Finite Group Databasebeta

← #29729776

Subgroups — #29729776

Order 15616.

764 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456195C211yesmaximal, characteristic, complemented
2#21455612C211yesmaximal, characteristic, complemented
3C2×#16839958C211yesmaximal, characteristic
4#21447178C212yesmaximal, complemented
5#21447075C212yesmaximal, complemented
6#21446729C212yesmaximal, complemented
7#21446659C211yesmaximal, characteristic, complemented
8#21446595C211yesmaximal, characteristic, complemented
9#21446481C212yesmaximal, complemented
10C4×#14519693C211yesmaximal, characteristic, complemented
11C61×#2962C211yesmaximal, characteristic, complemented
12C22×D488C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840128—22no—
16#16840110—21no—
17#16840110—21no—
18#16840059—22no—
19#16840049—21no—
20#16840039—21no—
21#16839958—21no—
22#16839958—21no—
23#16839958—22no—
24C2×#14519713C2212yes—
25#16839805C2212yes—
26#16839786C2211yescharacteristic
27#16839766C2212yes—
28#16839766C2212yes—
29#16839766C2212yes—
30C2×#14519699C2212yes—
31C2×#14519693C2212yes—
32C2×#14519693C2211yescharacteristic
33C2×#14519693C2211yescharacteristic
34C2×#14519693C2212yes—
35C4×#11758194C2211yescharacteristic
36#16839530C2211yescharacteristic
37#16839530C2211yescharacteristic
38#16839527C2212yes—
39C61×#533C2211yescharacteristic
40C61×#446—22no—
41C61×#446—21no—
42C8×Q8×C61—21no—
43C61×#424C2212yes—
44C2×C4⋊C8×C61C2211yescharacteristic
45C61×#405C2212yes—
46C4×M16×C61C2211yescharacteristic
47C22×D244—21no—
48C22×D244—21no—
49C22×D244C2312yes—
50C22×D244C2311yescharacteristic
51C22×C4×D61—21no—
52C2×D488—24no—
53C2×D488—22no—
54C2×D488—22no—
55C2×D488—42no—
56C2×#11758194C2311yescharacteristic
57C2×#11758194C2311yescharacteristic
58C2×#11758194C2312yes—
59#14519774—21no—
60#14519774—21no—
61#14519774—21no—
62#14519774—21no—
63C4×D244—21no—
64C4×D244—21no—
65C4×D244—21no—
66C4×D244—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×#11758198—21no—
69#14519713—42no—
70#14519710—22no—
71#14519708—22no—
72C4⋊#92418—21no—
73C4⋊#92418—21no—
74#14519699—22no—
75#14519699—22no—
76#14519697—22no—
77#14519697—22no—
78#14519697—22no—
79#14519697—22no—
80#14519697—22no—
81#14519697—22no—
82#14519693—22no—
83#14519693—22no—
84#14519693—24no—
85#14519693—42no—
86#14519688—21no—
87C4×#92418—21no—
88C2×M16×C61C2311yescharacteristic
89C2×M16×C61C2311yescharacteristic
90C22×C8×C61C2312yes—
91C4⋊Q8×C61D411yescharacteristic
92C4⋊Q8×C61D411yescharacteristic
93#125×C61D412yes—
94C22⋊Q8×C61—21no—
95C22⋊Q8×C61—21no—
96C4×Q8×C61—21no—
97C4×Q8×C61—21no—
98#117×C61C2312yes—
99C2×C42×C61C2311yescharacteristic
100C4⋊C8×C61—22no—
101C4⋊C8×C61—22no—
102C4⋊C8×C61D411yescharacteristic
103C4⋊C8×C61D411yescharacteristic
104C4⋊C8×C61D412yes—
105#97×C61—22no—
106#97×C61—21no—
107C4×C8×C61—22no—
108C4×C8×C61—21no—
109C23×D61—21no—
110C23×D61—22no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C22×C4×C61C2411yescharacteristic
116C22×Dic61—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—21no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—41no—
129C2×D244—41no—
130C2×D244—41no—
131C2×D244—41no—
132C2×D244—82no—
133C2×D244—82no—
134C2×D244#4112yes—
135C2×C4×D61—41no—
136C2×C4×D61—41no—
137C2×C4×D61—82no—
138M16×C61—21no—
139M16×C61—21no—
140M16×C61—21no—
141M16×C61#4112yes—
142C2×C8×C61—21no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—21no—
146C2×C8×C61C2×D412yes—
147C2×C8×C61C2×D412yes—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61C2×D412yes—
153C4⋊C4×C61C2×D412yes—
154C22⋊C4×C61—22no—
155C42×C61—22no—
156C42×C61C2×D411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160#11758198—41no—
161#11758198—41no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164#11758194—21no—
165#11758194—21no—
166#11758194—22no—
167#11758194—22no—
168#11758194—21no—
169#11758194—21no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—22no—
173D488—44no—
174D488—44no—
175C23×C61C2×#4111yescharacteristic
176C22×D61—22no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—42no—
180C22×D61—42no—
181C22×D61—42no—
182C22×D61—81no—
183C22×D61—81no—
184C22×D61—82no—
185C22×D61—82no—
186Q8×C61—42no—
187Q8×C61—81no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—41no—
192C2×C4×C61#12111yescharacteristic
193C2×C4×C61#12111yescharacteristic
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61H(5,𝔽2)11yescharacteristic
201C2×C4×C612−1+411yescharacteristic
202C2×Dic61—41no—
203C2×Dic61—81no—
204D244—22no—
205D244—22no—
206D244—42no—
207D244—41no—
208D244—42no—
209D244—42no—
210D244—41no—
211D244—42no—
212D244—82no—
213D244—82no—
214D244—82no—
215D244—82no—
216C4×D61—82no—
217C4×D61—82no—
218C8×C61—22no—
219C8×C61—22no—
220C8×C61—22no—
221C8×C61—41no—
222#92418—41no—
223#92418—41no—
224#92418—42no—
225#25312—611nomaximal, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61C2×#12111yescharacteristic
229C22×C61#52911yescharacteristic
230C22×C61#54211yescharacteristic
231C2×D61—42no—
232C2×D61—82no—
233C2×D61—81no—
234C2×D61—81no—
235C2×D61—82no—
236C2×D61—82no—
237C2×D61—82no—
238C2×D61—82no—
239C2×D61—162no—
240C2×D61—162no—
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—42no—
245C4×C61—42no—
246C4×C61#47212yes—
247C4×C61#54611yescharacteristic
248C4×C61#54811yescharacteristic
249C4×C61#54811yescharacteristic
250C4×C61#55011yescharacteristic
251Dic61—81no—
252#3138—611no—
253#3077—611no—
254C2×#459—611no—
255#2962D6111yescharacteristic, complemented
256#2063—612no—
257#2045—612no—
258#1978—612no—
259#1965—611no—
260#1950—611no—
261#1931—612no—
262C4×#100—611no—
263C2×C61—21no—
264C2×C61—21no—
265C2×C61#242611yescharacteristic
266C2×C61C2×#47211yescharacteristic
267C2×C61#306811yescharacteristic
268D61—82no—
269D61—162no—
270C22×D8—612no—
271#533C2×D6111yescharacteristic
272C2×#127—611no—
273C2×C4×D4—611no—
274#490—1222no—
275#486—1221no—
276#486—1221no—
277#478—1222no—
278#476—1221no—
279#474—1221no—
280#459—1221no—
281#459—1222no—
282#459—1221no—
283#446—22no—
284#446—21no—
285C8×Q8—21no—
286C2×#108—612no—
287#428—612no—
288#424C2×D6112yes—
289#423—611no—
290C2×C4⋊C8C2×D6111yescharacteristic
291#418—612no—
292#418—612no—
293#418—612no—
294C2×#102—612no—
295C2×#100—612no—
296C2×#100—611no—
297C2×#100—611no—
298C2×#100—612no—
299#405C2×D6112yes—
300C4×M16—611no—
301C4×M16C2×D6111yescharacteristic
302#343—611no—
303#343—611no—
304#341—612no—
305C61#2531211yescharacteristic, complemented, Sylow
306C22×D4—611no—
307C22×D4—612no—
308C22×D4—1221no—
309C22×D4—1221no—
310C23×C4—1221no—
311C2×D8—1224no—
312C2×D8—1222no—
313C2×D8—1222no—
314C2×D8—2442no—
315C2×M16—611no—
316C2×M16—612no—
317C2×M16—611no—
318C2×M16C22×D6111yescharacteristic
319C2×M16C22×D6111yescharacteristic
320C22×C8C22×D6112yes—
321C4⋊Q8D24411yescharacteristic
322C4⋊Q8D24411yescharacteristic
323#127—1221no—
324#127—1221no—
325#127—1221no—
326#127—1221no—
327#125D24412yes—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×D4—1221no—
333C4×D4—1221no—
334C4×D4—1221no—
335C4×D4—1221no—
336#117C22×D6112yes—
337C2×C4⋊C4—611no—
338C2×C22⋊C4—1221no—
339C2×C42C22×D6111yescharacteristic
340#108—2442no—
341#107—1222no—
342#106—1222no—
343C4⋊C8—22no—
344C4⋊C8—22no—
345C4⋊C8—1221no—
346C4⋊C8—1221no—
347C4⋊C8C22⋊D6111yescharacteristic
348C4⋊C8C22⋊D6111yescharacteristic
349C4⋊C8C22⋊D6112yes—
350#102—1222no—
351#102—1222no—
352#102—1222no—
353#102—1222no—
354#102—1222no—
355#102—1222no—
356#102—1222no—
357#102—1222no—
358#100—1222no—
359#100—1222no—
360#100—1224no—
361#100—2442no—
362#97—22no—
363#97—21no—
364#97—1221no—
365C4×C8—22no—
366C4×C8—21no—
367C4×C8—1221no—
368C24—1221no—
369C24—1222no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×D4—612no—
373C2×D4—1221no—
374C2×D4—1222no—
375C2×D4—1221no—
376C2×D4—1222no—
377C2×D4—1222no—
378C2×D4—1221no—
379C2×D4—1222no—
380C2×D4—1221no—
381C2×D4—1222no—
382C2×D4—1221no—
383C2×D4—1222no—
384C2×D4—4882no—
385C2×D4—4882no—
386C2×D4—2441no—
387C2×D4—2441no—
388C2×D4—2441no—
389C2×D4—2441no—
390C22×C4—1221no—
391C22×C4—4882no—
392C22×C4—2441no—
393C22×C4—2441no—
394C22×C4#1175822211yescharacteristic
395C22×C4#1175822211yescharacteristic
396C22×C4C23×D6111yescharacteristic
397D8—2444no—
398D8—2444no—
399M16—21no—
400M16—21no—
401M16—21no—
402M16—1221no—
403M16—1222no—
404M16—1221no—
405M16—1222no—
406M16—1221no—
407M16—1221no—
408M16#1175822612yes—
409C2×C8—21no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—21no—
413C2×C8—1221no—
414C2×C8—1221no—
415C2×C8—1222no—
416C2×C8C2×C22⋊D6112yes—
417C2×C8C2×C22⋊D6112yes—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4C2×D24412yes—
425C4⋊C4C2×D24412yes—
426C22⋊C4—22no—
427C22⋊C4—2441no—
428C22⋊C4—2441no—
429C42—22no—
430C42C2×D24411yescharacteristic
431C42C2×D24411yescharacteristic
432C42C2×C22⋊D6111yescharacteristic
433C42C2×C22⋊D6111yescharacteristic
434C23—1222no—
435C23—4882no—
436C23—4882no—
437C23—4881no—
438C23—4881no—
439C23—2442no—
440C23—2442no—
441C23—2442no—
442C23—2441no—
443C23—2441no—
444C23C2×#1175822211yescharacteristic
445Q8—42no—
446Q8—81no—
447D4—1222no—
448D4—1222no—
449D4—4882no—
450D4—4882no—
451D4—4882no—
452D4—4882no—
453D4—2442no—
454D4—2442no—
455D4—2441no—
456D4—2442no—
457D4—2442no—
458D4—2441no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—4882no—
464C2×C4—4882no—
465C2×C4—4881no—
466C2×C4—2441no—
467C2×C4#1451975111yescharacteristic
468C2×C4#1451975111yescharacteristic
469C2×C4#1451975111yescharacteristic
470C2×C4#1451975111yescharacteristic
471C2×C4C22×C22⋊D6111yescharacteristic
472C2×C4C22×D24411yescharacteristic
473C2×C4#41×D6111yescharacteristic
474C2×C4C2×#1175822611yescharacteristic
475C2×C4#1451982311yescharacteristic
476C2×C4#1451982511yescharacteristic
477C8—22no—
478C8—22no—
479C8—22no—
480C8—41no—
481C8—2441no—
482C8—2441no—
483C8—2442no—
484C22—21no—
485C22—21no—
486C22—9762no—
487C22—9762no—
488C22—4882no—
489C22—4882no—
490C22—4882no—
491C22—4882no—
492C22—4882no—
493C22—4881no—
494C22—4881no—
495C22—2442no—
496C22C2×#1451975111yescharacteristic
497C22#1684029911yescharacteristic
498C22#1684038411yescharacteristic
499C4—21no—
500C4—21no—
501C4—21no—
502C4—42no—
503C4—42no—
504C4—4881no—
505C4#1684003212yes—
506C4#1684040011yescharacteristic
507C4#1684042311yescharacteristic
508C4#1684042411yescharacteristic
509C4#1684043411yescharacteristic
510C2—21no—
511C2—21no—
512C2—9762no—
513C2—4882no—
514C2#2144937911yescharacteristic
515C2C2×#1684003211yescharacteristic
516C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2962C2
socleC22×C61C2×#121
prime radicalC2×#16839958C2
hypercenter#2962D61
hypocenterC61#25312
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
1. #21456195C2semidirect
2. #21455612C2semidirect
4. #21447178C2semidirect
5. #21447075C2semidirect
6. #21446729C2semidirect
7. #21446659C2semidirect
8. #21446595C2semidirect
9. #21446481C2semidirect
10. C4×#14519693C2semidirect
11. C61×#2962C2semidirect
255. #2962D61semidirect
305. C61#25312semidirect

Maximal subgroups

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

subgroupindex
#214561952
#214556122
C2×#168399582
#214471782
#214470752
#214467292
#214466592
#214465952
#214464812
C4×#145196932
C61×#29622
#2531261