Finite Group Databasebeta

← #19137912

Subgroups — #19137912

Order 5504.

1056 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10621898C211yesmaximal, characteristic, complemented
2C2×#13063295C211yesmaximal, characteristic, complemented
3C2×#13063293C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D43C211yesmaximal, characteristic, complemented
5#14582812C211yesmaximal, characteristic, complemented
6#14582811C211yesmaximal, characteristic, complemented
7#14582808C211yesmaximal, characteristic, complemented
8#14582807C211yesmaximal, characteristic, complemented
9#14582778C211yesmaximal, characteristic, complemented
10#14582777C211yesmaximal, characteristic, complemented
11#14582777C211yesmaximal, characteristic, complemented
12#14582775C211yesmaximal, characteristic, complemented
13#14582775C211yesmaximal, characteristic, complemented
14#14582761C211yesmaximal, characteristic, complemented
15C43×#386C211yesmaximal, characteristic, complemented
16C24×D43C2211yescharacteristic
17C2×#10621898—24no—
18C2×#10621898—24no—
19C2×#10621898—24no—
20C22×Dic86C2211yescharacteristic, complemented
21C22×D172C2211yescharacteristic
22C22×D172C2211yescharacteristic
23C22×C22⋊D43C2211yescharacteristic
24C22×C22⋊D43C2211yescharacteristic
25C22×C22⋊D43C2211yescharacteristic
26C22×C22⋊D43C2211yescharacteristic, complemented
27C22×C4×D43C2211yescharacteristic
28C22×C4×D43C2211yescharacteristic
29C22×C4×D43C2211yescharacteristic, complemented
30#13063295—24no—
31#13063293—24no—
32C2×C22⋊Dic43C2211yescharacteristic
33C2×C22⋊Dic43C2211yescharacteristic
34C2×C22⋊Dic43C2211yescharacteristic, complemented
35C2×(C2×D43)⋊C4C2211yescharacteristic
36C2×(C2×D43)⋊C4C2211yescharacteristic
37C2×(C2×D43)⋊C4C2211yescharacteristic
38C2×(C2×D43)⋊C4C2211yescharacteristic
39C2×(C2×D43)⋊C4C2211yescharacteristic
40C2×(C2×D43)⋊C4C2211yescharacteristic
41C2×(C2×D43)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D43)⋊C4C2211yescharacteristic, complemented
43C22⋊C4×D43—24no—
44C43⋊#95C2211yescharacteristic, complemented
45C43⋊#95C2211yescharacteristic, complemented
46C43⋊#95C2211yescharacteristic, complemented
47C43⋊#95C2211yescharacteristic, complemented
48C43⋊#95C2211yescharacteristic, complemented
49C43⋊#95C2211yescharacteristic, complemented
50C22×D4×C43C2211yescharacteristic
51C23×C4×C43C2211yescharacteristic, complemented
52C2×C22⋊C4×C43C2211yescharacteristic
53C2×C22⋊C4×C43C2211yescharacteristic
54C2×C22⋊C4×C43C2211yescharacteristic, complemented
55#95×C43C2211yescharacteristic, complemented
56#95×C43C2211yescharacteristic, complemented
57C24×C43C2311yescharacteristic
58C24×C43C2311yescharacteristic
59C23×D43—22no—
60C23×D43—22no—
61C23×D43—22no—
62C23×D43—22no—
63C23×D43—22no—
64C23×D43—41no—
65C23×D43—41no—
66C23×D43C2311yescharacteristic
67C23×D43C2311yescharacteristic
68C23×D43C2311yescharacteristic
69C2×D4×C43—22no—
70C2×D4×C43—22no—
71C2×D4×C43—41no—
72C22×C4×C43—22no—
73C22×C4×C43—22no—
74C22×C4×C43—22no—
75C22×C4×C43C2311yescharacteristic
76C22×C4×C43C2311yescharacteristic
77C22×C4×C43C2311yescharacteristic
78C22×C4×C43C2311yescharacteristic
79C22×C4×C43C2311yescharacteristic
80C2×C22⋊D43—22no—
81C2×C22⋊D43—22no—
82C2×C22⋊D43—22no—
83C2×C22⋊D43—22no—
84C2×C22⋊D43—22no—
85C2×C22⋊D43—22no—
86C2×C22⋊D43—22no—
87C2×C22⋊D43—22no—
88C2×C22⋊D43—22no—
89C2×C22⋊D43—41no—
90C2×C22⋊D43—41no—
91C2×C22⋊D43D414yescomplemented
92C22×Dic43C2311yescharacteristic
93C22×Dic43C2311yescharacteristic
94C22×Dic43C2311yescharacteristic
95C22×Dic43C2311yescharacteristic
96C22×Dic43C2311yescharacteristic
97#10621898—232no—
98C2×D172—22no—
99C2×D172—22no—
100C2×D172—22no—
101C2×D172—22no—
102C2×D172—41no—
103C2×D172D414yescomplemented
104C2×C4×D43—22no—
105C2×C4×D43—22no—
106C2×C4×D43—22no—
107C2×C4×D43—22no—
108C2×C4×D43—22no—
109C2×C4×D43—22no—
110C2×C4×D43—41no—
111C2×C4×D43—41no—
112C2×C4×D43D414yescomplemented
113C2×Dic86—22no—
114C2×Dic86—22no—
115C2×Dic86—22no—
116C22⋊C4×C43—22no—
117C22⋊C4×C43—22no—
118C22⋊C4×C43—41no—
119C22⋊Dic43—22no—
120C22⋊Dic43—22no—
121C22⋊Dic43—41no—
122(C2×D43)⋊C4—22no—
123(C2×D43)⋊C4—22no—
124(C2×D43)⋊C4—22no—
125(C2×D43)⋊C4—22no—
126(C2×D43)⋊C4—22no—
127(C2×D43)⋊C4—22no—
128(C2×D43)⋊C4—41no—
129(C2×D43)⋊C4—41no—
130C23×C43—21no—
131C23×C43—21no—
132C23×C43—21no—
133C23×C43—21no—
134C23×C43—21no—
135C23×C43—21no—
136C23×C43—21no—
137C23×C43—21no—
138C23×C43—21no—
139C23×C43—21no—
140C23×C43—21no—
141C23×C43—21no—
142C23×C43C2×D412yes—
143C23×C43#4112yescomplemented
144C23×C43C2411yescharacteristic
145C22×D43—21no—
146C22×D43—21no—
147C22×D43—21no—
148C22×D43—21no—
149C22×D43—28no—
150C22×D43—21no—
151C22×D43—21no—
152C22×D43—21no—
153C22×D43—21no—
154C22×D43—21no—
155C22×D43—21no—
156C22×D43—21no—
157C22×D43—21no—
158C22×D43—21no—
159C22×D43—21no—
160C22×D43—21no—
161C22×D43—21no—
162C22×D43—44no—
163C22×D43—44no—
164C22×D43—44no—
165C22×D43—44no—
166C22×D43—44no—
167C22×D43—44no—
168C22×D43C2×D412yes—
169C22×D43C2×D412yes—
170C22×D43C2×D412yes—
171C22×D43C2×D412yes—
172C22×D43#4112yescomplemented
173D4×C43—44no—
174C2×C4×C43—28no—
175C2×C4×C43—21no—
176C2×C4×C43—21no—
177C2×C4×C43—21no—
178C2×C4×C43—21no—
179C2×C4×C43—21no—
180C2×C4×C43—21no—
181C2×C4×C43—21no—
182C2×C4×C43—21no—
183C2×C4×C43—21no—
184C2×C4×C43—21no—
185C2×C4×C43—21no—
186C2×C4×C43—21no—
187C2×C4×C43—21no—
188C2×C4×C43C2×D412yes—
189C2×C4×C43C2×D412yes—
190C22⋊D43—28no—
191C22⋊D43—28no—
192C22⋊D43—44no—
193C22⋊D43—44no—
194C2×Dic43—21no—
195C2×Dic43—21no—
196C2×Dic43—21no—
197C2×Dic43—21no—
198C2×Dic43—21no—
199C2×Dic43—21no—
200C2×Dic43—21no—
201C2×Dic43—21no—
202C2×Dic43—21no—
203C2×Dic43—21no—
204C2×Dic43—21no—
205C2×Dic43—21no—
206C2×Dic43—21no—
207C2×Dic43C2×D412yes—
208C2×Dic43C2×D412yes—
209D172—28no—
210D172—44no—
211C4×D43—28no—
212C4×D43—28no—
213C4×D43—44no—
214Dic86—28no—
215C22×C43—22no—
216C22×C43—22no—
217C22×C43—22no—
218C22×C43—22no—
219C22×C43—22no—
220C22×C43—22no—
221C22×C43—41no—
222C22×C43—41no—
223C22×C43—41no—
224C22×C43—41no—
225C22×C43—41no—
226C22×C43—41no—
227C22×C43—41no—
228C22×C43—41no—
229C22×C43C22×D411yescharacteristic
230C22×C43C22×D411yescharacteristic
231C22×C43C22×D411yescharacteristic
232C22×C43C2×#4111yescharacteristic
233C22×C43C2×#4111yescharacteristic
234C22×C43H(5,𝔽2)11yescharacteristic
235C22×C43H(5,𝔽2)11yescharacteristic
236C2×D43—22no—
237C2×D43—22no—
238C2×D43—22no—
239C2×D43—22no—
240C2×D43—22no—
241C2×D43—22no—
242C2×D43—22no—
243C2×D43—22no—
244C2×D43—22no—
245C2×D43—22no—
246C2×D43—22no—
247C2×D43—22no—
248C2×D43—416no—
249C2×D43—41no—
250C2×D43—41no—
251C2×D43—41no—
252C2×D43—41no—
253C2×D43—41no—
254C2×D43—41no—
255C2×D43—41no—
256C2×D43—41no—
257C2×D43#12014yes—
258C4×C43—22no—
259C4×C43—22no—
260C4×C43—41no—
261C4×C43—41no—
262C4×C43—41no—
263Dic43—22no—
264Dic43—22no—
265Dic43—41no—
266Dic43—41no—
267Dic43—41no—
268#2404—431nomaximal, Sylow
269C2×C43—42no—
270C2×C43—42no—
271C2×C43C2×#12011yescharacteristic
272C2×C43#52511yescharacteristic
273C2×C43#53511yescharacteristic
274C2×C43#54611yescharacteristic
275C2×C43#54611yescharacteristic
276C2×C43#54611yescharacteristic
277C2×C43#54611yescharacteristic
278D43—24no—
279D43—42no—
280D43—42no—
281C22×#41—431no—
282C2×#120—431no—
283C2×#120—431no—
284C22×C22⋊C4—431no—
285#393—431no—
286#393—431no—
287#392—431no—
288#392—431no—
289#386—431no—
290#386—431no—
291#386—431no—
292#386—431no—
293#386—431no—
294#386D4311yescharacteristic, complemented
295#381—431no—
296C43#240411yescharacteristic, complemented, Sylow
297C25—431no—
298C2×#41—864no—
299C2×#41—864no—
300C2×#41—864no—
301C22×Q8—431no—
302C22×D4—431no—
303C22×D4—431no—
304C22×D4—431no—
305C22×D4—431no—
306C22×D4—431no—
307C22×D4—431no—
308C22×D4C2×D4311yescharacteristic
309C23×C4—431no—
310C23×C4—431no—
311C23×C4—431no—
312C23×C4C2×D4311yescharacteristic, complemented
313#120—864no—
314#120—864no—
315C2×C22⋊C4—431no—
316C2×C22⋊C4—431no—
317C2×C22⋊C4—431no—
318C2×C22⋊C4—431no—
319C2×C22⋊C4—431no—
320C2×C22⋊C4—431no—
321C2×C22⋊C4—431no—
322C2×C22⋊C4—431no—
323C2×C22⋊C4—431no—
324C2×C22⋊C4—431no—
325C2×C22⋊C4—431no—
326C2×C22⋊C4—864no—
327C2×C22⋊C4C2×D4311yescharacteristic
328C2×C22⋊C4C2×D4311yescharacteristic
329C2×C22⋊C4C2×D4311yescharacteristic, complemented
330#95—431no—
331#95—431no—
332#95—431no—
333#95—431no—
334#95—431no—
335#95—431no—
336#95C2×D4311yescharacteristic, complemented
337#95C2×D4311yescharacteristic, complemented
338C24—431no—
339C24—431no—
340C24—431no—
341C24—862no—
342C24—862no—
343C24—862no—
344C24—862no—
345C24—862no—
346C24—1721no—
347C24—1721no—
348C24C22×D4311yescharacteristic
349C24C22×D4311yescharacteristic
350#41—8632no—
351C2×Q8—862no—
352C2×Q8—862no—
353C2×Q8—862no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—41no—
357C2×D4—434no—
358C2×D4—434no—
359C2×D4—862no—
360C2×D4—862no—
361C2×D4—862no—
362C2×D4—862no—
363C2×D4—862no—
364C2×D4—862no—
365C2×D4—862no—
366C2×D4—862no—
367C2×D4—862no—
368C2×D4—862no—
369C2×D4—862no—
370C2×D4—862no—
371C2×D4—862no—
372C2×D4—1721no—
373C2×D4—1721no—
374C2×D4—1721no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—431no—
379C22×C4—431no—
380C22×C4—434no—
381C22×C4—431no—
382C22×C4—431no—
383C22×C4—431no—
384C22×C4—862no—
385C22×C4—862no—
386C22×C4—862no—
387C22×C4—862no—
388C22×C4—862no—
389C22×C4—862no—
390C22×C4—1721no—
391C22×C4—1721no—
392C22×C4C22×D4311yescharacteristic
393C22×C4C22×D4311yescharacteristic
394C22×C4C22×D4311yescharacteristic
395C22×C4C22×D4311yescharacteristic
396C22×C4C22×D4311yescharacteristic
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—41no—
400C22⋊C4—862no—
401C22⋊C4—862no—
402C22⋊C4—862no—
403C22⋊C4—862no—
404C22⋊C4—862no—
405C22⋊C4—862no—
406C22⋊C4—862no—
407C22⋊C4—862no—
408C22⋊C4—1721no—
409C22⋊C4—1721no—
410C22⋊C4—1721no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—432no—
424C23—432no—
425C23—432no—
426C23—432no—
427C23—432no—
428C23—868no—
429C23—861no—
430C23—861no—
431C23—861no—
432C23—861no—
433C23—861no—
434C23—861no—
435C23—861no—
436C23—861no—
437C23—861no—
438C23—861no—
439C23—861no—
440C23—861no—
441C23—861no—
442C23—861no—
443C23—861no—
444C23—861no—
445C23—1724no—
446C23—1724no—
447C23—1724no—
448C23—1724no—
449C23—1724no—
450C23—1724no—
451C23#1062189812yescomplemented
452C23D4×D4312yes—
453C23C23×D4311yescharacteristic
454Q8—868no—
455D4—44no—
456D4—868no—
457D4—868no—
458D4—868no—
459D4—1724no—
460D4—1724no—
461D4—1724no—
462C2×C4—28no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—432no—
477C2×C4—432no—
478C2×C4—868no—
479C2×C4—868no—
480C2×C4—861no—
481C2×C4—861no—
482C2×C4—861no—
483C2×C4—861no—
484C2×C4—861no—
485C2×C4—861no—
486C2×C4—861no—
487C2×C4—861no—
488C2×C4—861no—
489C2×C4—861no—
490C2×C4—861no—
491C2×C4—861no—
492C2×C4—861no—
493C2×C4—1724no—
494C2×C4D4×D4312yes—
495C2×C4D4×D4312yes—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—434no—
511C22—862no—
512C22—862no—
513C22—862no—
514C22—862no—
515C22—862no—
516C22—862no—
517C22—862no—
518C22—862no—
519C22—862no—
520C22—862no—
521C22—862no—
522C22—862no—
523C22—17216no—
524C22—1721no—
525C22—1721no—
526C22—1721no—
527C22—1721no—
528C22—1721no—
529C22—1721no—
530C22—1721no—
531C22—1721no—
532C22C2×D4×D4311yescharacteristic
533C22C2×D4×D4311yescharacteristic
534C22C2×D4×D4311yescharacteristic
535C22#41×D4311yescharacteristic
536C22C2×#1062189811yescharacteristic
537C22#1306337011yescharacteristic
538C22#1306337111yescharacteristic
539C4—22no—
540C4—22no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—862no—
545C4—862no—
546C4—1721no—
547C4—1721no—
548C4—1721no—
549C2—42no—
550C2—42no—
551C2—864no—
552C2—1722no—
553C2—1722no—
554C2#120×D4311yescharacteristic
555C2#1458340811yescharacteristic
556C2#1458346511yescharacteristic
557C2#1458353911yescharacteristic
558C2#1458354111yescharacteristic
559C2#1458354511yescharacteristic
560C2#1458354611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC43×#386C2
hypercenter#386D43
hypocenterC43#2404
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×#10621898C2semidirect
2. C2×#13063295C2semidirect
3. C2×#13063293C2semidirect
4. C2×C22⋊C4×D43C2semidirect
5. #14582812C2semidirect
6. #14582811C2semidirect
7. #14582808C2semidirect
8. #14582807C2semidirect
9. #14582778C2semidirect
10. #14582777C2semidirect
11. #14582777C2semidirect
12. #14582775C2semidirect
13. #14582775C2semidirect
14. #14582761C2semidirect
15. C43×#386C2semidirect
20. C22×Dic86C22semidirect
26. C22×C22⋊D43C22semidirect
29. C22×C4×D43C22semidirect
34. C2×C22⋊Dic43C22semidirect
41. C2×(C2×D43)⋊C4C22semidirect
42. C2×(C2×D43)⋊C4C22semidirect
44. C43⋊#95C22semidirect
45. C43⋊#95C22semidirect
46. C43⋊#95C22semidirect
47. C43⋊#95C22semidirect
48. C43⋊#95C22semidirect
49. C43⋊#95C22semidirect
51. C23×C4×C43C22semidirect
54. C2×C22⋊C4×C43C22semidirect
55. #95×C43C22semidirect
56. #95×C43C22semidirect
91. C2×C22⋊D43D4semidirect
103. C2×D172D4semidirect
112. C2×C4×D43D4semidirect
143. C23×C43#41semidirect
172. C22×D43#41semidirect
294. #386D43semidirect
296. C43#2404semidirect
312. C23×C4C2×D43semidirect
329. C2×C22⋊C4C2×D43semidirect
336. #95C2×D43semidirect
337. #95C2×D43semidirect
451. C23#10621898semidirect