Finite Group Databasebeta

← #29658169

Subgroups — #29658169

Order 15616.

1060 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456757C211yesmaximal, characteristic, complemented
2#21456734C211yesmaximal, characteristic, complemented
3#21456272C211yesmaximal, characteristic, complemented
4#21444891C214yesmaximal, complemented
5#21444884C214yesmaximal, complemented
6#21444481C211yesmaximal, characteristic, complemented
7#21444240C211yesmaximal, characteristic, complemented
8#21444236C211yesmaximal, characteristic, complemented
9C61×#3548C211yesmaximal, characteristic, complemented
10C2×#14519802—22no—
11C2×#14519774C2211yescharacteristic
12#16840152—22no—
13#16840138—21no—
14#16840138C2212yes—
15#16840136—21no—
16#16840136C2212yes—
17#16840110—21no—
18#16840110—21no—
19#16839962C2214yes—
20#16839957C2214yes—
21#16839912—22no—
22#16839895—22no—
23#16839815C2211yescharacteristic
24#16839815C2211yescharacteristic
25C4×#11758194C2211yescharacteristic
26#16839511C2212yescomplemented
27#16839506C2212yescomplemented
28Q8⋊#92418C2214yescomplemented
29D4⋊#92418C2214yescomplemented
30C61×#562—22no—
31C61×#554—21no—
32C61×#554C2212yescomplemented
33C61×#550—21no—
34C61×#550C2212yescomplemented
35C61×#547—21no—
36C61×#547—21no—
37C61×#540—22no—
38C61×#537—22no—
39C61×#537—22no—
40C61×#533C2211yescharacteristic
41C61×#532C2211yescharacteristic
42C61×#532C2211yescharacteristic
43C22×D244—21no—
44C22×D244—21no—
45C22×D244—21no—
46#14519802—48no—
47C2×#11758201—24no—
48C2×#11758201—24no—
49C2×#11758199—24no—
50C2×#11758199—24no—
51C2×#11758194—21no—
52#14519774—21no—
53#14519774—21no—
54#14519774—21no—
55#14519774C2312yes—
56C4⋊#92418C2312yes—
57C4⋊#92418C2312yes—
58#14519697—24no—
59#14519697—24no—
60C22⋊#92418—21no—
61C22⋊#92418—21no—
62#14519688—21no—
63C4×#92418—21no—
64C4×#92418—21no—
65C4×#92418C2312yes—
66C2×#41×C61—22no—
67C2×#41×C61—22no—
68C2×#41×C61—21no—
69C2×#41×C61D412yes—
70C4⋊Q8×C61D411yescharacteristic
71C4⋊Q8×C61D411yescharacteristic
72C4⋊Q8×C61D411yescharacteristic
73C4⋊Q8×C61C2311yescharacteristic
74#127×C61—22no—
75#127×C61D411yescharacteristic
76#127×C61C2311yescharacteristic
77#125×C61—22no—
78#125×C61—21no—
79#124×C61—21no—
80#124×C61D412yes—
81#123×C61—24no—
82#123×C61—24no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—24no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61—24no—
87C22⋊Q8×C61—21no—
88C22⋊Q8×C61D412yes—
89#121×C61—24no—
90#121×C61—24no—
91#121×C61—22no—
92#121×C61—24no—
93#121×C61—24no—
94#121×C61—21no—
95#121×C61—21no—
96#121×C61—21no—
97#121×C61—22no—
98#121×C61D412yes—
99C4×Q8×C61—21no—
100C4×Q8×C61C2312yes—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106C4×D4×C61C2312yes—
107#117×C61—21no—
108C2×C4⋊C4×C61—22no—
109C2×C4⋊C4×C61—22no—
110C2×C42×C61C2311yescharacteristic
111C23×D61—41no—
112#41×C61—44no—
113#41×C61—82no—
114#41×C61—84no—
115#41×C61—84no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61C2×D412yes—
119C2×Q8×C61C2×D412yes—
120C2×D4×C61—24no—
121C2×D4×C61—24no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61C2×D412yes—
129C2×D4×C61C2×D412yes—
130C22×C4×C61—24no—
131C22×C4×C61—24no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—44no—
141C2×D244—41no—
142C2×D244—41no—
143C2×D244—41no—
144C2×D244—82no—
145C2×D244—82no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61C2×D412yes—
157C4⋊C4×C61C2×D412yes—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—24no—
160C22⋊C4×C61—22no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C42×C61—22no—
166C42×C61C2×D411yescharacteristic
167C42×C61C2×D411yescharacteristic
168C42×C61C2×D411yescharacteristic
169C42×C61C2411yescharacteristic
170#11758201—48no—
171#11758201—84no—
172#11758199—48no—
173#11758199—84no—
174#11758194—24no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61#12011yescharacteristic
181C22×D61—42no—
182C22×D61—82no—
183C22×D61—82no—
184C22×D61—82no—
185Q8×C61—24no—
186Q8×C61—42no—
187Q8×C61—42no—
188Q8×C61—81no—
189D4×C61—24no—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—42no—
193D4×C61—42no—
194D4×C61—81no—
195D4×C61—82no—
196D4×C61—82no—
197D4×C61—84no—
198D4×C61—84no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—44no—
207C2×C4×C61—44no—
208C2×C4×C61—84no—
209C2×C4×C61—84no—
210C2×C4×C61#12011yescharacteristic
211C2×C4×C61#12011yescharacteristic
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216D244—44no—
217D244—44no—
218D244—44no—
219D244—82no—
220D244—82no—
221D244—82no—
222#92418—24no—
223#92418—42no—
224#92418—42no—
225#18716—611nomaximal, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61C2×#12011yescharacteristic
236C2×D61—82no—
237C2×D61—82no—
238C2×D61—82no—
239C2×D61—164no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—42no—
244C4×C61—42no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61#45312yes—
248C4×C61#57612yes—
249C4×C61#57712yes—
250#3548D6111yescharacteristic, complemented
251#3208—611no—
252#3205—611no—
253#3151—611no—
254#1622—614no—
255#1621—614no—
256#1543—611no—
257#1485—611no—
258#1484—611no—
259C2×C61—41no—
260C2×C61—82no—
261C2×C61—82no—
262C2×C61#300811yescharacteristic
263C2×C61#301011yescharacteristic
264C2×C61C2×#45311yescharacteristic
265D61—162no—
266C2×AGL(1,ℤ/8)—1222no—
267#562—22no—
268#554—21no—
269#554C2×D6112yescomplemented
270#550—21no—
271#550C2×D6112yescomplemented
272#547—21no—
273#547—21no—
274#540—22no—
275#537—22no—
276#537—22no—
277#533C2×D6111yescharacteristic
278#532C2×D6111yescharacteristic
279#532C2×D6111yescharacteristic
280C2×#127—611no—
281#496—1222no—
282#493—612no—
283#493—1221no—
284#492—612no—
285#492—1221no—
286#486—1221no—
287#486—1221no—
288#460—614no—
289#459—614no—
290#450—1222no—
291#447—1222no—
292#431—611no—
293#431—611no—
294C4×M16—611no—
295#333—612no—
296#331—612no—
297Q8⋊C8—614no—
298D4⋊C8—614no—
299C61#1871611yescharacteristic, complemented, Sylow
300C2×#41—22no—
301C2×#41—22no—
302C2×#41—21no—
303C2×#41C22⋊D6112yes—
304C22×D4—1221no—
305C22×D4—1221no—
306C22×D4—1221no—
307AGL(1,ℤ/8)—2448no—
308C2×SD16—1224no—
309C2×SD16—1224no—
310C2×D8—1224no—
311C2×D8—1224no—
312C2×M16—1221no—
313C4⋊Q8C22⋊D6111yescharacteristic
314C4⋊Q8C22⋊D6111yescharacteristic
315C4⋊Q8C22⋊D6111yescharacteristic
316C4⋊Q8C22×D6111yescharacteristic
317#127—22no—
318#127—612no—
319#127—1221no—
320#127—1221no—
321#127—1221no—
322#127C22⋊D6111yescharacteristic
323#127C22×D6111yescharacteristic
324#125—22no—
325#125—21no—
326#124—21no—
327#124C22⋊D6112yes—
328#123—24no—
329#123—24no—
330C22⋊Q8—22no—
331C22⋊Q8—24no—
332C22⋊Q8—22no—
333C22⋊Q8—24no—
334C22⋊Q8—21no—
335C22⋊Q8C22⋊D6112yes—
336#121—24no—
337#121—24no—
338#121—22no—
339#121—24no—
340#121—24no—
341#121—21no—
342#121—21no—
343#121—21no—
344#121—22no—
345#121C22⋊D6112yes—
346C4×Q8—21no—
347C4×Q8C22×D6112yes—
348C4×D4—22no—
349C4×D4—22no—
350C4×D4—21no—
351C4×D4—21no—
352C4×D4—21no—
353C4×D4C22×D6112yes—
354#117—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C42C22×D6111yescharacteristic
358C4⋊C8—612no—
359C4⋊C8—612no—
360#102—1224no—
361#102—1224no—
362C22⋊C8—1221no—
363C22⋊C8—1221no—
364#97—1221no—
365C4×C8—612no—
366C4×C8—1221no—
367C4×C8—1221no—
368C24—2441no—
369#41—44no—
370#41—82no—
371#41—84no—
372#41—84no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8C2×C22⋊D6112yes—
376C2×Q8C2×C22⋊D6112yes—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—1222no—
386C2×D4—1222no—
387C2×D4—1222no—
388C2×D4—4882no—
389C2×D4—4882no—
390C2×D4—2444no—
391C2×D4—2441no—
392C2×D4—2441no—
393C2×D4—2441no—
394C2×D4C2×C22⋊D6112yes—
395C2×D4C2×C22⋊D6112yes—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4C2×C22⋊D6111yescharacteristic
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403SD16—4884no—
404SD16—2448no—
405D8—4884no—
406D8—2448no—
407M16—1224no—
408C2×C8—1222no—
409C2×C8—1222no—
410C2×C8—1222no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4C2×C22⋊D6112yes—
422C4⋊C4C2×C22⋊D6112yes—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C42—22no—
431C42C2×C22⋊D6111yescharacteristic
432C42C2×C22⋊D6111yescharacteristic
433C42C2×C22⋊D6111yescharacteristic
434C42C23×D6111yescharacteristic
435C23—22no—
436C23—22no—
437C23—4882no—
438C23—4882no—
439C23—4882no—
440C23—2442no—
441C23#1451974611yescharacteristic
442Q8—24no—
443Q8—42no—
444Q8—42no—
445Q8—81no—
446D4—24no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—81no—
452D4—82no—
453D4—82no—
454D4—84no—
455D4—84no—
456D4—4882no—
457D4—4882no—
458D4—4882no—
459D4—2444no—
460D4—2444no—
461D4—2444no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—84no—
472C2×C4—84no—
473C2×C4#1451974611yescharacteristic
474C2×C4#1451974611yescharacteristic
475C2×C4#1451974611yescharacteristic
476C2×C4C22×C22⋊D6111yescharacteristic
477C2×C4C22×C22⋊D6111yescharacteristic
478C2×C4C22×C22⋊D6111yescharacteristic
479C8—1224no—
480C8—2442no—
481C8—2442no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—9764no—
492C22—4882no—
493C22—4882no—
494C22—4882no—
495C22C2×#1451974611yescharacteristic
496C4—21no—
497C4—21no—
498C4—21no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4#1683993012yes—
504C4#1684056512yes—
505C4#1684057012yes—
506C2—41no—
507C2—82no—
508C2—82no—
509C2—9762no—
510C2#2145508711yescharacteristic
511C2#2145510511yescharacteristic
512C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3548C2
socleC22×C61C2×#120
hypercenter#3548D61
hypocenterC61#18716
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

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

NH ≅ G/Ntype
1. #21456757C2semidirect
2. #21456734C2semidirect
3. #21456272C2semidirect
4. #21444891C2semidirect
5. #21444884C2semidirect
6. #21444481C2semidirect
7. #21444240C2semidirect
8. #21444236C2semidirect
9. C61×#3548C2semidirect
26. #16839511C22semidirect
27. #16839506C22semidirect
28. Q8⋊#92418C22semidirect
29. D4⋊#92418C22semidirect
32. C61×#554C22semidirect
34. C61×#550C22semidirect
250. #3548D61semidirect
269. #554C2×D61semidirect
271. #550C2×D61semidirect
299. C61#18716semidirect

Maximal subgroups

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

subgroupindex
#214567572
#214567342
#214562722
#214448912
#214448842
#214444812
#214442402
#214442362
C61×#35482
#1871661