Finite Group Databasebeta

← #29722546

Subgroups — #29722546

Order 15616.

858 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455443C212yesmaximal
2C2×#16840000C211yesmaximal, characteristic, complemented
3#21454511C211yesmaximal, characteristic
4C2×#108×D61C211yesmaximal, characteristic, complemented
5#21447182C212yesmaximal, complemented
6#21447115C212yesmaximal, complemented
7#21446735C211yesmaximal, characteristic, complemented
8#21446605C211yesmaximal, characteristic, complemented
9#21446284C212yesmaximal, complemented
10#21445883C211yesmaximal, characteristic, complemented
11C2×C61×#473C211yesmaximal, characteristic, complemented
12C2×M16×D61C2212yes—
13C22×C8×D61C2211yescharacteristic
14#16840269C2211yescharacteristic
15C2×#14519753C2212yes—
16#16840017—22no—
17#16840005—22no—
18#16840000—21no—
19#16840000—21no—
20#16840000—21no—
21#16840000—21no—
22#16840000—22no—
23#16839988—22no—
24#16839831—21no—
25#16839831—21no—
26C8×C22⋊D61—21no—
27C8×C22⋊D61—21no—
28C2×#14519713C2211yescharacteristic
29C2×#14519712C2212yes—
30#108×D61—24no—
31#16839804C2212yes—
32C2×#14519711C2211yescharacteristic
33#16839790C2211yescharacteristic
34#16839771C2212yes—
35#16839763C2212yes—
36C2×#14519701C2212yes—
37C2×#14519696C2212yes—
38#16839729C2211yescharacteristic
39#16839725C2211yescharacteristic
40#16839717C2211yescharacteristic
41#16839530C2212yes—
42#16839530C2212yes—
43C2×#137×C61C2212yes—
44C22×Q16×C61C2211yescharacteristic
45C2×#131×C61C2211yescharacteristic
46C61×#473—24no—
47C2×#108×C61C2211yescharacteristic
48C2×#101×C61C2212yes—
49C22×C4×D61C2311yescharacteristic
50C2×#11758194C2311yescharacteristic
51C2×#11758194C2312yes—
52C2×#11758189—22no—
53M16×D61—24no—
54M16×D61—42no—
55C22×#92418C2311yescharacteristic
56C2×C8×D61—22no—
57C2×C8×D61—22no—
58C2×C8×D61—22no—
59C2×C8×D61D411yescharacteristic
60C2×C8×D61D411yescharacteristic
61C2×C8×D61D411yescharacteristic
62C2×C8×D61D411yescharacteristic
63#14519758—21no—
64#14519753—22no—
65#14519753—22no—
66#14519753—22no—
67#14519753—22no—
68#14519753D411yescharacteristic
69#14519753D411yescharacteristic
70#14519748D411yescharacteristic
71#14519748D411yescharacteristic
72#14519745—21no—
73C4×C22⋊D61—21no—
74C4×C22⋊D61—21no—
75#14519736C2311yescharacteristic
76C2×C4⋊Dic61C2312yes—
77C2×#11758196—22no—
78C2×#11758198—22no—
79#14519713—22no—
80#14519712—24no—
81#14519711—21no—
82#14519711—21no—
83#14519711—22no—
84#14519709—22no—
85#14519706—21no—
86#14519706—21no—
87#14519701—22no—
88#14519701—22no—
89#14519701—22no—
90#14519701—22no—
91#14519701—22no—
92#14519698—22no—
93#14519696—42no—
94C22⋊#92418—21no—
95C22⋊#92418—21no—
96#14519689—21no—
97#14519689—21no—
98#14519687—21no—
99C8×Dic61—21no—
100C2×#41×C61C2311yescharacteristic
101C22×Q8×C61C2312yes—
102#137×C61—24no—
103#137×C61—42no—
104C2×Q16×C61—22no—
105C2×Q16×C61—21no—
106C2×Q16×C61—21no—
107C2×Q16×C61—22no—
108C2×Q16×C61—22no—
109C2×SD16×C61—22no—
110#131×C61—22no—
111#131×C61—22no—
112C2×M16×C61—21no—
113C2×M16×C61C2311yescharacteristic
114C2×M16×C61C2312yes—
115C22×C8×C61—21no—
116C22×C8×C61C2311yescharacteristic
117#108×C61D414yescomplemented
118#101×C61—24no—
119C23×D61#4111yescharacteristic
120#41×C61—21no—
121#41×C61—21no—
122#41×C61—21no—
123#41×C61—21no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61C2×D411yescharacteristic
131C2×D4×C61—21no—
132C2×D4×C61C2×D411yescharacteristic
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61C2411yescharacteristic
136C2×C22⋊D61—21no—
137C2×C22⋊D61—21no—
138C22×Dic61—22no—
139C22×Dic61#4111yescharacteristic
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61C2×D411yescharacteristic
145C2×C4×D61C2×D411yescharacteristic
146C2×C4×D61C2×D411yescharacteristic
147C2×C4×D61C2×D411yescharacteristic
148Q16×C61—44no—
149Q16×C61—42no—
150Q16×C61—42no—
151SD16×C61—44no—
152M16×C61—22no—
153M16×C61—21no—
154M16×C61—21no—
155M16×C61—42no—
156M16×C61C2×D414yes—
157C2×C8×C61—22no—
158C2×C8×C61—21no—
159C2×C8×C61—42no—
160C2×C8×C61C2×D411yescharacteristic
161C2×C8×C61C2×D411yescharacteristic
162C2×C8×C61C2×D412yes—
163C2×C8×C61#4112yes—
164C22⋊Dic61—21no—
165C22⋊Dic61—21no—
166C22⋊Dic61—21no—
167#11758198—21no—
168#11758198—21no—
169#11758198—42no—
170#11758198—42no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61C2×D411yescharacteristic
174C4⋊Dic61C2×D411yescharacteristic
175#11758196—21no—
176#11758196—21no—
177#11758196—42no—
178#11758196—42no—
179C4×Dic61—21no—
180#11758194—22no—
181#11758194—21no—
182#11758194—22no—
183#11758194—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—22no—
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190#11758189—44no—
191C8×D61—22no—
192C8×D61—22no—
193C8×D61—22no—
194C8×D61—22no—
195C8×D61—44no—
196C23×C61—21no—
197C23×C61C2×#4111yescharacteristic
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—22no—
201C22×D61—41no—
202C22×D61#12111yescharacteristic
203C22×D61#12111yescharacteristic
204Q8×C61—21no—
205Q8×C61—21no—
206Q8×C61—42no—
207Q8×C61—42no—
208Q8×C61—42no—
209Q8×C61—42no—
210D4×C61—21no—
211D4×C61—21no—
212D4×C61—41no—
213D4×C61—41no—
214C2×C4×C61—21no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61H(5,𝔽2)11yescharacteristic
225C2×C4×C612−1+411yescharacteristic
226C22⋊D61—42no—
227C22⋊D61—42no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—42no—
233C2×Dic61—82no—
234C2×Dic61#12111yescharacteristic
235C2×Dic61#12111yescharacteristic
236C4×D61—21no—
237C4×D61—21no—
238C4×D61—21no—
239C4×D61—21no—
240C4×D61—21no—
241C4×D61—21no—
242C4×D61—21no—
243C4×D61—21no—
244C8×C61—24no—
245C8×C61—41no—
246C8×C61#12112yes—
247C8×C61#12112yes—
248#92418—21no—
249#92418—21no—
250#92418—41no—
251#92418—42no—
252#24804—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61C2×#12111yescharacteristic
259C22×C61#54811yescharacteristic
260C22×C61#55311yescharacteristic
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61#47312yescomplemented
269C4×C61—41no—
270C4×C61—82no—
271C4×C61C2×#12111yescharacteristic
272C4×C61#53711yescharacteristic
273C4×C61D4211yescharacteristic
274C4×C61#54711yescharacteristic
275Dic61—21no—
276Dic61—21no—
277Dic61—41no—
278Dic61—41no—
279Dic61—82no—
280C2×#473D6111yescharacteristic, complemented
281#3056—612no—
282C2×#467—611no—
283#2929—611no—
284C22×#108—611no—
285#2064—612no—
286#2052—612no—
287#1979—611no—
288#1952—611no—
289#1901—612no—
290#1816—611no—
291C2×C61—21no—
292C2×C61—21no—
293C2×C61—81no—
294C2×C61#241611yescharacteristic
295C2×C61C2×#47311yescharacteristic
296C2×C61#306711yescharacteristic
297D61—22no—
298D61—41no—
299C2×#137C2×D6112yes—
300C22×Q16C2×D6111yescharacteristic
301C2×#131C2×D6111yescharacteristic
302C22×M16—612no—
303C23×C8—611no—
304#525—611no—
305C2×C22⋊Q8—612no—
306#473—24no—
307#470—1222no—
308#468—1222no—
309#467—1222no—
310#467—1221no—
311#467—1221no—
312#467—1221no—
313#467—1221no—
314#465—1222no—
315#435—1221no—
316#435—1221no—
317C8×D4—1221no—
318C8×D4—1221no—
319C2×#108—612no—
320C2×#108—611no—
321C2×#108—1224no—
322C2×#108C2×D6111yescharacteristic
323#428—612no—
324C2×#107—611no—
325#424—611no—
326#419—612no—
327#417—612no—
328C2×#103—612no—
329C2×#101—612no—
330C2×#101C2×D6112yes—
331#408—611no—
332#407—611no—
333#405—611no—
334#343—612no—
335#343—612no—
336C61#2480411yescharacteristic, complemented, Sylow
337C2×#41C22×D6111yescharacteristic
338C22×Q8C22×D6112yes—
339C23×C4—611no—
340#137—24no—
341#137—42no—
342C2×Q16—22no—
343C2×Q16—21no—
344C2×Q16—21no—
345C2×Q16—22no—
346C2×Q16—22no—
347C2×SD16—22no—
348#131—22no—
349#131—22no—
350C2×M16—21no—
351C2×M16—612no—
352C2×M16—611no—
353C2×M16—1222no—
354C2×M16—1224no—
355C2×M16—2442no—
356C2×M16C22×D6111yescharacteristic
357C2×M16C22×D6112yes—
358C22×C8—21no—
359C22×C8—611no—
360C22×C8—611no—
361C22×C8—611no—
362C22×C8—611no—
363C22×C8—611no—
364C22×C8—1222no—
365C22×C8—1222no—
366C22×C8—1222no—
367C22×C8C22×D6111yescharacteristic
368#123—1221no—
369C22⋊Q8—611no—
370C22⋊Q8—611no—
371C22⋊Q8—1222no—
372C22⋊Q8—1222no—
373C22⋊Q8—1222no—
374C22⋊Q8—1222no—
375#121—611no—
376#121—611no—
377#120—1221no—
378C4×D4—1221no—
379C4×D4—1221no—
380#117—611no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—1222no—
383C2×C22⋊C4—1222no—
384#108—1224no—
385#108—1222no—
386#108C22⋊D6114yescomplemented
387#107—1222no—
388#107—1221no—
389#107—1221no—
390#106—1222no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393#103—1222no—
394#103—1222no—
395#103—1222no—
396#103—1222no—
397#103—1222no—
398#102—1222no—
399#101—24no—
400#101—2442no—
401C22⋊C8—1221no—
402C22⋊C8—1221no—
403C22⋊C8—1221no—
404C22⋊C8—1221no—
405#97—1221no—
406C4×C8—1221no—
407C24—611no—
408#41—21no—
409#41—21no—
410#41—21no—
411#41—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8D4×D6111yescharacteristic
419C2×D4—21no—
420C2×D4—1221no—
421C2×D4—1221no—
422C2×D4D4×D6111yescharacteristic
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1222no—
435C22×C4C23×D6111yescharacteristic
436Q16—44no—
437Q16—42no—
438Q16—42no—
439SD16—44no—
440M16—22no—
441M16—21no—
442M16—21no—
443M16—42no—
444M16—1222no—
445M16—1221no—
446M16—1222no—
447M16—1221no—
448M16—2444no—
449M16C2×C22⋊D6114yes—
450C2×C8—22no—
451C2×C8—21no—
452C2×C8—42no—
453C2×C8—611no—
454C2×C8—611no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1222no—
458C2×C8—1222no—
459C2×C8—1222no—
460C2×C8—1222no—
461C2×C8—1221no—
462C2×C8—1222no—
463C2×C8—2444no—
464C2×C8D4×D6111yescharacteristic
465C2×C8D4×D6111yescharacteristic
466C2×C8#1175822412yes—
467C2×C8C2×C22⋊D6112yes—
468C4⋊C4—611no—
469C4⋊C4—611no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—2442no—
475C4⋊C4—2442no—
476C22⋊C4—1221no—
477C22⋊C4—1221no—
478C22⋊C4—1221no—
479C22⋊C4—1221no—
480C22⋊C4—1221no—
481C22⋊C4—2442no—
482C22⋊C4—2442no—
483C42—1221no—
484C23—21no—
485C23—611no—
486C23—611no—
487C23—1222no—
488C23—1221no—
489C23—1221no—
490C23—2441no—
491C23#41×D6111yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—42no—
495Q8—42no—
496Q8—42no—
497Q8—42no—
498D4—21no—
499D4—21no—
500D4—41no—
501D4—41no—
502D4—2442no—
503D4—2442no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—611no—
511C2×C4—611no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—4882no—
525C2×C4—2442no—
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C22×C22⋊D6111yescharacteristic
529C2×C4C2×#1175822411yescharacteristic
530C2×C4#1451982211yescharacteristic
531C2×C4#1451982411yescharacteristic
532C8—24no—
533C8—41no—
534C8—1221no—
535C8—1221no—
536C8—2441no—
537C8—2442no—
538C8#1451974812yes—
539C8#1451974812yes—
540C22—21no—
541C22—21no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—612no—
546C22—1221no—
547C22—1221no—
548C22—1221no—
549C22—1221no—
550C22—1221no—
551C22—2442no—
552C22—2442no—
553C22#121×D6111yescharacteristic
554C22#1684041911yescharacteristic
555C22#1684045911yescharacteristic
556C4—41no—
557C4—82no—
558C4—1221no—
559C4—1221no—
560C4—4882no—
561C4—2441no—
562C4—2441no—
563C4C2×#1451974811yescharacteristic
564C4#1684034111yescharacteristic
565C4D4×C22⋊D6111yescharacteristic
566C4#1684041211yescharacteristic
567C2—21no—
568C2—21no—
569C2—81no—
570C2—1222no—
571C2—2441no—
572C2#2144925511yescharacteristic
573C2#473×D6111yescharacteristic
574C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#473C2
socleC22×C61C2×#121
prime radical#16840269C22
hypercenterC2×#473D61
hypocenterC61#24804
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#473

Complements

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

NH ≅ G/Ntype
2. C2×#16840000C2semidirect
4. C2×#108×D61C2semidirect
5. #21447182C2semidirect
6. #21447115C2semidirect
7. #21446735C2semidirect
8. #21446605C2semidirect
9. #21446284C2semidirect
10. #21445883C2semidirect
11. C2×C61×#473C2semidirect
117. #108×C61D4semidirect
268. C2×D61#473semidirect
280. C2×#473D61semidirect
336. C61#24804semidirect
386. #108C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214554432
C2×#168400002
#214545112
C2×#108×D612
#214471822
#214471152
#214467352
#214466052
#214462842
#214458832
C2×C61×#4732
#2480461