—81no—
520C4—2442no—
521C4—2444no—
522C4—2442no—
523C4#1684031011yescharacteristic
524C4#1684032511yescharacteristic
525C4#1684043212yes—
526C2—82no—
527C2—161no—
528C2#2144943511yescharacteristic
529C2#2145505911yescharacteristic
530C2#2145513111yescharacteristic
531C2#2145785512yes—
532C2#2145788512yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2540C2
socleC23×C61C22×D4
hypercenter#2540D61
hypocenterC61#25580
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2540

Complements

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

NH ≅ G/Ntype
1. C2×#16840482C2semidirect
2. C2×#16840121C2semidirect
3. C2×#16839981C2semidirect
4. #21446827C2semidirect
5. #21446822C2semidirect
6. #21446767C2semidirect
7. #21446618C2semidirect
8. #21446142C2semidirect
9. #21446132C2semidirect
10. #21445652C2semidirect
55. #14519777D4semidirect
61. C4×Dic122D4semidirect
311. C61#25580semidirect

Maximal subgroups

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

subgroupindex
C2×#168404822
C2×#168401212
C2×#168399812
#214468272
#214468222
#214467672
#214466182
#214461422
#214461322
#214456522
C61×#25402
#2558061