Finite Group Databasebeta

← #29725142

Subgroups — #29725142

Order 15616.

620 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840035C211yesmaximal, characteristic
2#21454683C211yesmaximal, characteristic, complemented
3#21447287C211yesmaximal, characteristic, complemented
4#21447284C211yesmaximal, characteristic, complemented
5#21447099C211yesmaximal, characteristic
6#21446731C211yesmaximal, characteristic, complemented
7#21446606C211yesmaximal, characteristic, complemented
8#21446315C211yesmaximal, characteristic
9#21446310C211yesmaximal, characteristic
10#21445940C211yesmaximal, characteristic, complemented
11#21445932C211yesmaximal, characteristic, complemented
12#21445931C211yesmaximal, characteristic, complemented
13#21445871C211yesmaximal, characteristic
14#21445622C211yesmaximal, characteristic
15C61×#2049C211yesmaximal, characteristic
16C2×#14519807C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19#16840240C2211yescharacteristic
20#16840239C2211yescharacteristic
21#16840035—22no—
22#16840035—22no—
23#16839866—22no—
24#16839854—22no—
25#16839810C2211yescharacteristic
26C2×#14519712C2211yescharacteristic
27C2×#14519712C2211yescharacteristic
28#16839798C2211yescharacteristic
29#16839779C2211yescharacteristic
30#16839772C2211yescharacteristic
31#16839772C2211yescharacteristic
32#16839771C2211yescharacteristic
33C2×#14519696C2211yescharacteristic
34C2×#14519695C2211yescharacteristic
35C2×#14519695C2211yescharacteristic
36#16839736C2211yescharacteristic
37#16839720C2211yescharacteristic
38#16839720C2211yescharacteristic
39#16839534C2211yescharacteristic
40#16839534C2211yescharacteristic
41#16839534C2211yescharacteristic
42#16839533C2211yescharacteristic
43#16839533C2211yescharacteristic
44#16839533C2211yescharacteristic
45#16839532C2211yescharacteristic
46#16839519C2211yescharacteristic
47#16839518C2211yescharacteristic
48C2×#137×C61C2211yescharacteristic
49C61×#430C2211yescharacteristic
50C61×#421C2211yescharacteristic
51C61×#419C2211yescharacteristic
52C61×#413C2211yescharacteristic
53C2×#101×C61C2211yescharacteristic
54C61×#344C2211yescharacteristic
55C22×Dic122C2311yescharacteristic
56#14519807—22no—
57#14519807—41no—
58C2×#11758202—21no—
59C2×#11758202—21no—
60C2×#11758202—21no—
61C2×#11758202—21no—
62C2×#11758202—21no—
63C2×#11758202D412yes—
64C2×#11758202D412yes—
65C2×#11758200—21no—
66#14519784—21no—
67#14519784—21no—
68#14519784—21no—
69#14519784—21no—
70C2×#11758194—21no—
71C2×#11758194C2311yescharacteristic
72C2×#11758194C2311yescharacteristic
73C2×#11758194C2311yescharacteristic
74C22×#92418—21no—
75C22×#92418C2311yescharacteristic
76Q8×Dic61—22no—
77Q8×Dic61—41no—
78C4×Dic122—41no—
79#14519742—22no—
80#14519736—22no—
81#14519736C2311yescharacteristic
82#14519736C2311yescharacteristic
83#14519736C2311yescharacteristic
84#14519733—22no—
85#14519712—21no—
86#14519712—21no—
87#14519712—21no—
88#14519712—21no—
89#14519712—41no—
90#14519710—21no—
91#14519708—21no—
92#14519704—41no—
93#14519702—22no—
94#14519702—41no—
95#14519701—41no—
96#14519696—22no—
97#14519695—22no—
98#14519695—41no—
99#14519692—22no—
100#14519687—21no—
101#14519687—21no—
102C8×Dic61—21no—
103C8×Dic61—21no—
104C2×#41×C61C2311yescharacteristic
105C22×Q8×C61C2311yescharacteristic
106#137×C61—22no—
107#137×C61—41no—
108C2×Q16×C61—21no—
109C2×SD16×C61—21no—
110C2×M16×C61C2311yescharacteristic
111C2×M16×C61C2311yescharacteristic
112C2×M16×C61C2311yescharacteristic
113C2×M16×C61C2311yescharacteristic
114#117×C61C2311yescharacteristic
115#108×C61—41no—
116#104×C61—41no—
117#103×C61—22no—
118#101×C61—21no—
119#101×C61—41no—
120#100×C61—21no—
121#41×C61—21no—
122#41×C61—21no—
123#41×C61—21no—
124#41×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61C2×D412yes—
131C2×Q8×C61#4111yescharacteristic
132C2×D4×C61—21no—
133C2×D4×C61#4111yescharacteristic
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2411yescharacteristic
137C22×Dic61—21no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122C2×D412yes—
144Q16×C61—42no—
145SD16×C61—42no—
146M16×C61—21no—
147M16×C61—21no—
148M16×C61—21no—
149M16×C61—21no—
150M16×C61—21no—
151M16×C61—21no—
152M16×C61—21no—
153M16×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C4⋊C4×C61—21no—
159C22⋊C4×C61—21no—
160C42×C61—21no—
161C22⋊Dic61—21no—
162C22⋊Dic61—21no—
163C22⋊Dic61#4112yes—
164#11758202—22no—
165#11758202—22no—
166#11758202—22no—
167#11758202—22no—
168#11758202—42no—
169#11758200—42no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—42no—
174#11758196—42no—
175C4×Dic61—21no—
176C4×Dic61—21no—
177C4×Dic61—21no—
178C4×Dic61—42no—
179C4×Dic61—42no—
180#11758194—21no—
181#11758194—21no—
182#11758194—21no—
183#11758194—21no—
184#11758194—21no—
185#11758194—21no—
186#11758194—41no—
187#11758194—41no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—41no—
192C2×#92418—41no—
193C2×#92418C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195C2×#92418#4111yescharacteristic
196C2×#92418#4111yescharacteristic
197C2×#92418#4111yescharacteristic
198C2×#92418#4111yescharacteristic
199C23×C61—21no—
200C23×C61C2×#4111yescharacteristic
201Q8×C61—22no—
202Q8×C61—21no—
203Q8×C61—22no—
204Q8×C61—21no—
205Q8×C61—41no—
206Q8×C61—41no—
207D4×C61—21no—
208D4×C61—21no—
209D4×C61—41no—
210D4×C61—41no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C2×C4×C61H(5,𝔽2)11yescharacteristic
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230Dic122—22no—
231Dic122—22no—
232Dic122—41no—
233Dic122—41no—
234C8×C61—41no—
235C8×C61—41no—
236C8×C61—41no—
237C8×C61—41no—
238#92418—41no—
239#92418—41no—
240#92418—41no—
241#92418#12311yescharacteristic
242#92418#12311yescharacteristic
243#92418#12311yescharacteristic
244#92418#12311yescharacteristic
245#24989—611nomaximal, Sylow
246C22×C61—41no—
247C22×C61—41no—
248C22×C61—81no—
249C22×C61#52911yescharacteristic
250C22×C61#54611yescharacteristic
251C22×C61#55511yescharacteristic
252C4×C61—41no—
253C4×C61—42no—
254C4×C61—81no—
255C4×C61C2×#12311yescharacteristic
256C4×C61#54011yescharacteristic
257C4×C61#54611yescharacteristic
258C4×C61#55311yescharacteristic
259Dic61—42no—
260Dic61—42no—
261Dic61—81no—
262Dic61—81no—
263C2×#473—611no—
264#2950—611no—
265#2082—611no—
266#2082—611no—
267#2049—611no—
268#2049D6111yescharacteristic
269#1979—611no—
270#1953—611no—
271#1905—611no—
272#1905—611no—
273#1833—611no—
274#1830—611no—
275#1830—611no—
276#1814—611no—
277#1757—611no—
278C2×C61—21no—
279C2×C61—21no—
280C2×C61—21no—
281C2×C61—81no—
282C2×C61#242011yescharacteristic
283C2×#137—611no—
284C2×#137C2×D6111yescharacteristic
285C22×Q16—611no—
286C2×#131—611no—
287#519—611no—
288#519—611no—
289#473—1222no—
290#473—1222no—
291#441—1222no—
292C4×Q16—1222no—
293#430—611no—
294#430C2×D6111yescharacteristic
295C2×#108—611no—
296C2×#108—611no—
297#427—611no—
298#421—611no—
299#421C2×D6111yescharacteristic
300#419—611no—
301#419—611no—
302#419—611no—
303#419C2×D6111yescharacteristic
304#413C2×D6111yescharacteristic
305C2×#101—611no—
306C2×#101—611no—
307C2×#101—611no—
308C2×#101C2×D6111yescharacteristic
309#410—611no—
310#405—611no—
311#405—611no—
312#344—611no—
313#344—611no—
314#344—611no—
315#344—611no—
316#344—611no—
317#344—611no—
318#344—611no—
319#344C2×D6111yescharacteristic
320H(3,ℤ/4)—611no—
321H(3,ℤ/4)—611no—
322C61#2498911yescharacteristic, complemented, Sylow
323C2×#41C22×D6111yescharacteristic
324C22×Q8—611no—
325C22×Q8C22×D6111yescharacteristic
326#137—22no—
327#137—41no—
328#137—1222no—
329#137—2441no—
330C2×Q16—21no—
331C2×Q16—612no—
332C2×Q16—612no—
333C2×Q16—1221no—
334C2×Q16—1221no—
335C2×Q16—1221no—
336C2×Q16—1221no—
337C2×Q16—1221no—
338C2×SD16—21no—
339C2×SD16—1221no—
340#131—1221no—
341#131—1221no—
342#131—1221no—
343#131—1221no—
344C2×M16—611no—
345C2×M16—611no—
346C2×M16—611no—
347C2×M16—1221no—
348C2×M16C22×D6111yescharacteristic
349C2×M16C22×D6111yescharacteristic
350C2×M16C22×D6111yescharacteristic
351C2×M16C22×D6111yescharacteristic
352C22×C8—611no—
353C22×C8—1221no—
354C4×Q8—1222no—
355C4×Q8—1222no—
356C4×Q8—2441no—
357C4×Q8—2441no—
358#117—611no—
359#117—611no—
360#117—611no—
361#117—1222no—
362#117—1222no—
363#117C22×D6111yescharacteristic
364#108—41no—
365#108—1221no—
366#108—1221no—
367#108—1221no—
368#108—1221no—
369#108—2441no—
370#107—1221no—
371#106—1221no—
372#104—41no—
373#104—2441no—
374#103—22no—
375#103—1222no—
376#103—2441no—
377#103—2441no—
378#101—21no—
379#101—41no—
380#101—1222no—
381#101—1222no—
382#101—2441no—
383#100—21no—
384#99—1222no—
385#97—1221no—
386#97—1221no—
387C4×C8—1221no—
388C4×C8—1221no—
389#41—21no—
390#41—21no—
391#41—21no—
392#41—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—612no—
399C2×Q8—1221no—
400C2×Q8—1221no—
401C2×Q8—1221no—
402C2×Q8—1221no—
403C2×Q8—1221no—
404C2×Q8D4×D6112yes—
405C2×Q8#1175822411yescharacteristic
406C2×D4—21no—
407C2×D4#1175822411yescharacteristic
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—1221no—
411C22×C4C23×D6111yescharacteristic
412Q16—42no—
413Q16—1222no—
414Q16—1222no—
415Q16—1222no—
416Q16—1222no—
417Q16—2442no—
418SD16—42no—
419SD16—2442no—
420M16—21no—
421M16—21no—
422M16—21no—
423M16—21no—
424M16—21no—
425M16—21no—
426M16—21no—
427M16—21no—
428M16—1221no—
429M16—1221no—
430M16—1221no—
431M16—1221no—
432M16—1221no—
433M16—1221no—
434M16—2441no—
435M16—2441no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—611no—
441C2×C8—611no—
442C2×C8—611no—
443C2×C8—611no—
444C2×C8—611no—
445C2×C8—611no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—2441no—
450C2×C8—2441no—
451C4⋊C4—21no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—2442no—
456C4⋊C4—2442no—
457C22⋊C4—21no—
458C22⋊C4—612no—
459C22⋊C4—1221no—
460C22⋊C4—1221no—
461C42—21no—
462C42—1221no—
463C42—1221no—
464C42—1221no—
465C42—2442no—
466C42—2442no—
467C23—21no—
468C23#41×D6111yescharacteristic
469Q8—21no—
470Q8—22no—
471Q8—22no—
472Q8—21no—
473Q8—41no—
474Q8—41no—
475Q8—1222no—
476Q8—1222no—
477Q8—2441no—
478Q8—2441no—
479D4—21no—
480D4—21no—
481D4—41no—
482D4—41no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4C2×D4×D6111yescharacteristic
497C2×C4#41×D6111yescharacteristic
498C2×C4#41×D6111yescharacteristic
499C2×C4C2×#1175822411yescharacteristic
500C2×C4#1451982211yescharacteristic
501C2×C4#1451982211yescharacteristic
502C8—41no—
503C8—41no—
504C8—41no—
505C8—41no—
506C8—611no—
507C8—611no—
508C8—611no—
509C8—611no—
510C8—2441no—
511C8—2441no—
512C8—2441no—
513C22—41no—
514C22—41no—
515C22—81no—
516C22#1684028911yescharacteristic
517C22#1684040111yescharacteristic
518C22#1684047411yescharacteristic
519C4—41no—
520C4—42no—
521C4—81no—
522C4—4881no—
523C4—4881no—
524C4—2442no—
525C4—2442no—
526C4#123×D6111yescharacteristic
527C4#1684036211yescharacteristic
528C4#1684039911yescharacteristic
529C4#1684045511yescharacteristic
530C2—21no—
531C2—21no—
532C2—21no—
533C2—81no—
534C2#2144930411yescharacteristic
subgroupG/H
centerC2#21449304
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2049C2
socleC2×C61#2420
prime radicalC2×#41×C61C23
hypercenter#2049D61
hypocenterC61#24989
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2049

Complements

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

NH ≅ G/Ntype
2. #21454683C2semidirect
3. #21447287C2semidirect
4. #21447284C2semidirect
6. #21446731C2semidirect
7. #21446606C2semidirect
10. #21445940C2semidirect
11. #21445932C2semidirect
12. #21445931C2semidirect
322. C61#24989semidirect

Maximal subgroups

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

subgroupindex
C2×#168400352
#214546832
#214472872
#214472842
#214470992
#214467312
#214466062
#214463152
#214463102
#214459402
#214459322
#214459312
#214458712
#214456222
C61×#20492
#2498961