Finite Group Databasebeta

← #29733764

Subgroups — #29733764

Order 15616.

1294 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456272C211yesmaximal, characteristic, complemented
2#21456195C211yesmaximal, characteristic, complemented
3#21450765C211yesmaximal, characteristic, complemented
4#21446832C214yesmaximal, complemented
5#21446764C211yesmaximal, characteristic, complemented
6#21446739C211yesmaximal, characteristic, complemented
7#21446195C214yesmaximal, complemented
8C4×#14519694C211yesmaximal, characteristic, complemented
9C61×#3506C211yesmaximal, characteristic, complemented
10C2×#14519802C2214yes—
11C2×#14519774C2211yescharacteristic
12C2×#14519773C2212yes—
13C2×#14519751—22no—
14#16840152—22no—
15#16840138—21no—
16#16840136—21no—
17#16840128—22no—
18#16840110—21no—
19#16840110—21no—
20C2×#14519704C2214yes—
21#16839766C2214yes—
22C2×#14519694C2212yes—
23C4×#11758194C2211yescharacteristic
24C4×#11758194C2211yescharacteristic
25#16839671—22no—
26#16839663C2212yes—
27C4×C4⋊Dic61C2211yescharacteristic
28#16839531C2212yes—
29#16839521C2214yes—
30C61×#559—22no—
31C61×#556—21no—
32C61×#555—21no—
33C61×#553—22no—
34C61×#550—21no—
35C61×#547—21no—
36C61×#538—22no—
37C61×#535C2212yes—
38C61×#533C2211yescharacteristic
39C61×#532C2211yescharacteristic
40C61×#525—22no—
41C61×#520C2212yes—
42C2×C4×D4×C61C2211yescharacteristic
43C22×D244—21no—
44C22×D244—21no—
45C22×D244C2312yes—
46C22×C22⋊D61—24no—
47#14519802—28no—
48#14519802—44no—
49C2×#11758201—24no—
50C2×#11758199—24no—
51C2×#11758194C2312yes—
52C2×#11758194C2312yes—
53#14519774—21no—
54#14519774—21no—
55#14519774—21no—
56#14519774—21no—
57#14519773—44no—
58#14519751—48no—
59C2×C4⋊Dic61—21no—
60C2×#11758198—24no—
61C2×C4×Dic61C2312yes—
62#14519704—44no—
63#14519697—24no—
64#14519697—24no—
65#14519694—24no—
66#14519688—21no—
67#14519688—21no—
68C4×#92418—21no—
69C4×#92418—21no—
70#14519684—21no—
71C2×#41×C61—22no—
72C2×#41×C61C2312yes—
73C22×D4×C61C2311yescharacteristic
74C23×C4×C61D412yes—
75C4⋊Q8×C61D411yescharacteristic, complemented
76C4⋊Q8×C61D411yescharacteristic, complemented
77C4⋊Q8×C61D411yescharacteristic, complemented
78#127×C61—22no—
79#127×C61D411yescharacteristic, complemented
80#126×C61—24no—
81#125×C61D412yescomplemented
82#124×C61—24no—
83#124×C61—22no—
84#124×C61D412yescomplemented
85#123×C61—24no—
86#123×C61—24no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—21no—
89C22⋊Q8×C61—21no—
90#121×C61—24no—
91#121×C61—22no—
92#121×C61—21no—
93#121×C61—21no—
94#121×C61—22no—
95#121×C61—22no—
96#120×C61—24no—
97C4×Q8×C61—22no—
98C4×Q8×C61—21no—
99C4×Q8×C61—21no—
100C4×D4×C61—24no—
101C4×D4×C61—22no—
102C4×D4×C61—24no—
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C4×D4×C61—21no—
108C4×D4×C61—21no—
109#117×C61—22no—
110#117×C61—22no—
111#117×C61C2312yes—
112C2×C4⋊C4×C61—22no—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61D412yes—
115C2×C42×C61C2311yescharacteristic
116C24×C61C2×D412yes—
117C23×D61—22no—
118#41×C61—24no—
119#41×C61—24no—
120#41×C61—84no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61C2×D412yes—
124C2×D4×C61—22no—
125C2×D4×C61—24no—
126C2×D4×C61—24no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61C2×D412yes—
134C22×C4×C61—24no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—44no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C2×C22⋊D61—48no—
144C2×C22⋊D61—48no—
145C2×C22⋊D61—84no—
146C22×Dic61—22no—
147C2×D244—22no—
148C2×D244—22no—
149C2×D244—24no—
150C2×D244—24no—
151C2×D244—44no—
152C2×D244—41no—
153C2×D244—41no—
154C2×D244—41no—
155C2×D244—41no—
156C2×D244—82no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61C2×D412yes—
167C4⋊C4×C61C2×D412yes—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C42×C61—24no—
177C42×C61—22no—
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182#11758201—48no—
183#11758199—48no—
184#11758198—48no—
185C4⋊Dic61—24no—
186C4×Dic61—24no—
187#11758194—22no—
188#11758194—22no—
189#11758194—22no—
190#11758194#4114yescomplemented
191C2×#92418—22no—
192C2×#92418—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61C22×D411yescharacteristic
200C22×D61—44no—
201C22×D61—44no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—42no—
205Q8×C61—24no—
206Q8×C61—42no—
207Q8×C61—81no—
208D4×C61—24no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61—42no—
214D4×C61—81no—
215D4×C61—84no—
216D4×C61—84no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—41no—
223C2×C4×C61—44no—
224C2×C4×C61—44no—
225C2×C4×C61—42no—
226C2×C4×C61—82no—
227C2×C4×C61—84no—
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61#12011yescharacteristic
230C2×C4×C61#12011yescharacteristic
231C2×C4×C61#12011yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4112yes—
235C2×C4×C61H(5,𝔽2)12yes—
236C22⋊D61—816no—
237C2×Dic61—24no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240D244—44no—
241D244—44no—
242D244—82no—
243D244—82no—
244D244—82no—
245D244—82no—
246#92418—24no—
247#92418—42no—
248#25622—611nomaximal, Sylow
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—44no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—82no—
259C22×C61C2×#12011yescharacteristic
260C22×C61#53211yescharacteristic
261C22×C61#53511yescharacteristic
262C2×D61—84no—
263C2×D61—84no—
264C2×D61—82no—
265C2×D61—82no—
266C2×D61—82no—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—42no—
271C4×C61—42no—
272C4×C61—42no—
273C4×C61—42no—
274C4×C61#45312yes—
275C4×C61#54612yes—
276C4×C61#55012yes—
277Dic61—44no—
278#3506D6111yescharacteristic, complemented
279#3151—611no—
280#3138—611no—
281#2541—611no—
282#2002—614no—
283#1986—611no—
284#1980—611no—
285#1883—614no—
286C4×#100—611no—
287C2×C61—21no—
288C2×C61—21no—
289C2×C61—82no—
290C2×C61—82no—
291C2×C61#243111yescharacteristic
292C2×C61C2×#45311yescharacteristic
293C2×C61#302111yescharacteristic
294D61—162no—
295C2×AGL(1,ℤ/8)—614no—
296#559—22no—
297#556—21no—
298#555—21no—
299#553—22no—
300#550—21no—
301#547—21no—
302#538—22no—
303#535C2×D6112yes—
304#533C2×D6111yescharacteristic
305#532C2×D6111yescharacteristic
306C2×#127—612no—
307C2×#127—611no—
308#525—22no—
309C2×#121—1222no—
310#520C2×D6112yes—
311C2×C4×D4C2×D6111yescharacteristic
312#496—1222no—
313#493—1221no—
314#492—1221no—
315#490—1222no—
316#486—1221no—
317#486—1221no—
318C2×#104—614no—
319#418—614no—
320C2×#100—612no—
321C4×M16—611no—
322C4×M16—611no—
323#392—1222no—
324#390—612no—
325C4×C4⋊C4—611no—
326#343—612no—
327#339—614no—
328C61#2562211yescharacteristic, complemented, Sylow
329C2×#41—22no—
330C2×#41C22×D6112yes—
331C22×D4—612no—
332C22×D4—1224no—
333C22×D4—1221no—
334C22×D4—1221no—
335C22×D4C22×D6111yescharacteristic
336C23×C4C22⋊D6112yes—
337AGL(1,ℤ/8)—1228no—
338AGL(1,ℤ/8)—2444no—
339C2×SD16—1224no—
340C2×D8—1224no—
341C2×M16—612no—
342C2×M16—612no—
343C4⋊Q8C22⋊D6111yescharacteristic, complemented
344C4⋊Q8C22⋊D6111yescharacteristic, complemented
345C4⋊Q8C22⋊D6111yescharacteristic, complemented
346#127—22no—
347#127—1221no—
348#127—1221no—
349#127—1221no—
350#127—1221no—
351#127—2444no—
352#127C22⋊D6111yescharacteristic, complemented
353#126—24no—
354#125C22⋊D6112yescomplemented
355#124—24no—
356#124—22no—
357#124C22⋊D6112yescomplemented
358#123—24no—
359#123—24no—
360C22⋊Q8—22no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363#121—22no—
364#121—24no—
365#121—21no—
366#121—21no—
367#121—22no—
368#121—22no—
369#121—2448no—
370#120—24no—
371C4×Q8—22no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×D4—24no—
375C4×D4—22no—
376C4×D4—24no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383#117—22no—
384#117—22no—
385#117C22×D6112yes—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—1224no—
390C2×C22⋊C4C22⋊D6112yes—
391C2×C42—612no—
392C2×C42C22×D6111yescharacteristic
393#104—2444no—
394#102—1224no—
395#102—1224no—
396#100—1224no—
397#97—1221no—
398#97—1221no—
399C4×C8—1221no—
400C4×C8—1221no—
401#95—1221no—
402C24—1222no—
403C24C2×C22⋊D6112yes—
404#41—24no—
405#41—24no—
406#41—84no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8C2×C22⋊D6112yes—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1224no—
422C2×D4—1224no—
423C2×D4—4882no—
424C2×D4—4884no—
425C2×D4—2444no—
426C2×D4—2448no—
427C2×D4—2448no—
428C2×D4—2441no—
429C2×D4—2441no—
430C2×D4—2441no—
431C2×D4—2441no—
432C2×D4C2×C22⋊D6112yes—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—44no—
438C22×C4—1222no—
439C22×C4C2×C22⋊D6111yescharacteristic
440C22×C4C2×C22⋊D6112yes—
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C23×D6111yescharacteristic
443SD16—2448no—
444D8—2448no—
445M16—614no—
446M16—1222no—
447M16—1222no—
448M16—1222no—
449C2×C8—1222no—
450C2×C8—1222no—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—1224no—
461C4⋊C4C2×C22⋊D6112yes—
462C4⋊C4C2×C22⋊D6112yes—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—2448no—
472C42—24no—
473C42—22no—
474C42—1224no—
475C42C2×C22⋊D6111yescharacteristic
476C42C2×C22⋊D6111yescharacteristic
477C42C2×C22⋊D6111yescharacteristic
478C42C2×C22⋊D6111yescharacteristic
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—42no—
484C23—42no—
485C23—2444no—
486C23—2444no—
487C23—2442no—
488C23—2442no—
489C23—2442no—
490C23C22×C22⋊D6111yescharacteristic
491Q8—24no—
492Q8—42no—
493Q8—81no—
494D4—24no—
495D4—42no—
496D4—42no—
497D4—42no—
498D4—42no—
499D4—42no—
500D4—81no—
501D4—84no—
502D4—84no—
503D4—4882no—
504D4—4882no—
505D4—4882no—
506D4—48816no—
507D4—4882no—
508D4—2444no—
509D4—2444no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—82no—
520C2×C4—84no—
521C2×C4—1224no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4#1451974611yescharacteristic
525C2×C4#1451974611yescharacteristic
526C2×C4#1451974611yescharacteristic
527C2×C4#1451974611yescharacteristic
528C2×C4C22×C22⋊D6111yescharacteristic
529C2×C4C22×C22⋊D6111yescharacteristic
530C2×C4#41×D6112yes—
531C2×C4#1451982312yes—
532C8—1224no—
533C8—2442no—
534C22—21no—
535C22—21no—
536C22—44no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—82no—
544C22—4884no—
545C22—4884no—
546C22—4882no—
547C22—4882no—
548C22—4882no—
549C22C2×#1451974611yescharacteristic
550C22#1684030811yescharacteristic
551C22#1684032611yescharacteristic
552C4—21no—
553C4—21no—
554C4—21no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—2444no—
560C4#1683993012yes—
561C4#1684040012yes—
562C4#1684043112yes—
563C2—21no—
564C2—21no—
565C2—82no—
566C2—82no—
567C2—9762no—
568C2#2144943411yescharacteristic
569C2C2×#1683993011yescharacteristic
570C2#2145519111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3506C2
socleC22×C61C2×#120
hypercenter#3506D61
hypocenterC61#25622
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #21456272C2semidirect
2. #21456195C2semidirect
3. #21450765C2semidirect
4. #21446832C2semidirect
5. #21446764C2semidirect
6. #21446739C2semidirect
7. #21446195C2semidirect
8. C4×#14519694C2semidirect
9. C61×#3506C2semidirect
75. C4⋊Q8×C61D4semidirect
76. C4⋊Q8×C61D4semidirect
77. C4⋊Q8×C61D4semidirect
79. #127×C61D4semidirect
81. #125×C61D4semidirect
84. #124×C61D4semidirect
190. #11758194#41semidirect
278. #3506D61semidirect
328. C61#25622semidirect
343. C4⋊Q8C22⋊D61semidirect
344. C4⋊Q8C22⋊D61semidirect
345. C4⋊Q8C22⋊D61semidirect
352. #127C22⋊D61semidirect
354. #125C22⋊D61semidirect
357. #124C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214562722
#214561952
#214507652
#214468322
#214467642
#214467392
#214461952
C4×#145196942
C61×#35062
#2562261