Finite Group Databasebeta

← #29731020

Subgroups — #29731020

Order 15616.

884 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
1#21456276C211yesmaximal, characteristic
2C2×#16840034C212yesmaximal, complemented
3#21455291C211yesmaximal, characteristic
4#21454562C211yesmaximal, characteristic
5#21447119C212yesmaximal, complemented
6#21446938C212yesmaximal, complemented
7#21446664C211yesmaximal, characteristic, complemented
8#21446595C211yesmaximal, characteristic, complemented
9#21446337C212yesmaximal, complemented
10C4×#14519696C211yesmaximal, characteristic, complemented
11C2×C61×#462C211yesmaximal, characteristic, complemented
12C2×#14519808C2212yes—
13C2×#14519808C2212yes—
14C2×#14519808C2212yes—
15C2×#14519783C2212yes—
16#16840312C2211yescharacteristic
17#16840154—22no—
18#16840141—21no—
19#16840136—21no—
20#16840034—28no—
21#16839984—22no—
22#16839972—21no—
23#16839962—21no—
24#16839839—21no—
25#16839832—22no—
26#16839827—21no—
27C2×#14519713C2212yes—
28#16839786C2211yescharacteristic
29#16839761C2212yes—
30C2×#14519696C2212yes—
31C2×#14519696C2211yescharacteristic
32C2×#14519696C2212yes—
33C2×#14519696C2211yescharacteristic
34#16839728C2212yes—
35C4×#11758194C2211yescharacteristic
36C4×#11758194C2211yescharacteristic
37#16839531C2211yescharacteristic
38#16839531C2211yescharacteristic
39#16839527C2212yes—
40C22×Q16×C61C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C61×#462—21no—
44C61×#462—22no—
45C61×#462—21no—
46C2×C4⋊C8×C61C2211yescharacteristic
47C2×#103×C61C2212yes—
48C2×#11758222C2312yes—
49#14519808—24no—
50#14519808—24no—
51#14519808—24no—
52#14519808—42no—
53#14519808—42no—
54#14519808—42no—
55C2×#11758202—22no—
56C2×#11758202—22no—
57C2×#11758202—22no—
58C2×#11758201—22no—
59C2×#11758201—22no—
60C2×#11758201—22no—
61#14519783—24no—
62#14519783—42no—
63C2×#11758194C2311yescharacteristic
64C2×#11758194C2312yes—
65C2×#11758194C2311yescharacteristic
66C2×#11758194C2311yescharacteristic
67C2×#11758194C2311yescharacteristic
68C2×#11758189—22no—
69C2×C8×D61—22no—
70#14519777D411yescharacteristic
71#14519774D411yescharacteristic
72#14519765D412yes—
73#14519751—21no—
74#14519751—21no—
75C4×D244—21no—
76C4×D244—21no—
77#14519713—24no—
78C4⋊#92418—21no—
79C4⋊#92418—21no—
80#14519700—22no—
81#14519697—22no—
82#14519696—24no—
83#14519696—22no—
84#14519696—42no—
85#14519696D414yescomplemented
86#14519689—22no—
87#14519689—22no—
88#14519688—21no—
89#14519688—21no—
90C4×#92418—21no—
91C4×#92418—21no—
92C22×Q8×C61C2312yes—
93C22×Q8×C61C2311yescharacteristic
94C2×Q16×C61—24no—
95C2×Q16×C61—22no—
96C2×Q16×C61—22no—
97C2×Q16×C61—42no—
98C22×C8×C61C2312yes—
99C4⋊Q8×C61—21no—
100C4⋊Q8×C61—21no—
101C4⋊Q8×C61—21no—
102C4⋊Q8×C61—21no—
103C4×Q8×C61—21no—
104C4×Q8×C61—21no—
105C4×Q8×C61—21no—
106C4×Q8×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C42×C61—21no—
112C2×C42×C61C2311yescharacteristic
113C4⋊C8×C61D411yescharacteristic
114C4⋊C8×C61D412yes—
115C4⋊C8×C61D411yescharacteristic
116#103×C61—22no—
117#103×C61—22no—
118C2×Q8×C61—21no—
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61C2×D412yes—
130C22×C4×C61—21no—
131C22×C4×C61—22no—
132C22×C4×C61—21no—
133C22×C4×C61#4111yescharacteristic
134C22×C4×C61#4111yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C2×C22⋊D61—22no—
137#11758222—22no—
138#11758222—22no—
139#11758222—22no—
140#11758222—22no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244C2×D412yes—
144C2×C4×D61—22no—
145C2×Dic122C2×D412yes—
146Q16×C61—44no—
147Q16×C61—44no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61C2×D412yes—
151C2×C8×C61C2×D412yes—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—41no—
155C4⋊C4×C61—41no—
156C4⋊C4×C61—41no—
157C4⋊C4×C61—41no—
158C4⋊C4×C61—41no—
159C4⋊C4×C61—41no—
160C42×C61—41no—
161C42×C61—41no—
162C42×C61C2×D411yescharacteristic
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166#11758202—44no—
167#11758202—44no—
168#11758202—44no—
169#11758201—44no—
170#11758201—44no—
171#11758201—44no—
172#11758198—22no—
173#11758198—22no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176#11758194—21no—
177#11758194—21no—
178#11758194—21no—
179#11758194—22no—
180#11758194—21no—
181#11758194—22no—
182#11758194—21no—
183#11758194C2×D412yes—
184#11758194C2×D412yes—
185#11758194#4112yes—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—22no—
191#11758189—44no—
192C8×D61—44no—
193C23×C61C2×#4111yescharacteristic
194C22×D61—22no—
195Q8×C61—22no—
196Q8×C61—22no—
197Q8×C61—41no—
198Q8×C61—41no—
199Q8×C61—42no—
200Q8×C61—42no—
201Q8×C61—42no—
202Q8×C61—42no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—81no—
213C2×C4×C61#12111yescharacteristic
214C2×C4×C61#12111yescharacteristic
215C2×C4×C61#12111yescharacteristic
216C2×C4×C61#12111yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C22⋊D61—42no—
224C22⋊D61—42no—
225C2×Dic61—22no—
226D244—22no—
227D244—22no—
228D244—42no—
229D244—81no—
230C4×D61—42no—
231C4×D61—42no—
232Dic122—22no—
233Dic122—22no—
234C8×C61—22no—
235C8×C61—22no—
236#92418—22no—
237#92418—22no—
238#92418—22no—
239#92418—42no—
240#92418—41no—
241#25406—611nomaximal, Sylow
242C22×C61—21no—
243C22×C61—21no—
244C22×C61C2×#12111yescharacteristic
245C22×C61#53211yescharacteristic
246C22×C61#54011yescharacteristic
247C2×D61—42no—
248C2×D61—42no—
249C2×D61—42no—
250C4×C61—21no—
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—42no—
254C4×C61—82no—
255C4×C61—81no—
256C4×C61#47312yescomplemented
257C4×C61D4211yescharacteristic
258C4×C61D4211yescharacteristic
259C4×C61#54611yescharacteristic
260C4×C61#55011yescharacteristic
261Dic61—42no—
262#3152—611no—
263C2×#473—612no—
264#3037—611no—
265C2×#462D6111yescharacteristic, complemented
266#2935—611no—
267#2052—612no—
268#2020—612no—
269#1966—611no—
270#1950—611no—
271#1908—612no—
272C4×#101—611no—
273C2×C61—21no—
274C2×C61—21no—
275C2×C61#242811yescharacteristic
276C2×C61C2×#47311yescharacteristic
277C2×C61#306911yescharacteristic
278D61—82no—
279C2×#137—612no—
280C2×#137—612no—
281C2×#137—612no—
282C22×Q16C2×D6112yes—
283C2×#131—612no—
284#532—611no—
285C2×C4⋊Q8C2×D6111yescharacteristic
286C2×C4×Q8C2×D6111yescharacteristic
287#497—1222no—
288#494—1221no—
289#492—1221no—
290#473—1228no—
291#464—1222no—
292#462—21no—
293#462—22no—
294#462—21no—
295#462—1221no—
296#460—1221no—
297#436—1221no—
298#435—1222no—
299C8×D4—1221no—
300C2×#108—612no—
301#423—611no—
302C2×C4⋊C8C2×D6111yescharacteristic
303#417—612no—
304C2×#103C2×D6112yes—
305C2×#101—611no—
306C2×#101—612no—
307C2×#101—611no—
308C2×#101—612no—
309#408—612no—
310C4×M16—611no—
311C4×M16—611no—
312#343—611no—
313#343—611no—
314#341—612no—
315C61#2540611yescharacteristic, complemented, Sylow
316C2×#41—612no—
317C22×Q8C22×D6112yes—
318C22×Q8C22×D6111yescharacteristic
319#137—1224no—
320#137—1224no—
321#137—1224no—
322#137—2442no—
323#137—2442no—
324#137—2442no—
325C2×Q16—24no—
326C2×Q16—22no—
327C2×Q16—22no—
328C2×Q16—42no—
329C2×Q16—1222no—
330C2×Q16—1222no—
331C2×Q16—1222no—
332C2×SD16—1222no—
333C2×SD16—1222no—
334C2×SD16—1222no—
335#131—1224no—
336#131—2442no—
337C2×M16—611no—
338C2×M16—611no—
339C2×M16—611no—
340C2×M16—612no—
341C2×M16—611no—
342C2×M16—1222no—
343C22×C8—1222no—
344C22×C8C22×D6112yes—
345C4⋊Q8—21no—
346C4⋊Q8—21no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8—611no—
350#127—611no—
351#124—612no—
352#121—1221no—
353#121—1221no—
354C4×Q8—21no—
355C4×Q8—21no—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×D4—1221no—
359C4×D4—1221no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4C22×D6111yescharacteristic
364C2×C42—21no—
365C2×C42C22×D6111yescharacteristic
366#108—1224no—
367C4⋊C8—1221no—
368C4⋊C8—1221no—
369C4⋊C8C22⋊D6111yescharacteristic
370C4⋊C8C22⋊D6112yes—
371C4⋊C8C22⋊D6111yescharacteristic
372#103—22no—
373#103—22no—
374#103—1222no—
375#102—1222no—
376#101—614no—
377#101—1222no—
378#101—1224no—
379#101—2442no—
380C22⋊C8—1222no—
381C22⋊C8—1222no—
382#97—1221no—
383#97—1221no—
384C4×C8—1221no—
385C4×C8—1221no—
386#41—1222no—
387#41—1222no—
388#41—1222no—
389#41—1222no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—612no—
402C2×Q8D4×D6112yes—
403C2×D4—612no—
404C2×D4—1222no—
405C2×D4—1221no—
406C2×D4—1221no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—1222no—
411C22×C4#1175822411yescharacteristic
412C22×C4#1175822411yescharacteristic
413C22×C4C23×D6111yescharacteristic
414Q16—44no—
415Q16—44no—
416Q16—2444no—
417Q16—2444no—
418Q16—2444no—
419SD16—2444no—
420SD16—2444no—
421SD16—2444no—
422M16—612no—
423M16—612no—
424M16—612no—
425M16—1221no—
426M16—1221no—
427M16—1221no—
428M16—1222no—
429M16—1221no—
430M16—1221no—
431M16—1222no—
432M16—2444no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—1221no—
436C2×C8—1221no—
437C2×C8—1221no—
438C2×C8—1222no—
439C2×C8—1221no—
440C2×C8—2444no—
441C2×C8C2×C22⋊D6112yes—
442C2×C8C2×C22⋊D6112yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C22⋊C4—1222no—
454C22⋊C4—1222no—
455C42—41no—
456C42—41no—
457C42D4×D6111yescharacteristic
458C42D4×D6111yescharacteristic
459C42C2×C22⋊D6111yescharacteristic
460C42C2×C22⋊D6111yescharacteristic
461C23—1222no—
462C23C2×#1175822411yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—41no—
466Q8—41no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470Q8—42no—
471Q8—1222no—
472Q8—1222no—
473D4—1222no—
474D4—1222no—
475D4—4881no—
476D4—2442no—
477D4—2442no—
478D4—2442no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—81no—
489C2×C4—1222no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4#1451974811yescharacteristic
493C2×C4#1451974811yescharacteristic
494C2×C4#1451974811yescharacteristic
495C2×C4#1451974811yescharacteristic
496C2×C4C2×D4×D6111yescharacteristic
497C2×C4C2×D4×D6111yescharacteristic
498C2×C4C22×C22⋊D6111yescharacteristic
499C2×C4#41×D6111yescharacteristic
500C2×C4#1451982211yescharacteristic
501C2×C4#1451982311yescharacteristic
502C8—22no—
503C8—22no—
504C8—1222no—
505C8—1222no—
506C8—1222no—
507C8—2442no—
508C8—2441no—
509C22—21no—
510C22—21no—
511C22—2442no—
512C22—2442no—
513C22—2442no—
514C22C2×#1451974811yescharacteristic
515C22#1684030711yescharacteristic
516C22#1684036511yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—42no—
521C4—82no—
522C4—81no—
523C4—2442no—
524C4#1684003412yescomplemented
525C4D4×C22⋊D6111yescharacteristic
526C4#1684039611yescharacteristic
527C4#1684040911yescharacteristic
528C4#1684043111yescharacteristic
529C2—21no—
530C2—21no—
531C2—4882no—
532C2#2144940311yescharacteristic
533C2C2×#1684003411yescharacteristic
534C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#462C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenterC2×#462D61
hypocenterC61#25406
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
2. C2×#16840034C2semidirect
5. #21447119C2semidirect
6. #21446938C2semidirect
7. #21446664C2semidirect
8. #21446595C2semidirect
9. #21446337C2semidirect
10. C4×#14519696C2semidirect
11. C2×C61×#462C2semidirect
85. #14519696D4semidirect
256. C4×C61#473semidirect
265. C2×#462D61semidirect
315. C61#25406semidirect
524. C4#16840034semidirect

Maximal subgroups

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

subgroupindex
#214562762
C2×#168400342
#214552912
#214545622
#214471192
#214469382
#214466642
#214465952
#214463372
C4×#145196962
C2×C61×#4622
#2540661